On the p-adic height of Heegner cycles

Jan Nekovář

Introduction

(0.1) In [Gr - Za], Gross and Zagier proved a remarkable formula, which relates the first derivative of the L-function of a modular form f of weight 2 on Γ_{0}(N) (over a suitable imaginary quadratic field) and the Néron-Tate height of a “Heegner point” (over the same quadratic field) on the f-part of the Jacobian of the modular curve X_{0}(N). Later, Perrin-Riou [PR 2] proved a p-adic version of this formula.

Kolyvagin’s method of Euler systems [Ko], combined with the formula of Gross-Zagier, proves the conjecture of Birch and Swinnerton-Dyer (up to a controlled rational factor) for all modular elliptic curves over Q with analytic rank ≤ 1.

In [Ne 1], Kolyvagin’s method was applied to certain cohomological objects associated to a modular form f of arbitrary even weight 2r > 2. In the present work we prove, under suitable hypotheses, a p-adic version of the Gross-Zagier formula in this context; our Theorem A relates the first derivative of a p-adic L-function of f at the “central point” and the p-adic height of a “Heegner cycle” (as mentioned above, this result is due to Perrin-Riou in the weight two case). As a consequence of this formula (and results of [Ne 1]), we obtain a certain weak form (Theorem B) of the conjecture of Beilinson and Bloch [Be] in our situation (this conjecture replaces the conjecture of Birch and Swinnerton-Dyer when one deals with central values of L-functions associated to cohomology groups of degree 2r − 1 > 1).

(0.2) We now describe the results of the present work in more detail. Let N ≥ 1 be an integer and f = ∑_{n≥1} a_{n}q^{n} a normalized (a_{1} = 1) newform on Γ_{0}(N) of weight 2r > 2. Let K = Q(\sqrt{D}) be an imaginary quadratic field with an odd discriminant D < 0 such that all primes dividing N split in K. We further assume that, if N = 1, both 2 and 3 split in K. Denote by N the norm N_{K/Q} and write τ for the non-trivial element of G(K/Q). Fix an embedding \overline{Q} ↪ C; this enables us to view f as a form with Fourier coefficients in \overline{Q}. Let F (f) be the subfield of \overline{Q} generated by the coefficients a_{n}; it is a totally real extension of Q of finite degree. Let p be a prime number not dividing 2N that splits in K. We fix an embedding i_{p} : \overline{Q} ↪ \overline{Q}_{p} and assume that f is ordinary with respect to i_{p}, i.e. that i_{p}(a_{p}) is a p-adic unit. It is expected that the above conditions are satisfied for infinitely many primes p. Let F̂ (f) be the closure of i_{p}(F (f)). For reasons explained in 0.13 below we also have to assume that p > 2r − 1.

(0.3) Let Y (N) be the modular curve over Q classifying elliptic curves with a full level N structure and j : Y (N) ↪ X(N) its non-singular compactification. If N ≥ 3, then there is a universal elliptic curve f : E ⟶ Y (N). The (2r − 2)-fold fibre product of E with itself over Y (N) has a canonical non-singular compactification W ([De 1], [ScA 1]). There are natural maps

H_{et}^{2r−1}(W ⊗_{Q} \overline{Q}, Q_{p})(r) ⟶ H_{et}^{1}(X(N) ⊗_{Q} \overline{Q}, j_{∗}Sym^{2r−2}(R^{1}f_{∗}Q_{p}))(r) ⟶ V_{f},  (0.3.1)

where V_{f} is (a Tate twist of) the p-adic Galois representation associated to f: V_{f} is two-dimensional over F̂ (f), unramified outside pN and satisfies

det_{F (f)} (1 − tF r_{ℓ}|V_{f}) = 1 − a_{ℓ}ℓ^{−r}t + ℓ^{−1}t^{2}

for all primes ℓ |̸ pN (here F r_{ℓ} denotes a geometric Frobenius at ℓ).

Let H be the Hilbert class field of K = Q(\sqrt{D}). In II.3.6 we define a “Heegner cycle” εY (with rational coefficients) of dimension r − 1 supported in the fibre of W ⊗_{Q} H over a “Heegner point” in Y_{0}(N)(H).

The cohomology class of εY in H_{et}^{2r}(W ⊗_{Q} \overline{Q}, Q_{p})(r) vanishes, hence εY lies in the domain of the p-adic Abel-Jacobi map

Φ : CH^{r}(W ⊗_{Q} H)_{0} ⊗ Q ⟶ H^{1}(H, H_{et}^{2r−1}(W ⊗_{Q} \overline{Q}, Q_{p})(r))

Let Φ_{f} be the map induced by Φ and 0.3.1

Φ_{f} : CH^{r}(W ⊗_{Q} H)_{0} ⊗ Q ⟶ H^{1}(H, V_{f})

and put z_{f} = Φ_{f}(εY). In order to define z_{f} for N < 3, one chooses a suitable auxiliary integer M ≥ 3, performs the above construction for NM and takes an average over GL_{2}(Z/MZ); see II.3.8 for more details.

(0.4) Bloch and Kato [Bl - Ka] define a subspace H_{f}^{1}(H, V_{f}) ⊂ H^{1}(H, V_{f}) (their subscript f has nothing to do with our modular form f), which is a cohomological version of classical Selmer groups. Under our assumptions, Im(Φ_{f}) is contained in H_{f}^{1}(H, V_{f}) ^{1}. Fix a continuous homomorphism ℓ_{K} : A_{K}^{∗}/K^{∗} ⟶ Q_{p} (an “arithmetic logarithm”). The construction of [Ne 2] associates to ℓ_{H} = ℓ_{K} ∘ N_{H/K} : A_{H}^{∗}/H^{∗} ⟶ Q_{p} and the canonical splitting of certain Hodge filtration (this comes from the assumption that f is ordinary at p) a symmetric F̂ (f)-linear height pairing

⟨ , ⟩ : H_{f}^{1}(H, V_{f}) × H_{f}^{1}(H, V_{f}) ⟶ F̂ (f)

For a character C : G(H/K) ⟶ \overline{Q}^{∗}, we define

z_{f,C} = \frac{1}{h} ∑_{σ∈G(H/K)} C^{−1}(σ)z_{f}^{σ} ∈ H_{f}^{1}(H, V_{f}) ⊗_{F (f)} \overline{Q}_{p},

where h = [H : K] is the class number of K. Our aim is to compute the height (extended in a \overline{Q}_{p}-linear way)

⟨z_{f,C}, z_{f,\overline{C}}⟩ = \frac{1}{h} ∑_{σ∈G(H/K)}⟨z_{f}, z_{f}^{σ}⟩

(recall that ⟨x^{σ}, y^{σ}⟩ = ⟨x, y⟩ for all σ ∈ G(H/K)).

(0.5) The L-function of f has an Euler product

L(f, s) = ∑_{n=1}^{∞} a_{n}n^{−s} = ∏_{ℓ}[(1 − α_{ℓ}ℓ^{−s})(1 − β_{ℓ}ℓ^{−s})]^{−1}

(with α_{ℓ}β_{ℓ} = 0 for ℓ | N). Let 𝔣 ⊂ O_{K} be an ideal and K(𝔣) the corresponding ray class field of K. Suppose that W : G(K(𝔣)/K) ⟶ \overline{Q}^{∗} is a ray class character with conductor 𝔣. We view W as a character on ideals prime to 𝔣, via the reciprocity map (normalized in the old-fashioned way, using arithmetic Frobenius elements). The L-function of f over K, twisted by W, is given by

L(f ⊗ K, W, s) = ∏_{λ}[(1 − α_{N(λ)}W(λ)N(λ)^{−s})(1 − β_{N(λ)}W(λ)N(λ)^{−s})]^{−1},

where λ runs through prime ideals of O_{K} prime to 𝔣 and α_{N(λ)} = α_{ℓ}^{k}, β_{N(λ)} = β_{ℓ}^{k} if N(λ) = ℓ^{k}. Let K_{∞}/K be the unique Z_{p}^{2}-extension of K. In I.5.10 we define a p-adic L-function L_{p}(f ⊗K, C)(λ), defined on continuous characters λ : G(K_{∞}/K) ⟶ 1 + pZ_{p}, which has the following interpolation property (cf. 5.10, 6.1): If W : G(K_{∞}/K) ⟶ 1 + pZ_{p} is a character of finite order with conductor 𝔣, then

1 See 0.13, however

L_{p}(f ⊗ K, C)(W) = τ(CW)W(N)\overline{C}\overline{W}((\sqrt{D}))V_{p}(f, CW)\frac{|D|^{1/2}N(𝔣)^{r−1/2}}{α_{N(𝔣)}(f)} \frac{2((r − 1)!)^{2}}{(4π)^{2r}} \frac{L(f ⊗ K, \overline{C}\overline{W}, r)}{⟨f, f⟩_{N}},

where

V_{p}(f, CW) = ∏_{𝔭|p} (1 − \frac{\overline{C}\overline{W}(𝔭)p^{r−1}}{α_{p}(f)} ) (1 − \frac{CW(𝔭)p^{r−1}}{α_{p}(f)} ) ,

α_{p}(f) is the unique root of t^{2} − a_{p}t + p^{2r−1} which is a p-adic unit, τ(CW) ∈ \overline{Q}^{∗} is the root number for the L-function L(CW, s) (see I.3.3) and

⟨f, f⟩_{N} = ∫_{Γ_{0}(N)\H} |f(z)|^{2}y^{2r−2} dxdy

In the above formulas, \overline{W} = W^{−1}, \overline{C} = C^{−1}.

(0.6) Using the reciprocity map, we can view ℓ_{K} : A_{K}^{∗}/K^{∗} ⟶ Q_{p} as a homomorphism ℓ_{K} : G(K_{∞}/K) ⟶ Q_{p}. We have ℓ_{K} = p^{−n} log_{p} ∘λ, for a suitable integer n and a character λ : G(K_{∞}/K) ⟶ 1 + pZ_{p}. Here log_{p} : Q_{p}^{∗} ⟶ Z_{p} denotes the Iwasawa logarithm, characterized by log_{p}(p) = 0. We define the derivative of L_{p}(f ⊗ K, C) at the trivial character 11 in the direction of ℓ_{K} as

L′_{p}(f ⊗ K, C, 11) = p^{−n} \frac{d}{ds}L_{p}(f ⊗ K, C)(λ^{s})|_{s=0}

(0.7) We are now ready to state our main result.

Theorem A. For each character C : G(H/K) ⟶ \overline{Q}^{∗},

L′_{p}(f ⊗ K, C, 11) = − ∏_{𝔭|p} (1 − \frac{\overline{C}(𝔭)p^{r−1}}{α_{p}(f)} ) (1 − \frac{C(𝔭)p^{r−1}}{α_{p}(f)} ) \frac{h⟨z_{f,C}, z_{f,\overline{C}}⟩}{u^{2}(4|D|)^{r−1}} ,

where u = (♯O_{K}^{∗})/2.

Put z_{K,f} = cor_{H/K}(z_{f}) ∈ H_{f}^{1}(K, V_{f}). As res_{H/K}(z_{K,f}) = hz_{f,1} and h⟨x, y⟩_{K} = ⟨res_{H/K}(x), res_{H/K}(y)⟩_{H} for x, y ∈ H_{f}^{1}(K, V_{f}) (the subscripts K resp. H indicate the field over which the height is computed), we have

⟨z_{K,f}, z_{K,f}⟩_{K} = h⟨z_{f,1}, z_{f,1}⟩_{H}

Applying Theorem A for the trivial character, we obtain (omitting C = 1 from the notation)

Corollary.

L′_{p}(f ⊗ K, 11) = − (1 − \frac{p^{r−1}}{α_{p}(f)})^{4} \frac{⟨z_{K,f}, z_{K,f}⟩_{K}}{u^{2}(4|D|)^{r−1}}

The minus sign in the statement of Theorem A depends on the normalizations of the height pairing and the L-function, but not of the reciprocity map. It seems that [PR 2] contains a sign mistake; see II.6.4 for more details.

(0.8) In [Ne 1], we proved the following statement (in a slightly different formulation; see II.6.5 and 0.13 for more comments and precise assumptions).

Theorem. If z_{K,f} ≠ 0, then the image of the map

Φ_{K,f} : CH^{r}(W ⊗_{Q} K)_{0} ⊗ Q ⟶ H_{f}^{1}(K, V_{f})

satisfies

Q_{p} · Im(Φ_{K,f}) = F̂ (f) · z_{K,f} = H_{f}^{1}(K, V_{f})

Combined with the Corollary of Theorem A, this proves

Theorem B. If L′_{p}(f ⊗ K, 11) ≠ 0 for some ℓ_{K}, then

Q_{p} · Im(Φ_{K,f}) = F̂ (f) · z_{K,f} = H_{f}^{1}(K, V_{f})

(0.9) An arithmetic logarithm ℓ_{K} : A_{K}^{∗}/K^{∗} ⟶ Q_{p} is called cyclotomic (resp. anticyclotomic) if ℓ_{K} ∘τ = ℓ_{K} (resp. ℓ_{K} ∘ τ = −ℓ_{K}). Every ℓ_{K} can be written uniquely as a sum of a cyclotomic and an anticyclotomic logarithms. For an anticyclotomic ℓ_{K}, Theorem A is vacuous, as its statement becomes 0 = 0. Let Q_{∞}/Q be the unique Z_{p}-extension of Q. Write N : G(K_{∞}/K) ⟶ G(Q_{∞}/Q) for the restriction map and ⟨ ⟩ for the isomorphism G(Q_{∞}/Q) ⟶^{∼} 1 + pZ_{p} induced by the cyclotomic character. Then every cyclotomic ℓ_{K} is a scalar multiple of ⟨ ⟩ ∘ N and we have

L_{p}(f ⊗ K, ⟨ ⟩^{s} ∘ N) = \frac{⟨N⟩^{2s}(−1)^{r−1}|D|^{1/2}}{2^{2r−1}(2π)^{2}⟨f, f⟩_{N}} L_{p}^{MTT}(f, x^{r−1}⟨ ⟩^{s})L_{p}^{MTT}(f ⊗ (\frac{D}{·} ), x^{r−1}⟨ ⟩^{s}),  (0.9.1)

where L_{p}^{MTT}(f, λ) is the p-adic L-function of f over Q, in the normalization of [M - T - T] (cf. I.5.13). The complex functional equation

Λ(f, s) = N^{s/2}(2π)^{−s}Γ(s)L(f, s) = wΛ(f, 2r − s)

with w = (−1)^{r−1}λ_{N}(f) = ±1 (where λ_{N}(f) is the eigenvalue of f under the Fricke involution) has a p-adic counterpart ([M - T - T, 18.3])

L_{p}^{MTT}(f, x^{r−1}⟨ ⟩^{s}) = w⟨N⟩^{−s}L_{p}^{MTT}(f, x^{r−1}⟨ ⟩^{−s})
L_{p}^{MTT}(f ⊗ (\frac{D}{·} ), x^{r−1}⟨ ⟩^{s}) = −w⟨N|D|⟩^{−s}L_{p}^{MTT}(f ⊗ (\frac{D}{·} ), x^{r−1}⟨ ⟩^{−s})

(0.9.2)

(0.10) Assume that w = −1. In this case

z_{K,f} ∈ H_{f}^{1}(K, V_{f})^{G(K/Q)} = H_{f}^{1}(Q, V_{f})

and 0.9.1-2 imply that, for ℓ_{K} = ⟨ ⟩ ∘ N,

L′_{p}(f ⊗ K, 11) = \frac{(−1)^{r−1}|D|^{1/2}}{2^{2r−1}(2π)^{2}⟨f, f⟩_{N}} L_{p}^{MTT}(f ⊗ (\frac{D}{·} ), x^{r−1})\frac{d}{ds}L_{p}^{MTT}(f, x^{r−1}⟨ ⟩^{s})|_{s=0}

with

L_{p}^{MTT}(f ⊗ (\frac{D}{·} ), x^{r−1}) = (1 − \frac{p^{r−1}}{α_{p}(f)})^{2} \frac{(r − 1)!}{(−2πi)^{r−1}} L(f ⊗ (\frac{D}{·} ), r)

According to [Sk - Za], there is a non-zero Jacobi form ϕ = ∑ c(n, k)q^{n}ζ^{k} ∈ J_{r+1,N}^{cusp} (unique up to a scalar multiple) with the same Hecke eigenvalues as f. Multiplying by a constant, we can assume that all coefficients of ϕ are real. By [G - K - Z], we have

\frac{L(f ⊗ (\frac{D}{·} ), r)}{⟨f, f⟩_{N}} = \frac{2^{2r−1}π^{r}N^{r−1}}{(r − 1)!|D|^{r−1/2}} \frac{c(n, k)^{2}}{⟨ϕ, ϕ⟩} ,

whenever D = k^{2} − 4nN. It follows that

L′_{p}(f ⊗ K, 11) = (1 − \frac{p^{r−1}}{α_{p}(f)})^{2} \frac{(−i)^{r−1}c(n, k)^{2}}{2^{r+1}π|D|^{r−1}⟨ϕ, ϕ⟩} \frac{d}{ds}L_{p}^{MTT}(f, x^{r−1}⟨ ⟩^{s})|_{s=0}  (0.10.1)

Combining Cor. of Thm. A, Thm. 0.8 and the non-vanishing results of [Wa], we obtain the following statement.

Theorem C. Assume that

ord_{s=0}L_{p}^{MTT}(f, x^{r−1}⟨ ⟩^{s}) = 1

(which implies that w = −1). Then

(1) There exist infinitely many imaginary quadratic fields Q(\sqrt{D}) of odd discriminant, in which all primes dividing pN split and for which z_{Q(\sqrt{D}),f} ∈ H_{f}^{1}(Q, V_{f}) is non-zero.

(2) For any such Q(\sqrt{D}),

H_{f}^{1}(Q, V_{f}) = F̂ (f) · z_{Q(\sqrt{D}),f}

(3) If D_{1} = k_{1}^{2} − 4n_{1}N, D_{2} = k_{2}^{2} − 4n_{2}N, are odd discriminants of two imaginary quadratic fields in which all primes dividing pN split, then

c(n_{2}, k_{2})z_{Q(\sqrt{D_{1}}),f} = ±c(n_{1}, k_{1})z_{Q(\sqrt{D_{2}}),f}

One would expect that (3) always holds with the plus sign, as in [G - K - Z] in the weight two case (cf. also [ScC] for some calculations involving the complex Abel-Jacobi map).

(0.11) Assume that w = 1. In this case,

z_{K,f} ∈ H_{f}^{1}(K, V_{f})^{τ=−1}

and

L′_{p}(f ⊗ K, 11) = \frac{(−1)^{r−1}|D|^{1/2}}{2^{2r−1}(2π)^{2}⟨f, f⟩_{N}} L_{p}^{MTT}(f, x^{r−1})\frac{d}{ds}L_{p}^{MTT}(f ⊗ (\frac{D}{·} ), x^{r−1}⟨ ⟩^{s})|_{s=0}

with

L_{p}^{MTT}(f, x^{r−1}) = (1 − \frac{p^{r−1}}{α_{p}(f)})^{2} \frac{(r − 1)!}{(−2πi)^{r−1}} L(f, r)

In the absence of a p-adic analogue of the non-vanishing results of [B - F - H], [Iw], [Mu - Mu], we can prove only a conditional result.

Theorem D. Assume that L(f, r) ≠ 0 (which implies that w = 1). Suppose that there is an imaginary quadratic field Q(\sqrt{D}) with an odd discriminant, in which all primes dividing pN split and such that

\frac{d}{ds}L_{p}^{MTT}(f ⊗ (\frac{D}{·} ), x^{r−1}⟨ ⟩^{s})|_{s=0} ≠ 0

Then

H_{f}^{1}(Q, V_{f}) = 0

(0.12) The proof of Theorem A follows rather closely [PR 2]. Essentially, we compute the height ⟨x, T_{m}x^{σ}⟩ for sufficiently many integers m ≥ 1. Here x is a Heegner cycle, σ ∈ G(H/K) and T_{m} denotes the m-th Hecke correspondence on W. More precisely, if x and T_{m}x^{σ} have disjoint supports, then the height can be expressed in terms of local heights at all (non-archimedean) places v of H:

⟨x, T_{m}x^{σ}⟩ = ∑_{v}⟨x, T_{m}x^{σ}⟩_{v}

We compute the sum over places outside of p

∑_{v |̸p}⟨x, T_{m}x^{σ}⟩_{v},

which turns out to be almost equal to the m-th coefficient of a certain modular form G_{σ} ∈ \overline{M}_{2r}(Γ_{0}(Np^{∞}); Q_{p}) which controls the derivative of the p-adic L-function. To pass from G_{σ} to the L-function, one has to apply, among other things, Hida’s projector e onto the space of ordinary modular forms. A trick from [PR 2, 5.2] then shows that e kills the contribution of local heights at places v | p (at least when it comes to the f-component of the height). It is quite essential for the proof that

∑_{m≥1}⟨x, T_{m}x^{σ}⟩q^{m}

is known to be a modular form in S_{2r}(Γ_{0}(N); Q_{p}) (this follows from the fact that Φ commutes with Hecke correspondences and that the p-adic height factors through Φ). This helps to get around the fact that local heights are defined only for cycles with disjoint supports. In [Gr - Za] (and [PR 2]), a deformation argument was used to deal with local heights of divisors on a curve whose supports are not necessarily disjoint. However, we were unable to generalize this argument to our situation.

This is the main reason why an analogue of Theorem A for archimedean heights ⟨ , ⟩_{∞} is still lacking. Either we have to know a priori that

∑_{m≥1}⟨x, T_{m}x^{σ}⟩_{∞}q^{m}

is a modular form, or we have to find a way to compute all of its coefficients (at least for all m prime to N) and make this conclusion only a posteriori.

We used the ideas of [Br] and worked with sections of the sheaf Sym^{2r−2}R^{1}f_{∗}Q_{p}(r − 1) over divisors on the modular curve X_{0}(N), rather than with cycles on W. The point is that the Abel-Jacobi map and local heights for cycles contained in the fibres of the projection W ⟶ X_{0}(N) depend only on the cohomology of the given cycles in those fibres. Local heights at places v |̸ p are then computed by using an intersection formula from [Br] (with a corrected sign) and calculations of [Gr - Za].

It should be clear from this account how much the present work owes to [Gr - Za], [Br] and [PR 1], [PR 2]. When possible, we try to avoid repetition of calculations performed in these papers. The reader is therefore advised to consult them while reading the present work.

(0.13) The present paper and [Ne 1], [Ne 2] all make use of several deep results from p-adic Hodge theory. The ”crystalline conjecture” for smooth projective varieties with good reduction is invoked to ensure that V_{f} is a crystalline representation of G(\overline{Q}_{p}/Q_{p}) (so that the height machinery of [Ne 2] can be applied to H_{f}^{1}(H, V_{f})). The ”de Rham conjecture” for smooth separated (but not necessarily proper) varieties was used in [Ne 1] to show that the image of Φ_{f} is contained in H_{f}^{1}(H, V_{f}). However, in a letter to the author dated May 27, 1992, Faltings pointed out a gap in the proof of the ”de Rham conjecture” in [Fa] and sketched an idea how to modify the argument. To this day, however, a rigorous proof has not been written down. More recently, a remark of Berthelot [Ber] seems to call into question one step in the proof of the ”crystalline conjecture” in [Fa]. Fortunately, at least for sufficiently large primes p, one can rely on proofs of the crystalline conjecture in [Fo - Me] (unramified case) and [KaK - Me] (general case). It will be shown in a future publication that (again for sufficiently large p) the image of Φ_{f} is indeed contained in H_{f}^{1}(H, V_{f}) without making a reference to [Fa]. For the reader’s convenience, we sketch the argument in II.1.4. In our case the restriction is p > dim(W) = 2r − 1; if the results of [Fa] can be corroborated, then this condition can be dropped from the assumptions of Theorems A–D and Theorem 0.8. ^{1}

(0.14) During the work on this paper the author was supported by a fellowship from the Miller Institute for Basic Research in Science at the University of California at Berkeley. He would like to express his gratitude to the Miller Institute for its support and hospitality. It is also a pleasure to thank K. Ribet for helpful discussions.

I. Modular forms and p-adic L-functions

In this chapter we define p-adic L-functions referred to in 0.5. Most of the work has been done in the weight two case by Perrin-Riou in [PR 1]; we merely indicate basic changes that have to be made in the higher weight case. This occupies Sec. 1–5 and is carried out in the same generality as [PR 1] is (we could have appealed to the results of [Hi 1], [Hi 2], but the setting of [PR 1] was best suited for our work). In Sec. 6 we specialize to the situation of Thm. A and perform a calculation of Fourier coefficients of certain p-adic integrals after [PR 2].

1. Modular forms

(1.1) Let us recall the usual notation: H denotes the complex upper half plane with the standard action of

GL_{2}^{+}(R) by fractional linear transformations. For γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}^{+}(R) and any function f on H we

define, for k ∈ Z,

(f|_{k}γ)(z) := (det(γ))^{k/2}(cz + d)^{−k}f(γ(z))

(for odd k, we take the positive square root of det(γ)). If Γ is a subgroup of SL_{2}(Z) of finite index, we define M̃_{k}(Γ) to be the space of complex-valued C^{∞}-functions on H satisfying f|_{k}γ = f for all γ ∈ Γ and having at most polynomial growth at the cusps: |(f|_{k}α)(x + iy)| = O(y^{N}) as y ⟶ ∞ for each α ∈ SL_{2}(Z). We also write M_{k}(Γ) (resp. S_{k}(Γ)) for the space of holomorphic modular forms (resp. cusp forms) of weight k on Γ. If χ is a Dirichlet character modulo N > 1, we denote by M_{k}(Γ_{0}(N), χ) the set of f ∈ M_{k}(Γ_{1}(N)) satisfying

f|_{k}( \begin{matrix}a & b \\ c & d \end{matrix} ) = χ(d)f for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) and we put S_{k}(Γ_{0}(N), χ) = S_{k}(Γ_{1}(N)) ∩ M_{k}(Γ_{0}(N), χ). We

also denote by M̃_{k}(Γ_{0}(N), χ) the corresponding space of C^{∞}-functions. If f is a function on H, we define f^{τ} (z) = \overline{f(−\overline{z})}. For f ∈ S_{k}(Γ_{0}(N), χ), g ∈ M̃_{k}(Γ_{0}(N), χ), the Petersson scalar product is defined by

⟨f, g⟩_{N} := ∫_{Γ_{0}(N)\H} \overline{f(z)}g(z)y^{k−2} dx dy

(1.2) We shall need some facts about Shimura’s differential operators (cf. [Sh]), defined on H by the formulas

\begin{aligned}δ_{s} & = \frac{1}{2πi} (\frac{∂}{∂z} + \frac{s}{2iy} ) = q \frac{∂}{∂q} − \frac{s}{4πy} \\ δ_{s}^{r} & = \underbrace{δ_{s+2r−2} ∘ ⋯ ∘ δ_{s+2} ∘ δ_{s}}_{r times} = ∑_{t=0}^{r} \binom{r}{t}\frac{Γ(s + r)}{Γ(s + t)}(−4πy)^{t−r} (q \frac{∂}{∂q} )^{t} \end{aligned}  (1.2.1)

1 Added in proof. The assumption p > 2r − 1 is indeed unnecessary. An exchange of letters between Berthelot and Faltings (June–August 1994) helped to clarify several obscure points in [Fa]. In particular, the objection raised in [Ber] is not valid. A recent preprint of Tsuji gives a syntomic proof of the crystalline conjecture for all p > 2. This implies that the assumptions (a),(b) of Lemma II.1.4 can be replaced by p > 2.

(where z = x + iy, q = e^{2πiz}). They satisfy the relations

\begin{aligned}δ_{s+t}(fg) & = gδ_{s}(f) + fδ_{t}(g) \\ δ_{k}^{m}(f|_{k}γ) & = (δ_{k}^{m}f)|_{k+2m}γ  (∀γ ∈ GL_{2}^{+}(R)) \end{aligned}  (1.2.2)

It follows that for g ∈ M̃_{j}(Γ_{0}(N), χ) and h ∈ M̃_{k}(Γ_{0}(N), ψ),

gδ_{k}^{m}(h) ∈ M̃_{j+k+2m}(Γ_{0}(N), χψ)  (1.2.3)

We shall also need the formulas

δ_{k}^{m} ((cz + d)^{−k} (\frac{y}{|cz + d|^{2}} )^{s}) = (−4π)^{−m} \frac{Γ(k + s + m)}{Γ(k + s)}(cz + d)^{−(k+2m)} (\frac{y}{|cz + d|^{2}} )^{s−m}
δ_{1}^{r−1}(q^{n}) = \frac{(r − 1)!}{(−4πy)^{r−1}} q^{n} p_{r,r−1}(4πny)  (1.2.4)

where

p_{r,r−1}(t) = ∑_{j=0}^{r−1} \binom{r − 1}{j}\frac{(−t)^{j}}{j!}

(our notation follows [Gr - Za]).

(1.3) For every integer k ≥ 2 there is a holomorphic projection defined on a certain subspace M̃ ⊂ M̃_{k}(Γ_{0}(N), χ)

H : M̃ ⟶ S_{k}(Γ_{0}(N), χ)

with the properties (see [St], [Gr - Za])

\begin{aligned}⟨F, f⟩_{N} & = ⟨F, H(f)⟩_{N}  (∀F ∈ S_{k}(Γ_{0}(N), χ)) \\ H(f)|_{k}γ & = H(f|_{k}γ)  (∀γ ∈ GL_{2}^{+}(R)) \end{aligned}  (1.3.1)

In a special situation we are interested in, there is a simple formula for H (cf. [Gr - Za], where the case χ = 1 is discussed):

Suppose that f ∈ M̃_{2r}(Γ_{0}(N), χ) with 2r > 2 has a Fourier expansion

f(z) = ∑_{m∈Z} a_{m}(y) e^{2πimz}

and satisfies (f|_{2r}γ)(z) = O(y^{−ε}) as y = Im(z) ⟶ ∞ for some ε > 0 and every γ ∈ SL_{2}(Z). Then

H(f) = ∑_{m=1}^{∞} a_{m} e^{2πimz}

with

a_{m} = \frac{(4πm)^{2r−1}}{(2r − 2)!} ∫_{0}^{∞} a_{m}(y)e^{−4πmy}y^{2r−2} dy  (1.3.2)

For f = g δ_{k}^{m}(h) with g ∈ M_{j}(Γ_{0}(N), χ), h ∈ M_{k}(Γ_{0}(N), ψ) (hence f ∈ M̃_{j+k+2m}(Γ_{0}(N), χψ)), the holomorphic projection of f can be computed as follows (cf. [Hi 1], [Sh]): there is a unique decomposition

g δ_{k}^{m}(h) = ∑_{t=0}^{m} δ_{k−2t}^{t}(g_{t})

with g_{t} ∈ M_{j+k+2m−2t}(Γ_{0}(N), χψ) and H(f) = g_{0}.

(1.4) We now define non-holomorphic Eisenstein series that will be used in the sequel. Fix integers r ≥ 1, M ≥ 1. For any function ϕ : Z/MZ ⟶ C, define for z = x + iy ∈ H, Re(s) > 3/2 − r

\begin{aligned}E_{r}(z, s, ϕ) & = ∑_{{m,n∈Z \atop M|m}} ′ \frac{ϕ(n)}{(mz + n)^{2r−1}} (\frac{y}{|mz + n|^{2}} )^{s} \\ Ẽ_{r}(z, s, ϕ) & = ∑_{m,n∈Z}′ \frac{ϕ(m)}{(mz + n)^{2r−1}} (\frac{y}{|mz + n|^{2}} )^{s} \end{aligned}

Both series depend only on ϕ^{−}(n) = ((ϕ(n) − ϕ(−n))/2 and admit a holomorphic continuation to all s ∈ C ([Si, Cor. of Thm. I.5.3]). It makes sense, therefore, to define

E_{r}(z, ϕ) = E_{r}(z, 1 − r, ϕ),  Ẽ_{r}(z, ϕ) = Ẽ_{r}(z, 1 − r, ϕ)

Some properties of these Eisenstein series are summarized in the following Proposition.

(1.5) Proposition. (1) For γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(M) define ϕ^{∗}(n) = ϕ(an), ϕ′(m) = ϕ(dm). Then

\begin{aligned}E_{r}(z, ϕ)|_{2r−1}γ & = E_{r}(z, ϕ^{∗}) \\ Ẽ_{r}(z, ϕ)|_{2r−1}γ & = Ẽ_{r}(z, ϕ′) \end{aligned}

(2) If ϕ is a Dirichlet character modulo M, then

\begin{aligned}E_{r}(z, ϕ) & ∈ M̃_{2r−1}(Γ_{0}(M), \overline{ϕ}) \\ Ẽ_{r}(z, ϕ) & ∈ M̃_{2r−1}(Γ_{0}(M), ϕ) \end{aligned}

For general ϕ, E_{r}, Ẽ_{r} both lie in M̃_{2r−1}(Γ_{1}(M)). (3)

\begin{aligned}δ_{1}^{r−1}(E_{1}(z, ϕ)) & = \frac{(r − 1)!}{(−4π)^{r−1}} E_{r}(z, ϕ) \\ δ_{1}^{r−1}(Ẽ_{1}(z, ϕ)) & = \frac{(r − 1)!}{(−4π)^{r−1}} Ẽ_{r}(z, ϕ) \end{aligned}

(4) The Fourier expansions are

\begin{aligned}E_{r}(z, ϕ) & = (−\frac{2πi}{M}) y^{1−r}[\frac{1}{2}L̃(0, ϕ̂) + ∑_{k·m>0} ϕ̂(k)sgn(k)p_{r,r−1}(4πkmy)q^{km}] \\ Ẽ_{r}(z, ϕ) & = (−2πi)y^{1−r}[\frac{1}{2}L̃(0, ϕ) + ∑_{k·m>0} ϕ(k)sgn(k)p_{r,r−1}(4πkmy)q^{km}] \end{aligned}

where q = e^{2πiz} and

L̃(s, ϕ) = ∑_{0≠n∈Z} \frac{ϕ(n)sgn(n)}{|n|^{s}},  ϕ̂(k) = ∑_{n∈Z/MZ} ϕ(n)e^{2πikn/M}

(5) E_{r}(z, ϕ) = M^{−1}Ẽ(z, ϕ̂), E_{r}(z, ϕ̂) = −Ẽ(z, ϕ) (6)

\begin{aligned}E_{r}(z, ϕ)|_{2r−1}( \begin{matrix}0 & −1 \\ M & 0 \end{matrix} ) & = M^{−1/2}Ẽ_{r}(z, ϕ) \\ Ẽ_{r}(z, ϕ)|_{2r−1}( \begin{matrix}0 & −1 \\ M & 0 \end{matrix} ) & = −M^{−1/2}E_{r}(z, ϕ) \end{aligned}

Proof. (1) is a simple calculation; (2) follows from (1) and (3) from (1.2.4). (4) can be proved in a number of ways – e.g. by an explicit calculation based on the Poisson formula as in [Gr - Za]; or by a reduction to the well-known case r = 1 by using (3) and (1.2.4). (5) follows from (4) and (6) is again an easy calculation.

(1.6) The Rankin-Selberg convolution of two modular forms f ∈ M_{j}(Γ_{0}(M), χ), g ∈ M_{k}(Γ_{0}(M), ψ) with Fourier expansions

f = ∑_{n≥0} a(n)q^{n}, g = ∑_{n≥0} b(n)q^{n}

is defined as

D_{M}(f, g, s) := L_{M}(2s + 2 − j − k, χψ) ∑_{n≥1} \frac{a(n)b(n)}{n^{s}},

where

L_{M}(s, χψ) = ∑_{{n≥1 \atop (n,M)=1}} \frac{(χψ)(n)}{n^{s}} = ∏_{p |̸M}(1 − (χψ)(p)p^{−s})^{−1}

(1.7) If f (resp. g) is a normalized newform of level N_{f}|M (resp. N_{g}|M), then

\begin{aligned}L(f, s) & = ∑_{n≥1} a(n)n^{−s} = ∏_{p}[(1 − α_{p}p^{−s})(1 − α′_{p}p^{−s})]^{−1} \\ L(g, s) & = ∑_{n≥1} b(n)n^{−s} = ∏_{p}[(1 − β_{p}p^{−s})(1 − β′_{p}p^{−s})]^{−1} \end{aligned}

with α_{p}α′_{p} = χ(p)p^{2j−1}, β_{p}β′_{p} = ψ(p)p^{2k−1} for primes p |̸ M; α_{p}α′_{p}β_{p}β′_{p} = 0 for p|N = lcm(N_{f}, N_{g}) and (cf. [Sh, 3.1])

D_{M}(f, g, s) = ∏_{p}[(1−α_{p}β_{p}p^{−s})(1−α_{p}β′_{p}p^{−s})(1−α′_{p}β_{p}p^{−s})(1−α′_{p}β′_{p}p^{−s})]^{−1} ∏_{p|M/N}(1−α_{p}α′_{p}β_{p}β′_{p}p^{−2s}) (1.7.1)

(1.8) Suppose that, in the notation of 1.6, j = 2r is even, k = 1 and that f is a cusp form. Write ϕ for the character χψ extended by zero to Z/MZ. Then

E_{r}(z, s, ϕ) = L̃_{M}(2s + 2r − 1, ϕ) ∑_{γ∈Γ_{∞}\Γ_{0}(M)} ϕ(d) y^{s}|_{2r−1}γ

with γ = ( \begin{matrix}∗ & ∗ \\ c & d \end{matrix} ), Γ_{∞} = ±( \begin{matrix}1 & Z \\ 0 & 1 \end{matrix} ). The standard unfolding trick gives

⟨f^{τ} , gE_{r}(z, s, ϕ)⟩_{M} = L̃_{M}(2s + 2r − 1, ϕ) ∫_{Γ_{∞}\H} \overline{f^{τ} (z)}g(z)y^{s+2r−2} dx dy

with

∫_{Γ_{∞}\H} \overline{f^{τ} (z)}g(z)y^{s+2r−2} dx dy = \frac{Γ(s + 2r − 1)}{(4π)^{s+2r−1}} ∑_{n≥1} \frac{a(n)b(n)}{n^{s+2r−1}},

hence

\begin{aligned}⟨f^{τ} , gE_{r}(z, s, ϕ)⟩_{M} & = (1 − ϕ(−1))\frac{Γ(s + 2r − 1)}{(4π)^{s+2r−1}}D_{M}(f, g, s + 2r − 1) \\ ⟨f^{τ} , gE_{r}(z, ϕ)⟩_{M} & = (1 − ϕ(−1))\frac{(r − 1)!}{(4π)^{r}}D_{M}(f, g, r) \end{aligned}

Using Prop. 1.5.3, this can be rewritten as

⟨f^{τ} , gδ_{1}^{r−1}(E_{1}(z, ϕ))⟩_{M} = \frac{(1 − ϕ(−1))(−1)^{r−1}((r − 1)!)^{2}}{(4π)^{2r−1}} D_{M}(f, g, r)  (1.8.1)

(1.9) Proposition. Let f = ∑_{n≥1} a(n)q^{n} (resp. g = ∑_{n≥0} b(n)q^{n}) be a cusp form (resp. a holomorphic modular form) of weight one on Γ_{0}(N). Then H(fδ_{1}^{r−1}(g)) = ∑_{n≥1} c(n)q^{n} with

c(n) = \frac{(−1)^{r−1}}{\binom{2r−2}{r−1}} n^{r−1} ∑_{j+k=n} a(j)b(k)P_{r−1} (\frac{j − k}{j + k} ) ,

where

P_{k}(t) = \frac{1}{2^{k} · k!} (\frac{d}{dt})^{k} [(t^{2} − 1)^{k}]

is the Legendre polynomial of degree k.

Proof. It follows from (1.2.4) that

fδ_{1}^{r−1}(g) = \frac{(r − 1)!}{(−4π)^{r−1}} y^{1−r} ∑_{j≥1} ∑_{k≥0} a(j)b(k)p_{r,r−1}(4πky)q^{j+k}

The formula (1.3.2) then gives

c(n) = \frac{(r − 1)!}{(−4π)^{r−1}} · \frac{(4πn)^{2r−1}}{(2r − 2)!} ∑_{j+k=n} a(j)b(k) ∫_{0}^{∞}p_{r,r−1}(4πky)e^{−4π(j+k)y}y^{r−1} dy

and the value of the integral is equal to (cf. [Gr - Za, p.293])

∫_{0}^{∞}p_{r,r−1}(4πky)e^{−4π(j+k)y}y^{r−1} dy = \frac{(r − 1)!}{(−4π)^{r−1}} P_{r−1} (1 − \frac{2k}{j + k} )

(1.10) Let N > 1 be an integer and Q a divisor of N prime to N/Q. A Dirichlet character ψ modulo N decomposes uniquely as ψ = ψ_{Q} · ψ_{N/Q}, where ψ_{Q} (resp. ψ_{N/Q}) is a character modulo Q (resp. N/Q). The Atkin-Lehner operator (cf. [At - Li, sec. 1])

W_{Q} : M_{k}(Γ_{0}(N), ψ_{Q} · ψ_{N/Q}) ⟶ M_{k}(Γ_{0}(N), \overline{ψ}_{Q} · ψ_{N/Q})

is defined by

f|W_{Q} = ψ_{Q}(y)ψ_{N/Q}(x)f|_{k}( \begin{matrix}Qx & y \\ Nz & Qw \end{matrix} ),

where x, y, z, w are integers satisfying Qxw − (N/Q)yz = 1. These operators satisfy, among others, the following relations:

\begin{aligned}f|W_{Q}|W_{Q} & = ψ_{Q}(−1)\overline{ψ}_{N/Q}(Q)f \\ f|_{k}T_{p}|W_{Q} & = ψ_{Q}(p)f|W_{Q}|_{k}T_{p}, \end{aligned}  (1.10.1)

where T_{p} (for p |̸ N) denotes the p-th Hecke operator. If f is a normalized (a_{1} = 1) newform, then

f|W_{Q} = λ_{Q}(f)g

for a suitable normalized newform g. The complex number λ_{Q}(f) is algebraic and has absolute value equal to 1. If ψ_{Q} = 1 (i.e. cond(ψ) | N/Q), then g = f by (1.10.1) and

λ_{Q}(f)^{2} = \overline{ψ}_{N/Q}(Q)  (1.10.2)

2. p-adic modular forms

In this section we recall the basic setup of the theory of p-adic modular forms, following [Hi 1].

(2.1) As in the Introduction, we fix embeddings i_{∞} : \overline{Q} ↪ C, i_{p} : \overline{Q} ↪ \overline{Q}_{p} (for a fixed prime number p). For a subring A ⊂ \overline{Q} and Γ = Γ_{0}(N) or Γ_{1}(N), one defines M_{k}(Γ; A) as the set of all modular forms

f = ∑_{n≥0} a_{n}(f)q^{n} ∈ M_{k}(Γ)

with Fourier coefficients a_{n}(f) in A (and similarly for M_{k}(Γ_{0}(N), χ; A), where χ is any Dirichlet character modulo N with values in A). There is a natural p-adic norm on these spaces, given by

|f|_{p} = sup_{n} |i_{p}(a_{n}(F))|_{p},

where | · |_{p} is the standard valuation on \overline{Q}_{p} (normalized by |p|_{p} = p^{−1}). Denote by M_{k}(Γ; Â) (resp. M_{k}(Γ_{0}(N), χ; Â)) the corresponding completions; here Â denotes the closure of i_{p}(A). Finally, one defines

M_{k}(Γ_{i}(Np^{∞}), Â) = ⋃_{n=0}^{∞} M_{k}(Γ_{i}(Np^{n}), Â)

and writes \overline{M}_{k}(Γ_{i}(Np^{∞}), Â) for its completion under | · |_{p}.

(2.2) Let f = ∑ a_{n}q^{n} be a normalized newform in S_{k}(Γ_{0}(N), ψ) and suppose that Â contains all coefficients of f and values of ψ (under the fixed embedding i_{p}). The form f is called ordinary if |i_{p}(a_{p})|_{p} = 1. If this is the case, then precisely one of the roots of the polynomial X^{2} − a_{p}X + ψ(p)p^{k−1} (call it α_{p}(f)) satisfies |i_{p}(α_{p})|_{p} = 1. Hida [Hi 1, 3.3] defines

f_{0}(z) = f(z) − \frac{ψ(p)p^{k−1}}{α_{p}(f)} f(pz)

(recall that ψ(p) = 0 if p | N). This is a modular form in M_{k}(Γ_{0}(Np^{α}), ψ; Â) for

α = \{ \begin{matrix}1 & if p |̸ N \\ 0 & if p | N \end{matrix}

We have

f_{0}^{τ} (z) = f^{τ} (z) − \frac{\overline{ψ}(p)p^{k−1}}{\overline{α_{p}(f)}}

(2.3) Hida [Hi 1, p. 172] defines a projection

e : \overline{M}_{k}(Γ_{0}(Np^{∞}), ψ; Â) ⟶ M_{k}(Γ_{0}(Np^{α}), ψ; Â)

onto the space of ordinary forms. On M_{k}(Γ_{0}(Np^{n}), ψ; Â), it is equal to the limit

lim_{r→∞} U_{p}^{cp^{r}}

for a suitable integer c (depending on n). Here U_{p} is given by

U_{p} ( ∑_{m≥1} a(m)q^{m}) = ∑_{m≥1} a(mp)q^{m}

Hida [Hi 1, p. 174] also defines a projection (which he denotes by 11_{f})

11_{f_{0}} : M_{k}(Γ_{0}(Np^{α}), ψ; Â) ⟶ M_{k}(Γ_{0}(Np^{α}), ψ; c(f)^{−1}Â)

onto the f_{0}-part (here c(f) is a suitable non-zero element of Â). Following [PR 1, p.7], we define

l_{f_{0}} : M_{k}(Γ_{0}(Np^{α}), ψ; Â) ⟶ c(f)^{−1}Â

by l_{f_{0}}(g) = a_{1}(g|11_{f_{0}}) and

L_{f_{0}} = l_{f_{0}} ∘ e : \overline{M}_{k}(Γ_{0}(Np^{∞}), ψ; Â) ⟶ c(f)^{−1}Â

Note that our L_{f_{0}} is denoted by l_{f} in [Hi 1].

(2.4) [Hi 1, 4.5] gives an explicit formula for L_{f_{0}}: if g ∈ M_{k}(Γ_{0}(Np^{μ}), ψ; \overline{Q}) with μ ≥ α, then

L_{f_{0}}(g) = (\frac{p^{k/2−1}}{α_{p}(f)} )^{μ−α} \begin{matrix}⟨f_{0}^{τ} |_{k}( \begin{matrix}0 & −1 \\ Np^{μ} & 0 \end{matrix} ), g⟩ & Np^{μ} \\ ⟨f_{0}^{τ} |_{k}( \begin{matrix}0 & −1 \\ Np^{α} & 0 \end{matrix} ), f_{0}⟩ & Np^{α} \end{matrix}  (2.4.1)

and takes values in \overline{Q}. This implies that

L_{f_{0}}(f) = (1 − \frac{ψ(p)p^{2r−1}}{α_{p}(f)^{2}} )^{−1}  (2.4.2)

3. Galois groups and basic measures

(3.1) Let K = Q(\sqrt{D}), D < 0, be an imaginary quadratic field with discriminant D. Write O_{K} for the ring of integers in K and let h = ♯(Pic(O_{K})), w = ♯(O_{K}^{∗}) be the class number and the number of roots of unity of K respectively. We write N for the norm N_{K/Q}.

For a non-zero ideal 𝔣 ⊂ O_{K} we denote by K(𝔣) the ray class field of K (inside the fixed algebraic closure \overline{Q} of Q). The reciprocity map (= Artin symbol) defines an isomorphism

I𝔣/P𝔣 ⟶^{∼} G(K(𝔣)/K),

where I𝔣 is the multiplicative group of fractional ideals of K prime to 𝔣 and P𝔣 the group of principal ideals

(α) with α ≡ 1 (mod 𝔣)^{×}. For an integer m ≥ 1, let R_{m} ⊂ I_{(m)} be the group of principal ideals (β) with β ≡ b (mod mO_{K} )^{×} for some b ∈ Z prime to m. The subextension K ⊂ H_{m} ⊂ K((m)) satisfying G(K((m))/H_{m}) ⟶^{∼} R_{m}/P_{(m)} is the ring class field of K of conductor m. Write O_{m} := Z + mO_{K} for the subring of O_{K} of index m. An O_{m}-submodule A of K is an invertible O_{m}- module (= projective module of rank 1) iff O_{m} = {x ∈ K | xA ⊆ A}. A module with the latter property is called a proper O_{m}-ideal. The multiplicative group of proper O_{m}-ideals modulo principal O_{m}-ideals αO_{m} is canonically isomorphic to Pic(O_{m}). The restriction map I ↦ I ∩ O_{m}, defined on ideals I ⊂ O_{K} prime to m, extends to a (surjective) homomorphism I_{(m)} ⟶ Pic(O_{m}) with kernel R_{m}, defining thus an isomorphism

Pic(O_{m}) ⟶^{∼} I_{(m)}/R_{m} ⟶^{∼} G(H_{m}/K)

Write H for the Hilbert class field of K, H = H_{1} = K((1)).

(3.2) For every m ≥ 1, K((m)) is a Galois extension of Q. The Galois group G(K((m))/Q) is a semidirect product of G(K((m))/K) and G(K/Q). The non-trivial element τ ∈ G(K/Q) acts on the first group by conjugation: ^{τ} g = τ̃gτ^{−1}, where τ̃ is a lift of τ to G(K((m))/Q). We have

τ g = \{ \begin{matrix}g & if g ∈ G(K(μ_{m})/K) \\ g^{−1} & if g ∈ G(H_{m}/K) \end{matrix}

The compositum of all K((m)) is the maximal abelian extension K^{ab} of K. Denote by H_{∞} the compositum of all H_{m}. Fix a prime p > 2 and let K_{∞} be the compositum of all Z_{p}-extensions of K. It is contained in H_{∞}(μ_{∞}) (as the exponent of G(K^{ab}/H_{∞}(μ_{∞})) is equal to two) and G(K_{∞}/K) ⟶^{∼} Z_{p}^{2}. For a non-zero ideal 𝔣 ⊂ O_{K}, let m(𝔣) be the smallest integer m ≥ 1 such that 𝔣 | (m). An easy exercise in class field theory shows that

K(𝔣) ∩ H_{∞}(μ_{∞}) = K(𝔣) ∩ H_{n}(μ_{n})

with

n = m(𝔣) × \{ \begin{matrix}1 & if 8 |̸ m(𝔣) \\ 2 & if 8 | m(𝔣) \end{matrix}

This means that every character

Ω : G(K(𝔣) ∩ H_{∞}(μ_{∞})/K) ⟶ \overline{Q}^{∗}

can be decomposed into a product

Ω = η · (α ∘ N)  (3.2.1)

with

η : G(H_{n}/K) ⟶ \overline{Q}^{∗}, α : G(Q(μ_{n})/Q) ⟶ \overline{Q}^{∗},

where we use the notation of [PR 1] and write N for the restriction map G(\overline{Q}/K) ⟶ G(\overline{Q}/Q). We have ηη^{τ} = 1, α^{τ} = α, where W^{τ} (g) := W(^{τ} g). Note that a decomposition in (3.2.1) is not unique in general.

(3.3) Let W : G(K(𝔣)/K) ⟶ C^{∗} be a complex-valued ray class character of conductor 𝔣 (in other words, W does not factor through G(K(𝔤)/K) for any proper divisor 𝔤 | 𝔣). The theta function

Θ(W)(z) = ∑_{(𝔞,𝔣)=(1)} W(𝔞)q^{N(𝔞)} (+\frac{h}{w} if 𝔣 = (1))

lies in M_{1}(Γ_{0}(Δ(W)), (\frac{D}{·} )W|_{Z}) (in fact is a newform if 𝔣 ≠ (1)). Here Δ(W) = |D|N(𝔣) and W|_{Z} denotes the restriction of W to ideals in Z. If W = ρ · (χ ∘ N) as in (3.2), then W|_{Z} = χ^{2}. The L-function of Θ(W) is equal to the L-function of W over K:

L(Θ(W), s) = ∑_{(𝔞,𝔣)=(1)} W(𝔞)N(𝔞)^{−s} = L(W, s)

These functions satisfy the following functional equations

Λ(W, s) := Δ(W)^{s/2}(2π)^{−s}Γ(s)L(W, s) = τ(W)Λ(\overline{W}, 1 − s)

Θ(W)|_{1}( \begin{matrix}0 & −1 \\ Δ(W) & 0 \end{matrix} ) = −iτ(W)Θ(\overline{W})

with τ(W) ∈ \overline{Q}^{∗}, |τ(W)| = 1.

(3.4) Suppose f = ∑_{n≥1} a_{n}q^{n} ∈ S_{2r}(Γ_{0}(N), ψ) is a normalized (a_{1} = 1) newform of level N prime to |D|. The L-function of f has an Euler product

L(f, s) = ∑_{n≥1} a_{n}n^{−s} = ∏_{ℓ}(1 − a_{ℓ}ℓ^{−s} + ψ(ℓ)ℓ^{2r−1−s})^{−1} = ∏_{ℓ}[(1 − α_{ℓ}ℓ^{−s})(1 − β_{ℓ}ℓ^{−s})]^{−1}

(with ψ(ℓ) = 0 for ℓ | N, of course). The L-function of f over K, twisted by W, is defines by an Euler product

L(f ⊗ K, W, s) = ∏_{λ}[(1 − α_{N(λ)}W(λ)N(λ)^{−s})(1 − β_{N(λ)}W(λ)N(λ)^{−s})]^{−1},

where λ runs through prime ideals of O_{K}, α_{ℓ^{k}} = α_{ℓ}^{k}, β_{ℓ^{k}} = β_{ℓ}^{k} and W is extended by zero to ideals that are not prime to 𝔣. Note that

L(f ⊗ K, 11, s) = L(f, s)L(f ⊗ (\frac{D}{·} ), s)

with

L(f ⊗ (\frac{D}{·} ), s) = ∑_{n≥1} a_{n}(\frac{D}{n} )q^{n}

(one needs the assumption (N, |D|) = 1 at this place) and

L(f ⊗ K, W, s) = D_{lcm(N,Δ(W))}(f, Θ(W), s)

by 1.7.1.

(3.5) Following [PR 1, 2.1], we define basic measures associated to theta functions and Eisenstein series. Let p > 2 be a prime number. Fix an integer m ≥ 1 and a proper O_{m}-ideal A. Then Q_{A}(x) := N(x)/N(A) is a quadratic form on A. For a fixed integer c_{1} prime to p, define a distribution Θ_{A} on Z_{p,c_{1}}^{∗} = \underleftarrow{lim}_{ν}(Z/c_{1}p^{ν}Z)^{∗} by

Θ_{A}(a (mod c_{1}p^{ν} ))(z) = ∑_{x∈A_{Q_{A}(x)≡a (mod c_{1}p^{ν} )}} q^{Q_{A}(x)}

For n ≥ 1, denote by r_{A}(n) the number of proper ideals I ⊆ O_{m} with the class [I] = [A]. Note that r_{A}(n) = r_{A^{τ}} (n), where τ is the non-trivial element of G(K/Q). For n = 0, we put r_{A}(0) = w_{m}^{−1}. If ϕ is a function on Z/c_{1}p^{ν}Z with values in A, then

Θ_{A}(ϕ) = ∑_{x∈A} ϕ(Q_{A}(x))q^{Q_{A}(x)} = w_{m} ∑_{n≥1} ϕ(n)r_{A}(n)q^{n}

lies in M_{1}(Γ_{1}(M); A) for M = lcm(|D|m^{2}, c_{1}p^{2ν}) and depends only on the class [A] of A in Pic(O_{m}). If ϕ is a character, then Θ_{A}(ϕ) lies in M_{1}(Γ_{0}(M), (\frac{D}{·} )ϕ^{2}).

(3.6) For M ≥ 1 and α ∈ Z/MZ, define a function ϕ_{α,M} : Z/MZ ⟶ C by

ϕ_{α,M}(x) = e^{−2πiαx/M}

The formulas (for an integer C prime to pM)

E_{1}(α (mod Mp^{ν} ))(z) := (−2πi)^{−1}E_{1}(z, ϕ_{α,Mp^{ν}}) = \frac{1}{2}L̃(0, δ_{α (mod Mp^{ν} )}) + ∑_{{k·m>0 \atop k≡α (mod Mp^{ν} )}} sgn(k)q^{km}

E_{1}^{C}(α (mod Mp^{ν} ))(z) := E_{1}(α (mod Mp^{ν} ))(z) − CE_{1}(C^{−1}α (mod Mp^{ν} ))(z)

then define a distribution E_{1} (resp. a measure E_{1}^{C}) on Z_{p,M}^{∗} with values in \overline{M}_{1}(Γ_{1}(Mp^{∞})).

4. The convolution measures

In this section we slightly modify the construction of measures given in [PR 1, sec. 4] in order to obtain p-adic L-functions for modular forms of arbitrary even weight.

(4.1) Fix a prime number p > 2. Let f ∈ S_{2r}(Γ_{0}(N), ψ) be a newform of even weight 2r ≥ 2, ordinary at p. Decompose the level of f into N = N_{1}N_{2}N_{3} with N_{1} = cond(ψ), N_{2} = (N_{1}^{∞}, N/N_{1}). We write N = N′p^{γ} (resp. N_{i} = N′_{i}p^{γ_{i}}) with N′, N′_{i} prime to p. Let K = Q(\sqrt{D}) be an imaginary quadratic field with discriminant D prime to N. Let Ω : G(H_{∞}(μ_{∞})/K) ⟶ \overline{Q}^{∗} be a character of finite order with conductor 𝔣(Ω) prime to p. As in (3.2.1), it has a decomposition

Ω = η · (α ∘ N)

Let c_{1} be the conductor of the Dirichlet character α over Q and c_{2} the conductor of the ring class field character η. We put, as in [PR 1, 3.1],

c = N(𝔣(Ω)),  c_{3} = lcm(c_{1}, c_{2}),  Δ = lcm(|D′|c_{2}^{2}, c_{1}^{2}),

where D′ is the non-p part of D. We assume that (N, Δ) = 1. Denote by Δ_{1} the non-p part of cond((\frac{D}{·} )χ^{2}) and by Δ_{2} the non-p part of the level of Θ(Ω), i.e. of |D|c. Put Δ_{3} = Δ/Δ_{2} and ξ = ψ(\frac{D}{·} ). The p-adic measures defined in this section will be used to interpolate algebraic parts of the expressions D(f, Θ(W), r) for characters W of G(H_{∞}(μ_{∞})/K) such that Ω is equal to the non-p part of W. We decompose

W = ρ · (χ ∘ N)

as in (3.2.1). The level of Θ(W) is equal to Δ(W) = Δ_{2}p^{β}.

(4.2) Definition. In the notation of 4.1, we choose an integer C prime to NΔpc_{3} and put

Φ_{A,c_{1}}^{C}(a (mod c_{1}p^{ν} )) := H[ ∑_{α∈(Z/N_{1}Δ_{1}p^{ν}Z)^{∗}} ξ(α)Θ_{A}(α^{2}a (mod c_{1}p^{ν} ))(N_{1}N_{2}z)

δ_{1}^{r−1}(E_{1}^{C}(α (mod N_{1}Δ_{1}p^{ν} ))(N′_{3}z))]

This defines a distribution (in fact a measure – cf. [Hi 1, 5.1]) on Z_{p,c_{1}}^{∗} with values in \overline{M}_{2r}(Γ_{0}(NΔ p^{∞}), ψ; p^{−δ}Z_{p}) for some δ depending only on r. Note that the order of brackets matters: by (1.2.2) we have

δ_{1}^{r−1}(f(Az)) = A^{r−1}(δ_{1}^{r−1}f)(Az)

(4.3) Definition. We define a measure Ψ_{A,c_{1}}^{C} on Z_{p,c_{1}}^{∗} with values in \overline{M}_{2r}(Γ_{0}(Np^{∞}), ψ; p^{−δ}Z_{p}) by

Ψ_{A,c_{1}}^{C} := \frac{1}{2w_{m}} Φ_{A,c_{1}}^{C} |_{2r}T (Δ_{2})_{NΔp^{∞}/Np^{∞}},

where w_{m} = ♯(O_{m}^{∗}) (the index m was omitted in the Definition 6 in [PR 1]) and, for Md|M′, T (d) denotes the operator

M_{l}(Γ_{0}(M′), χ) ⟶ M_{l}(Γ_{0}(M), χ)

adjoint to g ↦ d^{l/2−1}g|_{l}( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ), i.e. satisfying

d^{l/2−1}⟨g|_{l}( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ), h⟩_{M′} = ⟨g, h|_{l}T (d)_{M′/M}⟩_{M}

for all g ∈ M_{l}(Γ_{0}(M), χ), h ∈ M_{l}(Γ_{0}(M′), χ)

(4.4) Definition. For a character ρ : G(H_{m}/K) ⟶^{∼} Pic(O_{m}) ⟶ \overline{Q}^{∗}, we define

Φ_{ρ,c_{1}}^{C} = ∑_{[A]∈Pic(O_{m})} ρ([A])^{−1}Φ_{A,c_{1}}^{C}
Ψ_{ρ,c_{1}}^{C} = ∑_{[A]∈Pic(O_{m})} ρ([A])^{−1}Ψ_{A,c_{1}}^{C}

We also define

Ψ_{f,ρ,c_{1}}^{C} := L_{f_{0}}(Ψ_{ρ,c_{1}}^{C})

This is a measure on Z_{p,c_{1}}^{∗} with values in c(f)^{−1}p^{−δ}O_{F (f)}, where F (f) is the field generated by coefficients

of f, F̂ (f) the closure of i_{p}(F (f)) and O_{F (f)} the ring of integers in F̂ (f).

(4.5) Definition. We view the Galois group G(K(μ_{c_{1}p^{∞}})/K) as a subgroup of Z_{p,c_{1}}^{∗} (of index 1 or 2) by the restriction and reciprocity maps

G(K(μ_{c_{1}p^{∞}})/K) ↪ G(Q(μ_{c_{1}p^{∞}})/Q) ⟶^{∼} Z_{p,c_{1}}^{∗}

All measures defined in (4.2)–(4.4) have support in G(K(μ_{c_{1}p^{∞}})/K) (by [PR 1, 5.1b]). We define a measure Ψ_{f,c_{1},c_{2}}^{C} on G(H_{c_{2}p^{∞}}/K) × G(K(μ_{c_{1}p^{∞}})/K) with values in c(f)^{−1}p^{−δ}O_{F (f)}.

Ψ_{f,c_{1},c_{2}}^{C}(σ (mod c_{2}p^{n}), τ (mod c_{1}p^{m})) := L_{f_{0}} (Ψ_{A,c_{1}}^{C}(a (mod c_{1}p^{m})))

for τ|_{Q(μ_{c_{1}p^{m}})} corresponding to a ∈ (Z/c_{1}p^{m}Z)^{∗} under the Artin symbol and σ (mod c_{2}p^{n}) corresponding to

[A]^{−1} ∈ Pic(O_{c_{2}p^{n}}). The argument of [PR 1, 5.1c] shows that Ψ_{f,c_{1},c_{2}}^{C} is in fact supported on the subgroup

G(H_{c_{2}p^{∞}}(μ_{c_{1}p^{∞}})/K) ⊂ G(H_{c_{2}p^{∞}}/K) × G(K(μ_{c_{1}p^{∞}})/K)

(4.6) Definition. We define modified measures Ψ̃_{A,c_{1}}^{C}, Ψ̃_{ρ,c_{1}}^{C}, Ψ̃_{f,ρ,c_{1}}^{C}, Ψ̃_{f,c_{1},c_{2}}^{C} by replacing the operator T (Δ_{2}) in (4.3) by T (1) – these measures are used (in a special case) in [PR 2].

5. Integrals of characters

In this section we sketch how the calculations of [PR 1] work in the higher weight case.

(5.1) Let χ : (Z/c_{1}p^{ν}Z)^{∗} ⟶ \overline{Q}^{∗} be a character such that its restriction to (Z/c_{1}Z)^{∗} is primitive and put M(ν) = N_{1}Δ_{1}p^{ν}. It follows from the definitions and Prop. 1.5.4 (cf. [PR 1, Lemma 7]) that

∫_{Z_{p,c_{1}}^{∗}} χ dΦ_{A,c_{1}}^{C} = (1 − Cξ(C)\overline{χ}^{2}(C))H [Θ_{A}(χ ∘ Q_{A})(N_{1}N_{2}z)δ_{1}^{r−1}(E_{1}(N′_{3}z, ϕ))] ,

where ϕ : Z/M(ν)Z ⟶ C satisfies (−\frac{2πi}{M(ν)}) ϕ̂ = \overline{ω} with ω = \overline{ψ}(\frac{D}{·} )χ^{2}.

(5.2) There is a bijection between pairs (x ∈ A (mod O_{m}^{∗}), Q_{A}(x)) and (I ⊂ O_{m} proper ideal, N(I)) given by I = xA^{−1}, hence

∫_{Z_{p,c_{1}}^{∗}} χ dΦ_{ρ,c_{1}}^{C} = w_{m}(1 − Cξ(C)\overline{χ}^{2}(C))H [Θ(W′′)(N_{1}N_{2}z)δ_{1}^{r−1}(E_{1}(N′_{3}z, ϕ))] ,

where W′′ = ρ · (χ ∘ N), considered as a character modulo 𝔣 = lcm(cond(ρ), cond(χ), p), and

Θ(W′′)(z) = ∑_{{𝔞⊂O \atop (𝔞,𝔣)=1}} W′′(𝔞)𝔮^{N(A)}

(note that [PR 1, Lemma 7’] gives an incorrect factor 2 instead of w_{m}).

(5.3) Put h = Θ(W′′)(N_{1}N_{2}z)δ_{1}^{r−1}(E_{1}(N′_{3}z, ϕ)). Hida’s formula (2.4.1) and the basic property (1.3.1) of the holomorphic projection imply that we have, for μ >> 0,

∫_{Z_{p,c_{1}}^{∗}} χ dΨ_{f,ρ,c_{1}}^{C} = \frac{1}{2}(1 − Cξ(C)\overline{χ}^{2}(C))\frac{(Δ_{2}p^{μ−α})^{r−1}}{α_{p}(f)^{μ−α}} \begin{matrix}⟨f_{0}^{τ} |_{2r}( \begin{matrix}0 & −1 \\ NΔ_{2}p^{μ} & 0 \end{matrix} ), h⟩ & NΔp^{μ} \\ ⟨f_{0}^{τ} |_{2r}( \begin{matrix}0 & −1 \\ Np^{α} & 0 \end{matrix} ), f_{0}⟩ & Np^{α} \end{matrix}  (5.3.1)

(5.4) Following [PR 1, 4.3] we use the involution

F = ( \begin{matrix}0 & −1 \\ NΔp^{μ} & 0 \end{matrix} )( \begin{matrix}N′_{3} & y \\ NΔp^{μ}t & N′_{3} \end{matrix} )

with N′_{3}xw − Δp^{μ}tyN/N′_{3} = 1 to compute the numerator in (5.3.1). We have (by [PR 1, Lemma 16,17, Cor. 20])

f_{0}|_{2r}( \begin{matrix}0 & −1 \\ NΔ_{2}p^{μ} & 0 \end{matrix} )|_{2r}F = ψ_{0}(N′_{3}w)λ_{N′_{3}}(f)f_{0}|_{2r}( \begin{matrix}Δ_{3} & 0 \\ 0 & 1 \end{matrix} )

Θ(W′′)(N_{1}N_{2}z)|_{1}F = (N_{1}N_{2})^{−1/2}((\frac{D}{·} )\overline{χ}^{2})(w)Θ(W′′)|_{1}( \begin{matrix}0 & −1 \\ Δp^{μ+γ_{3}} & 0 \end{matrix} )

E_{1}(N′_{3}z, ϕ)|_{1}F = \frac{ω_{0}(N′_{3}w)}{2πi} (\frac{NΔp^{μ}}{N′_{3}} )^{1/2} E_{1}(z, \overline{ω} (mod NΔp^{μ}))

Here ω_{0} is the primitive character associated to ω = \overline{ψ}(\frac{D}{·} )χ^{2} : (Z/N_{1}Δ_{1}p^{ν}Z)^{∗} ⟶ C^{∗}. Putting this and (5.3.1) together (and using (1.2.2)), we obtain

∫_{Z_{p,c_{1}}^{∗}} χ dΨ_{f,ρ,c_{1}}^{C} = (1 − Cξ(C)\overline{χ}^{2}(C))\frac{((\frac{D}{·} )χ^{2})(N′_{3})λ_{N′_{3}}(f)(Δp^{γ_{3}})^{1/2}}{(4πi)α_{p}(f)^{−α}} (\frac{Δ_{2}}{p^{α}} )^{r−1} \frac{Λ_{μ}(W′′)}{⟨f_{0}^{τ} |_{2r}( \begin{matrix}0 & −1 \\ Np^{α} & 0 \end{matrix} ), f_{0}⟩_{Np^{α}}}

with

Λ_{μ}(W′′) = \frac{p^{μ(r−1/2)}}{α_{p}(f)^{μ}} ⟨f_{0}^{τ} |_{2r}( \begin{matrix}Δ_{3} & 0 \\ 0 & 1 \end{matrix} ), g⟩_{NΔp^{μ}}

g = Θ(W′′)|_{1}( \begin{matrix}0 & −1 \\ Δp^{μ+γ_{3}} & 0 \end{matrix} )δ_{1}^{r−1}(E_{1}(z, \overline{ω} (mod NΔp^{μ})))

(5.5) The theta function Θ(W) satisfies the functional equation

Θ(W)|_{1}( \begin{matrix}0 & −1 \\ Δ_{2}p^{β} & 0 \end{matrix} ) = −iτ(W)Θ(\overline{W})

A calculation similar to [PR 1, 4.4] gives

Λ_{μ}(W′′) = −iτ(W)Δ_{3}^{1/2}Γ(W) ∑_{𝔞|p_{N(𝔞)=p^{s}}} μ̃(𝔞)W(𝔞)Λ_{μ,s}

with

Λ_{μ,s} = p^{μ(r−1/2)−s/2}α_{p}(f)^{−μ}⟨f_{0}^{τ} , Θ(\overline{W})|_{1}( \begin{matrix}p^{x} & 0 \\ 0 & 1 \end{matrix} )δ_{1}^{r−1}(E_{1}(z, \overline{ω} (mod NΔ_{2}p^{μ})))⟩_{NΔ_{2}p^{μ}}

x = μ + γ_{3} − β − s and (cf. [PR 1, Lemma 24])

Γ(W) = ∑_{𝔟|c_{3}_{N(𝔟)=m}} \frac{μ̃(𝔟)W(𝔟)a_{m}(Θ(\overline{W}))}{m^{r}} = ∏_{l|c_{3}} (1 − \frac{|a_{l}(Θ(W))|^{2}}{l^{r}} ) = ∏_{{l|(c_{1},c_{2}) \atop l |̸c/c_{1}}} (1 − \frac{1}{l^{r}} )

(5.6) Finally, a rather tedious calculation along the lines of [PR 1, pp. 23–25] (which will not be reproduced here) yields

Λ_{μ} = V_{p}(f, W) (\frac{p^{r−1/2}}{α_{p}(f)} )^{β−γ_{3}} \frac{2((r − 1)!)^{2}(−1)^{r−1}}{(4π)^{2r−1}} D_{NΔ(W)}(f, Θ(\overline{W}), r)

with

V_{p}(f, W) = ∏_{𝔭|p} (1 − \frac{ψ(N(𝔭))\overline{W}(𝔭)}{α_{N(𝔭)}(f)}N(𝔭)^{r−1}) (1 − \frac{W(𝔭)}{α_{N(𝔭)}(f)}N(𝔭)^{r−1})

The constant involving factorials comes from 1.8.1, which replaces Prop. 22 in [PR 1]. (5.7) The denominator in (5.4) is given by (cf. [Hi 2, 9.5])

α_{p}(f)^{−α}⟨f_{0}^{τ} |_{2r}( \begin{matrix}0 & −1 \\ Np^{α} & 0 \end{matrix} ), f_{0}⟩_{Np^{α}} = λ_{N}(f)p^{α(1−r)}H_{p}(f)⟨f, f⟩_{N}

with

H_{p}(f) = (1 − \frac{ψ(p)p^{2r−2}}{α_{p}(f)^{2}} ) (1 − \frac{ψ(p)p^{2r−1}}{α_{p}(f)^{2}} )

(5.8) Putting 5.4-7 together, we obtain a higher weight version of [PR 1, Thm. B]: Theorem. In the notation of 4.1, we have

Δ_{3}^{−1}(1 − Cψ(C)(\frac{D}{C} )\overline{W}(C))^{−1} ∫_{Z_{p,c_{1}}^{∗}} χ dΨ_{f,ρ,c_{1}}^{C} = \frac{L_{p}(f, W)Γ(Ω)t_{p}(f)V_{p}(f, W)Δ(W)^{r−1/2}}{α_{p}(f)^{β}H_{p}(f)p^{γ_{3}(r−1)}}

with t_{p}(f) = λ_{N′_{3}}(f)λ_{N}(f)^{−1}α_{p}(f)^{γ_{3}}, Δ(W) = |D|N(𝔣(W)) = the level of Θ(W),

\begin{aligned}L_{p}(f, W) & = (\frac{D}{−N′_{3}} )W(N′_{3})τ(W)\frac{2(−1)^{r−1}((r − 1)!)^{2}}{(4π)^{2r}} \frac{D_{NΔ(W)}(f, Θ(\overline{W}), r)}{⟨f, f⟩_{N}} \\ Γ(Ω) & = ∏_{{l|(c_{1},c_{2}) \atop l |̸c/c_{1}}} (1 − \frac{1}{l^{r}} ) \end{aligned}

(5.9) For the modified measures, the same calculation gives

∫_{Z_{p,c_{1}}^{∗}} χ dΨ̃_{f,ρ,c_{1}}^{C} = \frac{1}{2}(1 − Cξ(C)\overline{χ}^{2}(C))\frac{(p^{μ−α})^{r−1}}{α_{p}(f)^{μ−α}} \begin{matrix}⟨f_{0}^{τ} |_{2r}( \begin{matrix}0 & −1 \\ Np^{μ} & 0 \end{matrix} ), h⟩ & NΔp^{μ} \\ ⟨f_{0}^{τ} |_{2r}( \begin{matrix}0 & −1 \\ Np^{α} & 0 \end{matrix} ), f_{0}⟩ & Np^{α} \end{matrix}

∫_{Z_{p,c_{1}}^{∗}} χ dΨ̃_{f,ρ,c_{1}}^{C} = (1 − Cξ(C)\overline{χ}^{2}(C))\frac{((\frac{D}{·} )χ^{2})(N′_{3})λ_{N′_{3}}(f)(Δp^{γ_{3}})^{1/2}}{(4πi)α_{p}(f)^{−α}} (\frac{1}{p^{α}} )^{r−1} \frac{Λ̃_{μ}(W′′)}{⟨f_{0}^{τ} |_{2r}( \begin{matrix}0 & −1 \\ Np^{α} & 0 \end{matrix} ), f_{0}⟩_{Np^{α}}}

with

Λ̃_{μ}(W′′) = \frac{p^{μ(r−1/2)}}{α_{p}(f)^{μ}} ⟨f_{0}^{τ} |_{2r}( \begin{matrix}Δ & 0 \\ 0 & 1 \end{matrix} ), g⟩_{NΔp^{μ}}
g = Θ(W′′)|_{1}( \begin{matrix}0 & −1 \\ Δp^{μ+γ_{3}} & 0 \end{matrix} )δ_{1}^{r−1}(E_{1}(z, \overline{ω} (mod NΔp^{μ})))

Λ̃_{μ}(W′′) = −iτ(W)Δ_{3}^{1/2}Γ(Ω)V_{p}(f, W) (\frac{p^{r−1/2}}{α_{p}(f)} )^{β−γ_{3}} \frac{2((r − 1)!)^{2}(−1)^{r−1}}{(4π)^{2r−1}} D_{NΔ(W)}(f|_{2r}( \begin{matrix}Δ_{2} & 0 \\ 0 & 1 \end{matrix} ), Θ(\overline{W}), r)

As Δ_{2} divides the level of Θ(\overline{W}), we have

\frac{D(f|_{2r}( \begin{matrix}Δ_{2} & 0 \\ 0 & 1 \end{matrix} ), Θ(\overline{W}), r)}{D(f, Θ(\overline{W}), r)} = Δ_{2}^{r} \frac{∑_{n≥1} a(n, f)a(nΔ_{2}, Θ(\overline{W}))(nΔ_{2})^{−r}}{∑_{n≥1} a(n, f)a(n, Θ(\overline{W}))n^{−r}} = a(Δ_{2}, Θ(\overline{W})) = \overline{W}(D′),

where D′ is the non-p-part of D = (\sqrt{D}). It follows that

∫_{Z_{p,c_{1}}^{∗}} χ dΨ̃_{f,ρ,c_{1}}^{C} = Δ_{2}^{1−r}\overline{W}(D′) ∫_{Z_{p,c_{1}}^{∗}} χ dΨ_{f,ρ,c_{1}}^{C}

(5.10) In terms of measures on Galois groups, this can be rephrased as the following

Theorem. If W : G(H_{c_{2}p^{∞}}(μ_{c_{1}p^{∞}})/K) ⟶ \overline{Q}_{p}^{∗} is a character of finite order with non-p part equal to Ω, then

Δ_{3}^{−1}(1−Cψ(C)(\frac{D}{C} )\overline{W}(C))^{−1} ∫_{G(H_{c_{2}p^{∞}} (μ_{c_{1}p^{∞}})/K)}W dΨ̃_{f,c_{1},c_{2}}^{C} = \overline{W}(D′)\frac{L_{p}(f, W)Γ(Ω)t_{p}(f)V_{p}(f, W)Δ(W)^{r−1/2}}{α_{p}(f)^{β}H_{p}(f)(Δ_{2}p^{γ_{3}})^{r−1}}

Definition. For any continuous character ϕ : G(H_{c_{2}p^{∞}}(μ_{c_{1}p^{∞}})/K) ⟶ \overline{Q}_{p}^{∗} with (ϕ)^{non−p} = Ω, we define (following [PR 2, Def. 2.4])

L_{p}(f ⊗ K, ϕ) := (−1)^{r−1}H_{p}(f)Δ_{3}^{−1}(\frac{D}{−N′_{3}} )(1 − Cψ(C)(\frac{D}{C} )ϕ(C)^{−1})^{−1} ∫_{G(H_{c_{2}p^{∞}} (μ_{c_{1}p^{∞}})/K)} ϕ dΨ̃_{f,c_{1},c_{2}}^{C}

Fix a character C : G(H_{c_{2}p^{∞}}(μ_{c_{1}p^{∞}})/K) ⟶ \overline{Q}_{p}^{∗} of finite order and consider

L_{p}(f ⊗ K, C)(λ) := L_{p}(f ⊗ K, Cλ)

as a function on characters λ : G(H_{c_{2}p^{∞}}(μ_{c_{1}p^{∞}})/K) ⟶ (1 + pZ_{p}) (such characters automatically factor through G(K_{∞}/K)). It follows from [PR 1, Prop. 29] that L_{p}(f ⊗ K, C) is an Iwasawa function with values in c(f)^{−1}p^{−δ}O_{F (f)}, provided p |̸ cond(ψ).

(5.11) Following [PR 1, 5.2], we show that the p-adic L-function L_{p}(f ⊗K, C) satisfies a functional equation, provided the following condition is satisfied:

(N, Δ(Ω)) = 1, p |̸ cond(ψ)

If this is the case, then (because f is ordinary) either (a) p |̸ N, |i_{p}(a_{p})|_{p} = 1, or (b) p∥N, p |̸ cond(ψ). We have γ = γ_{3} = 0 (resp. = 1) in the case (a) (resp. (b)).

We use the fact that, for a character W of G(H_{c_{2}p^{∞}}(μ_{c_{1}p^{∞}})/K) of finite order, we have

\frac{a_{p}(f^{τ})}{a_{p}(f)} = \frac{α_{p}(f^{τ})}{α_{p}(f)} = \overline{ψ}_{0}(p), λ_{N}(f^{τ}) = λ_{N}(f)^{−1}

V_{p}(f^{τ} , \overline{W}) = V_{p}(f, W), H_{p}(f^{τ}) = H_{p}(f), ⟨f^{τ} , f^{τ} ⟩_{N} = ⟨f, f⟩_{N}
λ_{N′_{3}}(f)/λ_{N′_{3}}(f^{τ}) = λ_{N′_{3}}(f)^{2} = \overline{ψ}_{0}(N′_{3}), t_{p}(f)/t_{p}(f^{τ}) = \overline{ψ}_{0}(N′_{3})ψ_{0}(p)^{γ}λ_{N}(f)^{−2}

and the functional equation for the Rankin-Selberg convolution (which can be proved, e.g., by a computation of local constants; see [PR 1, 5.2] for the weight two case)

D_{NΔ(W)}(f, Θ(\overline{W}), r) = (\frac{D}{−N′} )\overline{W}(N′)λ_{N}(f)^{2}ψ_{0}(Δ(W)p^{−γ})τ(W)^{−2}D_{NΔ(W)}(f^{τ} , Θ(W), r)  (5.11.1)

which holds for β = ord_{p}Δ(W) big enough. This implies that

\begin{aligned}\frac{L_{p}(f, W)}{L_{p}(f^{τ} , \overline{W})} & = W^{2}(N′_{3})(\frac{D}{−N′} )\overline{W}(N′)ψ_{0}(Δ(W)p^{−γ})λ_{N}(f)^{2} \\ \frac{L_{p}(f ⊗ K, W)}{L_{p}(f^{τ} ⊗ K, \overline{W})} & = \overline{W}^{2}(D′)W^{2}(N′_{3})(\frac{D}{−N′} )\overline{W}(N′)ψ_{0}(Δ(Ω))\overline{ψ}_{0}(N′_{3}) \end{aligned}

(5.12) In the notation of 5.11, we define

Λ_{p}(f ⊗ K, C)(λ) := (Cλ)(D′N′_{3}^{−1})λ(N′)^{1/2}L_{p}(f ⊗ K, C)(λ)

It follows from (5.11.1) that

Proposition. Λ_{p} satisfies the functional equation

Λ_{p}(f ⊗ K, C)(λ) = \overline{ψ}_{0}(N′_{3})(\frac{D}{−N′} )ψ_{0}(Δ(C)^{non−p})\overline{C}(N′)Λ_{p}(f^{τ} ⊗ K, \overline{C})(λ^{−1})

Corollary. Suppose that ψ = 1, (\frac{D}{N′} ) = 1 and both C and λ are anticyclotomic (i.e. CC^{τ} = λλ^{τ} = 1). Then L_{p}(f ⊗ K, C) = 0.

Proof. Under our assumptions, we have f^{τ} = f. Noting that Λ_{p}(f ⊗ K, C)(λ) = Λ_{p}(f ⊗ K, C^{τ})(λ^{τ}), the functional equation reads as follows:

Λ_{p}(f ⊗ K, C)(λ) = (\frac{D}{−N′} )\overline{C}(N′)Λ_{p}(f ⊗ K, \overline{C}^{τ})(λ^{−τ}) = −Λ_{p}(f ⊗ K, C)(λ)

(5.13) We now specialize the p-adic L-function to cyclotomic λ’s. Let χ be a Dirichlet character of conductor c_{1}p^{ν} and put W = \overline{χ} ∘ N. Assume that (pN, |D|c_{1}) = 1. In this case we have c_{2} = 1, c_{3} = c_{1}, c = c_{1}^{2}, Δ = Δ_{2} = |D|c_{1}^{2}, Δ_{3} = 1, Δ(W) = |D|p^{2ν}, β = 2ν, Γ(Ω) = 1. Under our assumptions, χ(\frac{D}{·} ) has conductor |D|c_{1}p^{ν} and

D_{NΔ(χ∘N)}(f, Θ(χ ∘ N), s) = L(f ⊗ χ, s)L(f ⊗ χ(\frac{D}{·} ), s),

where

\begin{matrix}L(f ⊗ χ, s) = ∑_{n≥1} a_{n}χ(n)n^{−s} \\ L(f ⊗ χ(\frac{D}{·} ), s) = ∑_{n≥1} a_{n}χ(n)(\frac{D}{n} )n^{−s} \end{matrix}

We also have the relations

α_{p}(f ⊗ (\frac{D}{·} )) = (\frac{D}{p} )α_{p}(f)

τ(\overline{χ} ∘ N) = −iG(\overline{χ})G(\overline{χ}(\frac{D}{·} ))/(c_{1}p^{ν}) = −i\overline{χ}(|D|)(\frac{D}{c_{1}p^{ν}} )G(\overline{χ})^{2}G((\frac{D}{·} ))/(c_{1}p^{ν}) =

= \overline{χ}(|D|)(\frac{D}{c_{1}p^{ν}} )G(\overline{χ})^{2}/(c_{1}p^{ν}) = \overline{χ}(|D|)(\frac{D}{c_{1}p^{ν}} )(c_{1}p^{ν})G(χ)^{−2},

where

G(χ) = ∑_{x∈(Z/cond(χ)Z)^{∗}} χ(x)e^{2πix/cond(χ)}

is the Gauss sum. Theorem 5.10 tells us that

L_{p}(f⊗K, \overline{χ}∘N) = \overline{χ}(N′_{3})^{2}t_{p}(f)V_{p}(f, \overline{χ}∘N)\frac{|D|^{1/2}p^{2ν}}{p^{γ_{3}(r−1)}c_{1}^{2r−1}α_{p}(f)^{2ν}} (\frac{D}{c_{1}p^{ν}} )\frac{2((r − 1)!)^{2}}{(4π)^{2r}} \frac{D_{N|D|p^{2ν}} (f, Θ(χ ∘ N), r)}{⟨f, f⟩_{N}G(χ)^{2}}

(5.13.1)

Over Q, there exist several constructions of p-adic L-functions associated to modular forms. We adopt the normalization of [M - T - T]. Their L-function is defined on continuous characters λ : Z_{p}^{∗} ⟶ C_{p}^{∗} and satisfies the following interpolation property. For χ as above and x : Z_{p}^{∗} ⟶ Z_{p}^{∗} the identity,

L_{p}^{MTT}(f, x^{r−1}\overline{χ}) = \frac{p^{rν}}{α_{p}(f)^{ν}} (1 − \frac{χ(p)ψ(p)}{α_{p}(f)}p^{r−1}) (1 − \frac{\overline{χ}(p)p^{r−1}}{α_{p}(f)} ) \frac{(r − 1)!}{(−2πi)^{r−1}} \frac{L(f ⊗ χ, r)}{G(χ)}

We define

L_{p}^{MTT}(f ⊗ K, λ) = L_{p}^{MTT}(f, λ)L_{p}^{MTT}(f ⊗ (\frac{D}{·} ), λ)

Then

L_{p}^{MTT}(f ⊗ K, x^{r−1}\overline{χ}) = \frac{p^{2rν}}{α_{p}(f)^{2ν}} (\frac{D}{p^{ν}} )V_{p}(f, \overline{χ} ∘ N)\frac{(−1)^{r−1}((r − 1)!)^{2}}{(2π)^{2r−2}} \frac{D_{N|D|p^{2ν}} (f, Θ(χ ∘ N), r)}{G(χ)^{2}} (5.13.2)

Comparing 5.13.1-2. we obtain

L_{p}(f ⊗ K, \overline{χ} ∘ N) = (\frac{D}{c_{1}} )\frac{\overline{χ}(N′_{3})^{2}(−1)^{r−1}t_{p}(f)|D|^{1/2}}{(2c_{1})^{2r−1}p^{γ_{3}(r−1)}} \frac{L_{p}^{MTT}(f ⊗ K, x^{r−1}\overline{χ})}{(2π)^{2}⟨f, f⟩_{N}}  (5.13.3)

Consequently,

L_{p}(f ⊗ K, λ ∘ N) = (\frac{D}{c_{1}} )\frac{λ(N′_{3})^{2}(−1)^{r−1}t_{p}(f)|D|^{1/2}}{(2c_{1})^{2r−1}p^{γ_{3}(r−1)}} \frac{L_{p}^{MTT}(f ⊗ K, x^{r−1}λ)}{(2π)^{2}⟨f, f⟩_{N}}  (5.13.4)

for any continuous character λ : G(Q_{∞}/Q) ⟶ 1 + pZ_{p}.

6. The Fourier expansion

(6.1) In this section we shall investigate more closely the p-adic L-function defined in 5.10. In order to prove Theorem A, it is sufficient to specialize the constants in Sec. 5 as follows: Ω = 1, ψ = 1, p |̸ N|D|, m = 1, 2 |̸ |D|. This implies that (|D|, 2N) = 1, f is a normalized newform in S_{2r}(Γ_{0}(N)), N_{1} = N_{2} = 1, N_{3} = N′_{3} = N, c_{1} = c_{2} = c = 1, Δ = Δ_{1} = Δ_{2} = |D|, Δ_{3} = 1, γ = γ_{3} = 0. Under these assumptions, the interpolation formula from 5.10 (with W replaced by CW) reduces to that of 0.5. In particular, for an ideal A ⊂ O_{K} and an integer C prime to N|D|p we have the following measures on Z_{p}^{∗}:

Φ_{A}^{C}(β (mod p^{ν} )) = H[  ∑_{α∈(Z/|D|p^{ν}Z)^{∗}} (\frac{D}{α} )Θ_{A}(α^{2}β (mod p^{ν} ))(z)δ_{1}^{r−1}(E_{1}^{C}(α (mod |D|p^{ν} ))(Nz))]

Ψ̃_{A}^{C}(β (mod p^{ν} )) = \frac{1}{2w} Φ_{A}^{C}(β (mod p^{ν} ))|_{2r}T (1)_{N|D|p^{ν}/Np^{ν}} ,

where

g|_{2r}T (1)_{N|D|p^{ν}/Np^{ν}} = ∑_{γ∈Γ_{0}(N|D|p^{ν})\Γ_{0}(Np^{ν})} g|_{2r}γ

is the trace from M_{2r}(Γ_{0}(N|D|p^{ν})) to M_{2r}(Γ_{0}(Np^{ν})).

(6.2) For a fixed character C : G(H/K) ⟶ \overline{Q}_{p}^{∗} (recall that H is the Hilbert class field of K) we have the p-adic L-function L_{p}(f ⊗ K, C) from 5.10, defined on continuous characters

λ : G(K_{∞}/K) ⟶ (1 + pZ_{p})

If (\frac{D}{N} ) = 1, then, according to Cor. of Prop. 5.12, L_{p}(f ⊗ K, C) vanishes on characters satisfying λλ^{τ} = 1. If λ = λ^{τ}, then λ factors through G(Q_{∞}/Q), where Q_{∞}/Q is the cyclotomic Z_{p}-extension. Write λ̃ for the composite map

Z_{p}^{∗} ↪ Q_{p}^{∗} ⟶ A_{Q}^{∗}/Q^{∗} ⟶ G(Q^{ab}/Q) ⟶ G(Q_{∞}/Q)⟶^{λ}1 + pZ_{p}

(we have λ = λ̃ ∘ N in the notation of 3.2.1). Then

L_{p}(f ⊗ K, C)(λ) = (−1)^{r−1}H_{p}(f)(\frac{D}{−N} )(1 − C(\frac{D}{C} )λ̃^{−2}(C))^{−1}L_{f_{0}} [ ∑_{[A]∈Pic(O_{K})} C(A)^{−1} ∫_{Z_{p}^{∗}} λ̃ dΨ̃_{A}^{C}] (6.2.1)

(this is true regardless of the value of (\frac{D}{N} )).

(6.3) The rest of this section will be devoted to the calculation of (some) Fourier coefficients of ∫_{Z_{p}^{∗}} λ dΨ̃_{A}^{C} for

an arbitrary continuous function on Z_{p}^{∗}. Since D is odd by assumption, |D| is square-free. This means that one can take as representatives of Γ_{0}(N|D|p^{ν})\Γ_{0}(Np^{ν}) the set of matrices

|D_{1}|^{−1}W_{D_{1}}^{(ν)}( \begin{matrix}1 & j \\ 0 & |D_{1}| \end{matrix} ),

where |D_{1}| runs through positive divisors of D, j ∈ Z/|D_{1}|Z and

W_{D_{1}}^{(ν)} = ( \begin{matrix}|D_{1}|a & b \\ N|D|p^{ν}c & |D_{1}|d \end{matrix} ), a, b, c, d ∈ Z

with det(W_{D_{1}}^{(ν)}) = |D_{1}| is fixed. We normalize the sign of D_{1} by D_{1} ≡ 1 (mod 4 ) and put D_{2} = D/D_{1}. Let λ be a function on (Z/p^{ν}Z)^{∗} with values in C_{p}. We define

h_{D_{1}}(λ) := ∑_{β∈(Z/p^{ν}Z)^{∗}} ∑_{α∈(Z/|D|p^{ν}Z)^{∗}} (\frac{D}{α} )λ(β)H[Θ_{A}(α^{2}β (mod p^{ν} ))(z)|_{1}W_{D_{1}}^{(ν)}×

×δ_{1}^{r−1}(E_{1}(α (mod |D|p^{ν} ))(Nz)|_{1}W_{D_{1}}^{(ν)}]

Then

∫_{Z_{p}^{∗}} λ dΨ̃_{A} = \frac{1}{2w} ∑_{D=D_{1}·D_{2}} ∑_{j∈Z/|D_{1}|Z} h_{D_{1}}(λ)|_{2r}( \begin{matrix}1 & j \\ 0 & |D_{1}| \end{matrix} )  (6.3.1)

Note that we have

∫_{Z_{p}^{∗}} λ dΨ̃_{A}^{C} = ∫_{Z_{p}^{∗}} λ_{C} dΨ̃_{A}  (6.3.2)

with λ_{C}(β) = λ(β) − C(\frac{D}{C} )λ(C^{−2}β).

(6.4) Proposition. We have

Θ_{A}(α (mod p^{ν} ))(z)|_{1}W_{D_{1}}^{(ν)} = (\frac{D_{1}}{cp^{ν}N} )(\frac{D_{2}}{aN(A)})κ(D_{1})^{−1}Θ_{AD_{1}^{−1}}(|D_{1}|a^{2}α (mod p^{ν} ))(z)

∑_{α_{1} (mod |D_{1}|)} (\frac{D_{1}}{α_{1}} )E_{1}((α_{1}, α_{2}) (mod (|D_{1}|, |D_{2}|p^{ν}) ))(Nz)|_{1}W_{D_{1}}^{(ν)} =

= (\frac{D_{1}}{bN} )κ(D_{1}) ∑_{n=1}^{∞} ( ∑_{{k|n \atop k≡aα_{2} (mod |D_{2}|p^{ν} )}} sgn(k)(\frac{D_{1}}{n/k} )) q^{nN} + \{ \begin{matrix}0 & if D_{1} ≠ 1 \\ \frac{1}{2} L̃(0, δ_{aα_{2} (mod |D_{2}|p^{ν} )}) & if D_{1} = 1 \end{matrix}

where: κ(D_{1}) = 1 (resp. i) if D_{1} > 0 (resp. D_{1} < 0), D_{1} is the unique ideal of O_{K} with N(D_{1}) = |D_{1}| and (α_{1}, α_{2}) (mod (D_{1}|, |D_{2}|p^{ν}) ) denotes the unique residue class modulo |D|p^{ν} congruent to α_{1} modulo |D_{1}| and to α_{2} modulo |D_{2}|p^{ν}.

Proof. [PR 2, Prop. 3.8, 3.15]

(6.5) Writing a_{m}(g) for the m-th Fourier coefficient of a holomorphic modular form g, we obtain from Prop. 6.4 and Prop. 1.9

a_{m}(h_{D_{1}}(λ)) = \frac{(−1)^{r−1}}{\binom{2r−2}{r−1}} wm^{r−1}(\frac{D_{1}}{bcp^{ν}} )(\frac{D_{2}}{N(A)}) ∑_{{α_{2}∈(Z/|D_{2}|p^{ν}Z)^{∗} \atop β∈(Z/p^{ν}Z)^{∗}}} (\frac{D_{2}}{aα_{2}} )λ(β) ∑_{{j+nN=m \atop j≡|D_{1}|a^{2}α_{2}^{2}β (mod p^{ν} )}}

r_{AD_{1}^{−1}}(j) ∑_{{k|n \atop k≡aα_{2} (mod |D_{2}|p^{ν} )}} sgn(k)(\frac{D_{1}}{n/k} )P_{r−1} (\frac{j − nN}{j + nN} ) = (\frac{D_{1}}{−|D_{2}|N} )(\frac{D_{2}}{N(A)})×

× ∑_{{j+nN=m \atop p |̸j}} ∑_{{k|n \atop p |̸k}} r_{AD_{1}^{−1}}(j)(\frac{D_{2}}{k} )(\frac{D_{1}}{n/k} )sgn(k)λ (\frac{j}{|D_{1}|k^{2}} ) P_{r−1} (1 − \frac{2nN}{m} )

plus the contribution of an extra term for D_{1} = 1 in Prop. 6.4. Applying (6.3.1) and

a_{m}( ∑_{j (mod |D_{1}|)} g|_{2r}( \begin{matrix}1 & j \\ 0 & |D_{1}| \end{matrix} )) = |D_{1}|^{1−r}a(|D_{1}|m, g),

we see that

a_{m} (∫_{Z_{p}^{∗}} λ dΨ̃_{A}) = \frac{1}{2w} ∑_{D=D_{1}·D_{2}} |D_{1}|^{1−r}a(|D_{1}|m, h_{D_{1}}(λ)) = \frac{(−1)^{r−1}}{\binom{2r−2}{r−1}} m^{r−1}(\frac{D}{−N} ) ∑_{D=D_{1}·D_{2}} (\frac{D_{2}}{N(A)})×

× ∑_{{j+nN=|D_{1}|m \atop p |̸j}} ∑_{{p |̸k|n \atop k>0}} r_{AD_{1}^{−1}}(j)(\frac{D_{2}}{−kN} )(\frac{D_{1}}{|D_{2}|n/k} )λ (\frac{m|D_{1}| − nN}{|D_{1}|k^{2}} ) P_{r−1} (1 − \frac{2nN}{m|D_{1}|}) (6.5.1)

plus an extra term for D_{1} = 1. (6.6) We define, for integers n, k and an ideal A ⊂ O_{K},

ε_{A}(n, k) := \{ \begin{matrix}0 & if (k, n/k, |D|) ≠ 1 \\ (\frac{D_{1}}{k} )(\frac{D_{2}}{−nN/k} )(\frac{D_{2}}{N(A)}) & if \{ \begin{matrix}(k, n/k, |D|) = 1 \\ (k, |D|) = |D_{2}| \\ D_{1}D_{2} = D \end{matrix} \} \end{matrix}

Proposition. If p | m, then

a_{m} (∫_{Z_{p}^{∗}} λ dΨ̃_{A}) = \frac{(−1)^{r−1}}{\binom{2r−2}{r−1}} m^{r−1}(\frac{D}{−N} ) ∑_{{1≤n≤ \frac{m|D|}{N} \atop p |̸n}} r_{A}(m|D| − nN)P_{r−1} (1 − \frac{2nN}{m|D|}) ×

× ∑_{{k|n \atop k>0}} ε_{A}(n, k)λ (\frac{m|D| − nN}{|D|} · \frac{k^{2}}{n^{2}} )

Proof. As p | m, the extra term with D_{1} = 1 in (6.5.1) vanishes (cf. [PR 2, p. 488]). We introduce new variables n_{new} := n|D_{2}| and k_{new} := n|D_{2}|/k. Observing that r_{AD_{1}^{−1}}(j) = r_{AD^{−1}}(j|D_{2}|) = r_{A}(j|D_{2}|) (the last equality because D = (\sqrt{D}) is a principal ideal), we obtain from (6.5.1) that the L.H.S. is equal to

\frac{(−1)^{r−1}}{\binom{2r−2}{r−1}} m^{r−1}(\frac{D}{−N} ) ∑_{D=D_{1}·D_{2}} ∑_{1≤n≤ \frac{m|D|}{N} ; {|D_{2}||n \atop p |̸m|D|−nN}} r_{A}(m|D| − nN) ∑_{|D_{2}||k|n ; {p |̸n/k \atop k>0}} (\frac{D_{1}}{k} )(\frac{D_{2}}{−nN/k} )(\frac{D_{2}}{N(A)})×

×λ (\frac{m|D| − nN}{|D|} · \frac{k^{2}}{n^{2}} ) P_{r−1} (1 − \frac{2nN}{m|D|})

The product (\frac{D_{1}}{k} )(\frac{D_{2}}{−nN/k} ) is non-zero (for |D_{2}| | k | n) iff (k, \frac{n}{k}, |D|) = 1 and (k, |D|) = |D_{2}|. Finally, p |̸ m|D| − nN iff p |̸ n.

Corollary. If (\frac{D}{N} ) = 1 and p | m, then

a_{m} (∫_{Z_{p}^{∗}} log_{p} dΨ̃_{A}) = \frac{(−1)^{r}}{\binom{2r−2}{r−1}}m^{r−1} ∑_{{n=1 \atop p |̸n}}^{\frac{m|D|}{N}} r_{A}(m|D| − nN)σ_{A}(n)P_{r−1} (1 − \frac{2nN}{m|D|}) ,

where

σ_{A}(n) = ∑_{{k|n \atop k>0}} ε_{A}(n, k) log_{p} (\frac{n}{k^{2}} )

Proof. As in [PR 2, Prop. 3.18].

(6.7) Let σ ∈ G(H/K) correspond to [A]^{−1} under the reciprocity isomorphism G(H/K) ⟶^{∼} Pic(O_{K}). For every integer m ≥ 1 we define

\begin{aligned}B_{m}^{σ} & := m^{r−1} ∑_{{n=1 \atop p |̸n}}^{\frac{m|D|}{N}} r_{A}(m|D| − nN)σ_{A}(n)P_{r−1} (1 − \frac{2nN}{m|D|}) \\ C_{m}^{σ} & := m^{r−1} ∑_{n=1}^{\frac{m|D|}{N}} r_{A}(m|D| − nN)σ_{A}(n)P_{r−1} (1 − \frac{2nN}{m|D|}) \end{aligned}

Note that, for (\frac{D}{N} ) = 1 and p | m, we have

a_{m} (∫_{Z_{p}^{∗}} log_{p} dΨ̃_{A}) = \frac{(−1)^{r}}{\binom{2r−2}{r−1}}B_{m}^{σ}

by the previous corollary.

Proposition. Suppose that pO_{K} = 𝔭\overline{𝔭} with 𝔭 ≠ \overline{𝔭}. Then

C_{mp^{4}}^{σ} − 2p^{r−1}(C_{mp^{3}}^{σσ_{𝔭}} + C_{mp^{3}}^{σσ\overline{𝔭}}) + p^{2(r−1)}(C_{mp^{2}}^{σσ_{𝔭}^{2}} + 4C_{mp^{2}}^{σ} + C_{mp^{2}}^{σσ2 ; \overline{𝔭}}) − 2p^{3(r−1)}(C_{mp}^{σσ_{𝔭}} + C_{mp}^{σσ\overline{𝔭}}) + C_{m}^{σ} =

= B_{mp^{4}}^{σ} − p^{2(r−1)}B_{mp^{2}}^{σ},

where σ_{𝔭} denotes the Frobenius element associated to 𝔭.

Proof. A calculation along the lines of [PR 2, Prop. 3.20]. The details are omitted. In more suggestive terms, the above result can be stated as

(U_{p} − p^{r−1}σ_{𝔭})^{2}(U_{p} − p^{r−1}σ\overline{𝔭})^{2} ( ∑_{m≥1} C_{m}^{σ}q^{m}) = (U_{p}^{4} − p^{2(r−1)}U_{p}^{2}) ( ∑_{m≥1} B_{m}^{σ}q^{m})  (6.7.1)

II. Heegner cycles and heights

The contents of this chapter is the following. In Sec. 1 we recall basic facts about p-adic heights constructed in [Ne 2]. Sec. 2 deals with the geometry of Kuga-Sato varieties. In Sec. 3 we define Heegner cycles and in Sec. 4–5 we compute their local heights. The final section contains the proof of Thm A.

1. Abel-Jacobi maps and mixed extensions (1.1) Let X be a proper smooth scheme over a field F of characteristic zero. For an integer i ≥ 0, let CH^{i}(X) be the Chow group of algebraic cycles of codimension i on X modulo rational equivalence. For each prime number p, there is a cycle class into p-adic étale cohomology

CH^{i}(X) ⟶ H_{et}^{2i}(X ⊗_{F} \overline{F} , Q_{p}(i))^{G(\overline{F} /F)}  (1.1.1)

and we denote by CH^{i}(X)_{0} its kernel (it does not depend on p, by the Lefschetz principle and a comparison between étale and singular cohomology over C). This is a slightly non-standard notation, as we include all torsion into CH^{i}(X)_{0}.

(1.2) The cycle class map (1.1.1) factors through H_{et}^{2i}(X, Q_{p}(i)). The Hochschild-Serre spectral sequence

H^{i}(G(\overline{F} /F), H_{et}^{j}(X ⊗_{F} \overline{F} , Q_{p}(k))) ⟹ H_{et}^{i+j}(X, Q_{p}(k))

then induces a p-adic analogue of the Abel-Jacobi map (cf. [Ja 1])

Φ : CH^{i}(X)_{0} ⟶ H^{1}(G(\overline{F} /F), H_{et}^{2i−1}(X ⊗_{F} \overline{F} , Q_{p}(k)))

Here one must interpret both the Galois and étale cohomology as a continuous cohomology ([Ja 1], [Ta]) – a convention we adopt for the rest of the paper. We shall also omit the subscript “et” and use the abbreviation H^{i}(F, A) for H^{i}(G(\overline{F} /F), A). There is a geometric definition of the Abel-Jacobi map Φ ([Ja 2]): let Y be a cycle on X representing an element of CH^{i}(X)_{0}. Consider the exact cohomology sequence (in which \overline{X} = X ⊗_{F} \overline{F} etc.)

0 ⟶ H^{2i−1}(\overline{X}, Q_{p}(i)) ⟶ H^{2i−1}(\overline{X} − \overline{Y} , Q_{p}(i)) ⟶ H2i ; \overline{Y}(\overline{X}, Q_{p}(i)) ⟶ H^{2i}(\overline{X}, Q_{p}(i))

of (continuous, linear) p-adic representations of G(\overline{F} /F). Its pull-back by

cl(\overline{Y}) : Q_{p} ⟶ H2i ; \overline{Y}(\overline{X}, Q_{p}(i))

gives a short exact sequence

0 ⟶ V ⟶ E ⟶ Q_{p} ⟶ 0  (1.2.1)

The class of E in Ext_{G(\overline{F} /F)}^{1}(Q_{p}, V) = H^{1}(F, V) is equal to Φ(Y).

(1.3) Suppose that F is a finite extension of Q_{ℓ} and V an arbitrary (linear, continuous) p-adic representation of G(\overline{F} /F). Bloch and Kato [Bl - Ka] defined a subspace H_{f}^{1}(F, V) ⊆ H^{1}(F, V). If W is the largest subrepresentation of V “with good reduction”, then H_{f}^{1}(F, V) is equal to the Ext-group Ext^{1}(Q_{p}, W) in the category of representations with good reduction, i.e. unramified (resp. crystalline) representations if ℓ ≠ p (resp. if ℓ = p) – cf. [Fo - PR]. For a number field F, one defines

H_{f}^{1}(F, V) = {x ∈ H^{1}(F, V) | x_{v} ∈ H_{f}^{1}(F_{v}, V) ∀v}

(1.4) The main result of [Ne 1], as well as the application of the height machinery of [Ne 2] to algebraic cycles ([Ne 2, Sec. 5]) were both based on the following statement (cf. [Ne 2, 5.4]):

”Would-be Lemma”. Let F be a number field, X a smooth projective scheme over F with good reduction at all places dividing p. Assume that V := H^{2i−1}(\overline{X}, Q_{p}(i)) satisfies the monodromy conjecture at all places of bad reduction of X. Then:

(1) V is a crystalline representation of G(\overline{F}_{v}/F_{v}) for each place v|p of F.

(2) The image of Φ : CH^{i}(X)_{0} ⟶ H^{1}(F, V) is contained in H_{f}^{1}(F, V).

Here (1) is the ”crystalline conjecture” and (2) was deduced from (1), [Bl - Ka, 3.8.4], the crystalline Weil conjectures [KaN - Me] and the ”de Rham conjecture” [Fa]. For reasons explained in 0.13, it would be desirable to have a proof that would not refer to [Fa]. We sketch an argument that works for sufficiently large p, leaving the details for a future publication [Ne 3].

Lemma. Let F be a number field, X a proper smooth scheme over F with good reduction at all places dividing p. Assume that V := H^{2i−1}(\overline{X}, Q_{p}(i)) satisfies the monodromy conjecture at all places of bad reduction of X and that either (a) p > 2dim(X) + 1; or (b) p > dim(X) and, for each place v|p of F, X ⊗_{F} F_{v} is obtained by base change from a proper smooth scheme with good reduction over an unramified extension of Q_{p}. Then:

(1) V is a crystalline representation of G(\overline{F}_{v}/F_{v}) for each place v|p of F.

(2) The image of Φ : CH^{i}(X)_{0} ⟶ H^{1}(F, V) is contained in H_{f}^{1}(F, V).

Proof. (1) Under the assumption (a) (resp. (b)) this is proved in [KaK - Me] (resp. [Fo - Me]).

(2) In fact, we need H^{1}(F_{v}, V) = 0 for all v |̸ p, and this follows from our assumptions (cf. [Ne 2, 2.1.3]). Fix v|p and a proper smooth model X of X ⊗_{F} F_{v} over the ring of integers of F_{v}. Recall that the Abel-Jacobi map over F_{v} is given by the composition

CH^{i}(X ⊗_{F} F_{v})_{0} ⊗ Q ⟶ H^{2i}(X ⊗_{F} F_{v}, Q_{p}(i))_{0} := Ker [H^{2i}(X ⊗_{F} F_{v}, Q_{p}(i))⟶^{γ}

H^{0}(F_{v}, H^{2i}(X ⊗_{F} \overline{F}_{v}, Q_{p}(i)))] ⟶^{δ}H^{1}(F_{v}, H^{2i−1}(X ⊗_{F} \overline{F}_{v}, Q_{p}(i))) = H^{1}(F_{v}, V),

where the maps γ, δ come from the Hochschild-Serre spectral sequence. Define syntomic cohomology with Q_{p}-coefficients of the formal scheme X̂ = \underrightarrow{lim} (X ⊗ Z/p^{n}Z) as

H^{∗}(X̂_{syn}, s_{∞}(r)) := (\underleftarrow{lim}_{n} H^{∗}(X̂_{syn}, s_{n}(r))) ⊗ Q,

where s_{n}(r) is the sheaf denoted by S_{n}^{r} in [Fo - Me]. In our case there is no difference between continuous and naive p-adic étale cohomology, hence the Fontaine-Messing maps [Fo - Me, III.5] define, in the limit, homomorphisms

β : H^{∗}(X̂_{syn}, s_{∞}(r)) ⟶ H^{∗}(X ⊗_{F} F_{v}, Q_{p}(r))

(one needs p > 2 at this point, but for dim(X) = 1 the statement (2) follows from [Bl - Ka]). Chern classes c_{i} into syntomic cohomology [Gro] (resp. étale cohomology) - divided by (−1)^{i−1}(i−1)! - define cycle classes, which fit to a commutative diagram (the compatibility of the Chern classes on K_{0} follows from the compatibility of c_{1}, proved in [Fo - Me, p. 205], and the compatibility of the Fontaine-Messing maps β with products; recall also that i ≤ dim(X) < p)

CH^{i}(X) ⊗ Q ⟶ CH^{i}(X ⊗_{F} F_{v}) ⊗ Q
↓^{≀}  ↓^{≀}
(K_{0}(X) ⊗ Q)^{(i)} ⟶ (K_{0}(X ⊗_{F} F_{v}) ⊗ Q)^{(i)}
↓  ↓
H^{2i}(X̂_{syn}, s_{∞}(i)) ⟶^{β} H^{2i}(X ⊗_{F} F_{v}, Q_{p}(i))

(the superscript (i) denotes the weight i eigenspace for the Adams operations). As the upper horizontal arrow is surjective, it follows that

Im(Φ) ⊆ Im [Ker(γ ∘ β)⟶^{δ∘β}H^{1}(F_{v}, V)]

As shown in [Ne 3], under the assumptions (a) or (b), the latter group is equal to H_{f}^{1}(F_{v}, V).

(1.5) In the rest of Sec. 1 we shall assume that X satisfies the assumptions of Lemma 1.4 and is equidimensional of dimension d. Fix integers i, j ≥ 1 with i + j = d + 1. The Poincaré duality

H^{2i−1}(\overline{X}, Q_{p}(i)) × H^{2j−1}(\overline{X}, Q_{p}(j))⟶^{∪}H^{2d}(\overline{X}, Q_{p}(d + 1))⟶^{Tr}Q_{p}(1)

identifies W = H^{2j−1}(\overline{X}, Q_{p}(j)) with the twisted dual V^{∗}(1) of V = H^{2i−1}(\overline{X}, Q_{p}(i)). The Abel-Jacobi maps are

\begin{matrix}Φ_{i} : CH^{i}(X)_{0} & ⟶ H_{f}^{1}(F, V) \\ Φ_{j} : CH^{j}(X)_{0} & ⟶ H_{f}^{1}(F, V^{∗}(1)) \end{matrix}

According to [Ne 2], there are height pairings

⟨ , ⟩ : H_{f}^{1}(F, V) × H_{f}^{1}(F, V^{∗}(1)) ⟶ Q_{p}

depending on the following data: (A) A continuous homomorphism ℓ_{F} : A_{F}^{∗} /F^{∗} ⟶ Q_{p}, i.e. a collection of continuous homomorphisms

ℓ_{v} : F_{v}^{∗} ⟶ Q_{p} such that ∑_{v} ℓ_{v}(a) = 0 for all a ∈ F^{∗}. We view each ℓ_{v} as a map

H^{1}(F_{v}, Q_{p}(1)) = F_{v}^{∗} ˆ⊗Q_{p} := (\underleftarrow{lim}_{n} F_{v}^{∗} ˆ⊗Z/p^{n}Z) ⊗_{Z_{p}} Q_{p} ⟶ Q_{p}

(B) For each place v | p, a Q_{p}-linear splitting of the Hodge filtration

F^{i}H_{dR}^{2i−1}(X ⊗_{F} F_{v}/F_{v})⟶^{⟵}H_{dR}^{2i−1}(X ⊗_{F} F_{v}/F_{v})  (1.5.1)

(1.6) We recall a definition of ⟨ , ⟩ in terms of local heights ([Ne 2, Sec. 4]). Given a_{1} ∈ H_{f}^{1}(F, V) and a_{2} ∈ H_{f}^{1}(F, V^{∗}(1)), represent them by extensions of Galois representations

\begin{matrix}e_{1} : & 0 ⟶ V ⟶ E_{1} ⟶ Q_{p} ⟶ 0 \\ e_{2} : & 0 ⟶ V^{∗}(1) ⟶ E_{2}^{∗}(1) ⟶ Q_{p} ⟶ 0 \end{matrix}

We claim that the Yoneda product of [e_{1}] ∈ Ext^{1}(Q_{p}, V) and [e_{2}^{∗}(1)] ∈ Ext^{1}(V, Q_{p}(1)) in Ext^{2}(Q_{p}, Q_{p}(1)) = H^{2}(F, Q_{p}(1)) vanishes. Indeed, it is given (up to a sign) by the cup product a_{1}∪a_{2}; we note that H^{2}(F, Q_{p}(1)) injects into ∏_{v} H^{2}(F_{v}, Q_{p}(1)) and each localization (a_{1})_{v} ∪ (a_{2})_{v} ∈ H^{2}(F_{v}, Q_{p}(1)) vanishes by [Bl - Ka, 3.8]. This implies that there is a mixed extension E of e_{1}, e_{2}^{∗}(1), i.e. a Galois representation E fitting into an exact diagram

0  0
↓  ↓
0 ⟶ Q_{p}(1) ⟶ E_{2} ⟶ V ⟶ 0
∥  ↓  ↓
0 ⟶ Q_{p}(1) ⟶ E ⟶ E_{1} ⟶ 0
↓  ↓
Q_{p} = Q_{p}
↓  ↓
0  0

In fact, the set of all such E’s is an H^{1}(F, Q_{p}(1))-torsor (cf. [Ne 2, 4.4]). Fix one such E. If v is a place of F not dividing p, then H^{1}(F_{v}, V) = H^{1}(F_{v}, V^{∗}(1)) = 0 by our assumptions, hence E splits, as a representation of G(\overline{F_{v}}/F_{v}), into

E = V ⊕ U_{v},

where

0 ⟶ Q_{p}(1) ⟶ U_{v} ⟶ Q_{p} ⟶ 0  (1.6.1)

is a “Kummer extension” with class [U_{v}] ∈ H^{1}(F_{v}, Q_{p}(1)). The local height of a_{1}, a_{2} (depending on E) is defined as

⟨a_{1}, a_{2}⟩_{E,v} = −ℓ_{v}([U_{v}]) ∈ Q_{p}

(1.7) For v | p, there is an exact sequence

0 ⟶ H_{f}^{1}(F_{v}, Q_{p}(1)) ⟶ H_{f}^{1}(F_{v}, E_{2}) ⟶ H_{f}^{1}(F_{v}, V) ⟶ 0  (1.7.1)

with a splitting

H_{f}^{1}(F_{v}, Q_{p}(1))⟵^{r_{v}}H_{f}^{1}(F_{v}, E_{2})

determined by the splitting (1.5.1) (see [Ne 2]). If E is crystalline as a representation of G(\overline{F_{v}}/F_{v}), then its class lies in H_{f}^{1}(F_{v}, E_{2}) and the local height is

⟨a_{1}, a_{2}⟩_{E,v} = −ℓ_{v}(r_{v}([E]))  (1.7.2)

If E is not crystalline, one takes instead of (1.7.1) its push-out by j : H_{f}^{1}(F_{v}, Q_{p}(1)) ↪ H^{1}(F_{v}, Q_{p}(1))

0 ⟶ H^{1}(F_{v}, Q_{p}(1)) ⟶ j_{∗}H_{f}^{1}(F_{v}, E_{2}) ⟶ H_{f}^{1}(F_{v}, V) ⟶ 0,

again equipped with a splitting

H^{1}(F_{v}, Q_{p}(1))⟵^{r_{v}}j_{∗}H_{f}^{1}(F_{v}, E_{2})

The local height is given by the same formula (1.7.2). If a_{1,v} vanishes in H^{1}(F_{v}, V), then [E] lies in the image of H^{1}(F_{v}, Q_{p}(1)) in H^{1}(F_{v}, E_{2}), which implies that

⟨a_{1}, a_{2}⟩_{E,v} = −ℓ_{v}([E])  (1.7.3)

does not depend on the choice of the splittings (1.5.1). The difference between two splittings of (1.5.1) is given by a map

s : H_{f}^{1}(F_{v}, V) × H_{f}^{1}(F_{v}, V^{∗}(1)) ⟶ Q_{p}

(using the exponential map of [Bl - Ka], which is in our case an isomorphism). The difference between the corresponding local heights is then equal to

1⟨a_{1}, a_{2}⟩_{E,v} −^{2}⟨a_{1}, a_{2}⟩_{E,v} = s(a_{1,v}, a_{2,v})

(1.8) The global height

⟨a_{1}, a_{2}⟩ = ∑_{v} ⟨a_{1}, a_{2}⟩_{E,v}

does not depend on the choice of E – if we replace E by its translation by x ∈ H^{1}(F, Q_{p}(1)) = F^{∗} ˆ⊗Q_{p}, then the local pairing at v is changed by ℓ_{v}(x) and the sum ∑_{v} ℓ_{v}(x) vanishes.

(1.9) Mixed extensions arise in nature as follows: suppose that Y (resp. Z) is a cycle of codimension i (resp.

j) on X representing an element of CH^{i}(X)_{0} (resp. CH^{j}(X)_{0}) and assume that Y and Z have disjoint supports. By definition, the Abel-Jacobi image Φ_{i}(Y) is represented by the extension

0 ⟶ V ⟶ E_{1} ⟶ Q_{p} ⟶ 0

obtained from

0 ⟶ H^{2i−1}(\overline{X}, Q_{p}(i)) ⟶ H^{2i−1}(\overline{X} − \overline{Y} , Q_{p}(i)) ⟶ H2i ; \overline{Y}(\overline{X}, Q_{p}(i))

by pull-back by cl(\overline{Y}) : Q_{p} ⟶ H2i ; \overline{Y}(\overline{X}, Q_{p}(i)). Similarly, the push-out of the sequence

H^{2i−2}(\overline{Z}, Q_{p}(i)) ⟶ H^{2i−1}(\overline{X} rel \overline{Z}, Q_{p}(i)) ⟶ H^{2i−1}(\overline{X}, Q_{p}(i)) ⟶ 0

by −Tr\overline{Z} : H^{2i−2}(\overline{Z}, Q_{p}(i)) ⟶ Q_{p}(1) gives an extension

0 ⟶ Q_{p}(1) ⟶ E_{2} ⟶ V ⟶ 0,

the dual of which represents Φ_{j}(Z) ([ScA 2, 7.5]). The mixed extension E is obtained from H^{2i−1}(\overline{X} − \overline{Y} rel \overline{Z}, Q_{p}(i)) (sitting in two exact cohomology sequences) by applying both cl(\overline{Y})^{∗} and (−Tr\overline{Z})_{∗} (see [Ne 2]. [ScA 2] for more details). We shall write ⟨Y, Z⟩_{v} for the local height ⟨Φ_{i}(Y), Φ_{j}(Z)⟩_{E,v} associated to this mixed extension. It has the following compatibility property (cf. [PR 2, 4.3a]): if F′ is a finite extension of F, Y a cycle of codimension i on X and Z a cycle of codimension j on X ⊗_{F} F′, then

⟨Y, N_{F′/F} (Z)⟩_{v} = ⟨Y, Z⟩_{w},  (1.9.1)

where v is a place of F, w a place of F′ over v. The logaritm ℓ′_{F} over F′ is equal to ℓ′_{F} = ℓ_{F} ∘ N_{F′/F} and the splitting (1.5.1) over F′_{w} (if v | p) is induced from the splitting over F_{v}.

(1.10) We shall need, later on, basic integrality properties of the Abel-Jacobi maps and local heights. If Y is a cycle representing an element in CH^{i}(X)_{0}, then its cohomology class in H^{2i}(\overline{X}, Z_{p}(i)) is torsion. The torsion subgroup H^{2i}(\overline{X}, Z_{p}(i))_{tors} is finite, of order p^{a}. The construction of the extension class associated to p^{a}Y works with cohomology with Z_{p}-coefficients. This implies that p^{a}Φ_{i} factors through H^{1}(F, H^{2i−1}(\overline{X}, Z_{p}(i))). In the same way, p^{b}Φ_{j} factors through H^{1}(F, H^{2j−1}(\overline{X}, Z_{p}(j))), where p^{b} is the order of H^{2j}(\overline{X}, Z_{p}(j))_{tors}. Replacing Y, Z by p^{a}Y and p^{b}Z, we see that the construction (1.9) works in Z_{p}-cohomology, giving rise to a mixed extension T_{E} of extensions

0 ⟶ T ⟶ T_{1} ⟶ Z_{p} ⟶ 0
0 ⟶ Z_{p}(1) ⟶ T_{2} ⟶ T ⟶ 0

representing a_{1} = Φ_{i}(p^{a}Y) ∈ H_{f}^{1}(F, T), a_{2} = Φ_{j}(p^{b}Z) ∈ H_{f}^{1}(F, T^{∗}(1)). Here

T = Im (H^{2i−1}(\overline{X}, Z_{p}(i)) ⟶ H^{2i−1}(\overline{X}, Q_{p}(i)))

Assume that v |̸ p. Then the group H^{1}(F_{v}, T^{∗}(1)) is finite, of order p^{c}. As p^{c}[T_{2}] = 0∈ Ext_{G(\overline{F}_{v}/F_{v})}^{1}(T, Z_{p}(1)), we have

\begin{aligned}p^{c}⟨a_{1}, a_{2}⟩_{E,v} & ∈ ℓ_{v}(F_{v}^{∗}) \\ ⟨Y, Z⟩_{v} & ∈ p^{−a−b−c}ℓ_{v}(F_{v}^{∗}) \end{aligned}  (1.10.1)

Assume now that v | p. The torsion subgroup H^{1}(F_{v}, T)_{tors} is finite, of order p^{d}. If a_{1,v} goes to zero in H_{f}^{1}(F_{v}, V), then p^{d}[T_{E}] lies in the image of H^{1}(F_{v}, Z_{p}(1)) = F_{v}^{∗} ˆ⊗Z_{p} in H^{1}(F_{v}, T_{1}). It then follows from (1.7.3) that

\begin{aligned}p^{d}⟨a_{1}, a_{2}⟩_{E,v} & ∈ ℓ_{v}(F_{v}^{∗} ˆ⊗Z_{p}) \\ ⟨Y, Z⟩_{v} & ∈ p^{−a−b−d}ℓ_{v}(F_{v}^{∗} ˆ⊗Z_{p}) \end{aligned}  (1.10.2)

(1.11) We need a generalization of (1.10.2). Assume the following conditions are satisfied: (A) ℓ_{v} : F_{v}^{∗} ⟶ Q_{p} is ramified, i.e. ℓ_{v} does not vanish identically on the group of units O_{v}^{∗}. (B) As a representation of G(\overline{F}_{v}/F_{v}), V is a direct sum V = V′ ⊕ V′′, where V′ satisfies the “Pančiškin condition” of [Ne 2], [PR 3] (this happens, for example, when V′ is ordinary in the sense of [Gre]). Put T′ = T ∩ V′, T′′ = T ∩ V′′. Then p^{d_{0}}T ⊆ T′ ⊕ T′′ ⊆ T for some integer d_{0}. (C) The V′-component of the splitting (1.5.1) is equal to the canonical splitting V′ acquires from (B). (D) The image of a_{2,v} in H_{f}^{1}(F_{v}, V^{′′∗}(1)) vanishes.

The local reciprocity map F_{v}^{∗} ⟶ G(F_{v}^{ab}/F_{v}) takes Ker(ℓ_{v}) to G(F_{v}^{ab}/F_{v,∞}), where F_{v,∞}/F_{v} is a ramified Z_{p}-extension, by (A). Write F_{v,n} for its n-th layer, G(F_{v,n}/F_{v}) ⟶^{∼} Z/p^{n}Z. Let

N_{∞}H_{f}^{1}(F_{v}, T′) = ⋂_{n≥1} Im (cor_{n} : H_{f}^{1}(F_{v,n}, T′) ⟶ H_{f}^{1}(F_{v}, T′))

be the subgroup of universal norms in H_{f}^{1}(F_{v}, T′). According to [Ne 2, 6.9], it is a subgroup of finite index.

Proposition. Let p^{d_{1}} be the order of H^{1}(F_{v}, T^{′′∗}(1))_{tors} and p^{d_{2}} the order of H_{f}^{1}(F_{v}, T′)/N_{∞}H_{f}^{1}(F_{v}, T′). Then

p^{d_{0}+d_{1}+d_{2}}⟨a_{1}, a_{2}⟩_{E,v} ∈ ℓ_{v}(F_{v}^{∗} ˆ⊗Z_{p})

Proof. Replace a_{2} by p^{d_{0}+d_{1}}a_{2}. Then a_{2,v} ∈ H_{f}^{1}(F_{v}, T^{′∗}(1)) by (B) and (D), hence T_{2} (for this new a_{2}) splits as T_{2} = T′_{2} ⊕ T′′ with

0 ⟶ Z_{p}(1) ⟶ T′_{2} ⟶ T′ ⟶ 0

Suppose first that E is crystalline as a representation of G(\overline{F}_{v}/F_{v}). Under this assumption, [T_{E}] ∈ H_{f}^{1}(F_{v}, T_{2}). Consider the following exact sequences

\begin{matrix}0 & ⟶ & H_{f}^{1}(F_{v}, Z_{p}(1)) & ⟶ & H_{f}^{1}(F_{v}, T′_{2}) & ⟶^{β} & H_{f}^{1}(F_{v}, T′) & ⟶ & 0 \\ 0 & ⟶ & H_{f}^{1}(F_{v,n}, Z_{p}(1)) & ⟶ & H_{f}^{1}(F_{v,n}, T′_{2}) & ⟶^{β_{n}} & H_{f}^{1}(F_{v,n}, T′) & ⟶ & 0 \end{matrix}  (1.11.1)

By definition, the local height ⟨a_{1}, a_{2}⟩_{E,v} is equal to the image of [T_{E}] under

H_{f}^{1}(F_{v}, T_{2}) ⟶ H_{f}^{1}(F_{v}, E_{2})⟶^{r_{v}}H_{f}^{1}(F_{v}, Q_{p}(1))⟶^{−ℓ_{v}}Q_{p},

which is the same as the image of [T_{E}] under

H_{f}^{1}(F_{v}, T_{2}) ⟶ H_{f}^{1}(F_{v}, T′_{2})⟶^{r′_{v}}H_{f}^{1}(F_{v}, Q_{p}(1))⟶^{−ℓ_{v}}Q_{p},

where r′_{v} is associated, by our assumption (C), to the canonical splitting. The map ℓ_{v} ∘ r′_{v} can be described in terms of universal norms as follows (see [Ne 2, 6.11.2]). Let x ∈ H_{f}^{1}(F_{v}, T′_{2}). For each n ≥ 1, we have p^{d_{2}}β(x) = cor_{n}(y_{n}) for some y_{n}. Write y_{n} = β_{n}(x_{n}) with x_{n} ∈ H_{f}^{1}(F_{v,n}, T′_{2}). Then p^{d_{2}}x − x_{n} lies in the image of H_{f}^{1}(F_{v}, Z_{p}(1)); put

a_{n} := p^{−d_{2}}ℓ_{v}(p^{d_{2}}x − x_{n}) ∈ p^{−d_{2}}ℓ_{v}(F_{v}^{∗} ˆ⊗Z_{p})

If we make another choice of y_{n} and x_{n}, then a_{n} changes by an element of

p^{−d_{2}}ℓ_{v} ∘ N_{F_{v,n}/F_{v}}(F_{v,n}^{∗} ˆ⊗Z_{p}) = p^{n−d_{2}}ℓ_{v}(F_{v}^{∗} ˆ⊗Z_{p})

The sequence a_{n} is convergent and its limit is equal to ℓ_{v} ∘ r′_{v}(x). It follows that

Im (ℓ_{v} ∘ r′_{v} : H_{f}^{1}(F_{v}, T′_{2}) ⟶ Q_{p}) ⊆ p^{−d_{2}}ℓ_{v}(F_{v}^{∗} ˆ⊗Z_{p})  (1.11.2)

If E is not crystalline, we replace the sequences (1.11.1) by their push-outs by j, as in (1.7). The rest of the argument remains unchanged and shows that (1.11.2) holds in this case as well. Recalling that we have multiplied the original a_{2} by p^{d_{0}+d_{1}}, (1.11.2) proves the Proposition.

2. Kuga-Sato varieties

(2.1) For an integer N ≥ 3, let Y (N) be the affine curve over Q parametrizing elliptic curves with a full

level N structure and let Y (N) ↪^{j} X(N) be its smooth projective compactification classifying generalized elliptic curves. We have the universal generalized elliptic curve E ⟶ X(N) which restricts to the universal elliptic curve f : E ⟶ Y (N). For an integer w ≥ 1, let E^{w} be the w-fold fibre product over X(N) of E with itself and let W be its canonical desingularization described in [De 1], [ScA 1]. The finite group

Γ = S_{w}× (μ_{2}×(Z/NZ)^{2})^{w}

acts on E^{w} (along the fibres): the symmetric group S_{w} permutes the factors, (Z/NZ)^{2} acts by translations by points of order N and μ_{2} by multiplication by ±1 in the given factor. This action extends canonically to W.

(2.2) Scholl [ScA 1] defines a projector

ε ∈ Z [\frac{1}{2N · w!}] [Γ]

corresponding to the following representation of Γ: trivial on (Z/NZ)^{2}, the product map on μ_{2}^{w} and the sign character on S_{w}. The main result of [ScA 1] says that (for any prime number p) there is a canonical isomorphism

H^{1}(X(N) ⊗_{Q} \overline{Q}, j_{∗}Sym^{w}(R^{1}f_{∗}Q_{p})) ⟶^{∼} εH^{w+1}(W ⊗_{Q} \overline{Q}, Q_{p}) = εH^{∗}(W ⊗_{Q} \overline{Q}, Q_{p})  (2.2.1)

The expression (2.2.1) is a p-adic representation of G(\overline{Q}/Q), which contains representations associated to all cusp forms of weight w + 2 on Γ(N).

(2.3) Suppose, from now on, that w = 2r − 2 ≥ 2 is even. Let F be an extension of Q and ỹ a closed point of Y (N) ⊗_{Q} F. The fibre (W ⊗_{Q} F)_{y} of W ⊗_{Q} F over ỹ is equal to E2r−2 ; _{y}, where E_{y} is an elliptic curve over k(ỹ), the residue field of ỹ. If Y is a cycle of dimension r−1 supported in (W ⊗_{Q} F)_{y}, we can form εY (a cycle with rational coefficients), also supported in (W ⊗_{Q} F)_{y}. It lies in ε (CH^{r}(W ⊗_{Q} F)_{0} ⊗ Q), as

εH^{2r}(W ⊗_{Q} \overline{F} , Q_{p}(r)) = εH^{2r}(W ⊗_{Q} \overline{Q}, Q_{p}(r)) = 0

by (2.2.1). Denote by b(Y) the cohomology class of ε(Y ⊗_{F} \overline{F}) in

(W ⊗_{Q} F)_{y} ⊗_{F} \overline{F} = (W ⊗_{Q} \overline{F} )_{_{y}⊗_{F} \overline{F}}

It lies in

εH^{2r−2}((W ⊗_{Q} \overline{F} )_{_{y}⊗_{F} \overline{F}} , Q_{p}(r − 1))^{G(\overline{F} /F)} = H^{0}(ỹ ⊗_{F} \overline{F} , B)^{G(\overline{F} /F)},

where

B = Sym^{2r−2} (R^{1}f_{∗}Q_{p}) (r − 1)

is a smooth p-adic sheaf on Y (N), equipped with a non-degenerate symmetric pairing B × B ⟶ Q_{p} coming from the Poincaré duality.

(2.4) Proposition. (1) There is a canonical isomorphism

H^{1}(X(N) ⊗_{Q} \overline{F} , j_{∗}B)(1) ⟶^{∼} εH^{2r−1}(W ⊗_{Q} \overline{F} , Q_{p}(r))

(2) The Abel-Jacobi image of εY in H^{1}(F, H^{2r−1}(W ⊗_{Q} \overline{F} , Q_{p}(r))) depends only on b(Y). It is represented by the pull-back of the extension (of p-adic representations of G(\overline{F} /F))

0 ⟶ H^{1}(X(N) ⊗_{Q} \overline{F} , j_{∗}B)(1) ⟶ H^{1}(X(N) ⊗_{Q} \overline{F} − (ỹ ⊗_{F} \overline{F}), j_{∗}B)(1) ⟶ H^{0}(ỹ ⊗_{F} \overline{F} , B) ⟶ 0

by the map Q_{p} ⟶ H^{0}(ỹ ⊗_{F} \overline{F} , B) sending 1 to b(Y).

Proof. (1) is just a reminder of Scholl’s result (2.2.1).

(2) Consider the commutative diagram with exact rows

0 ⟶ εH^{2r−1}(W ⊗_{Q} \overline{F} , Q_{p}(r)) ⟶ εH^{2r−1}(W ⊗_{Q} \overline{F} − ε(Y ⊗_{F} \overline{F}), Q_{p}(r)) ⟶ εH_{ε(_{y}⊗_{F} \overline{F})}^{2r}(W ⊗_{Q} \overline{F} , Q_{p}(r)) ⟶ 0
∥  ↓  ↓
0 ⟶ εH^{2r−1}(W ⊗_{Q} \overline{F} , Q_{p}(r)) ⟶  εH^{2r−1}(W ⊗_{Q} \overline{F} − U, Q_{p}(r))  ⟶  εH_{U}^{2r}(W ⊗_{Q} \overline{F} , Q_{p}(r))  ⟶ 0

where U = (Y ⊗_{F} \overline{F} )_{_{y}⊗_{F} \overline{F}}. There is a canonical isomorphism

εH_{U}^{2r}(W ⊗_{Q} \overline{F} , Q_{p}(r)) ⟶^{∼} εH^{2r−2}(U, Q_{p}(r − 1)) = H^{0}(ỹ ⊗_{F} \overline{F} , B)

The composite map

Q_{p} ⟶^{cl(ε(Y ⊗_{F} \overline{F}))}εH_{ε(_{y}⊗_{F} \overline{F})}^{2r}(W ⊗_{Q} \overline{F} , Q_{p}(r)) ⟶ εH_{U}^{2r}(W ⊗_{Q} \overline{F} , Q_{p}(r)) = H^{0}(ỹ ⊗_{F} \overline{F} , B)

takes 1 to b(Y), so it remains to show that the bottom sequence of the diagram is canonically isomorphic to the exact sequence in the statement of the Proposition. As Γ acts along the fibres, we can apply ε to the Leray spectral sequence for g : W ⊗_{Q} \overline{F} ⟶ X(N) ⊗_{Q} \overline{F}, obtaining

H^{k}(X(N) ⊗_{Q} \overline{F} , εR^{m}g_{∗}Q_{p}(r − 1)) ⟹ εH^{k+m}(W ⊗_{Q} \overline{F} , Q_{p}(r − 1))

The sheaves C_{m} := εR^{m}g_{∗}Q_{p}(r − 1) satisfy

j^{∗}C_{m} = \{ \begin{matrix}B & if m = 2r − 2 \\ 0 & if m ≠ 2r − 2 \end{matrix}

The spectral sequence then gives

0 ⟶ H^{1}(X(N) ⊗_{Q} \overline{F} , C_{q−1}) ⟶ εH^{q}(W ⊗_{Q} \overline{F} , Q_{p}(r − 1)) ⟶ H^{0}(X(N) ⊗_{Q} \overline{F} , C_{q}) ⟶ 0

for all q < 2r. Combined with (2.2.1), this implies that H^{0}(X(N)⊗_{Q} \overline{F} , C_{q}) = 0 for q ≤ 2r −2, hence C_{q} = 0 for q < 2r − 2 (as the sheaves C_{q} are punctual for q ≠ 2r − 2). For q = 2r − 1 we get an exact sequence

0 ⟶ H^{1}(X(N) ⊗_{Q} \overline{F} , C_{2r−2}) ⟶ H^{1}(X(N) ⊗_{Q} \overline{F} , j_{∗}B) ⟶ H^{0}(X(N) ⊗_{Q} \overline{F} , C_{2r−1}) ⟶ 0

As both the kernel and cokernel of the canonical map C_{2r−2} ⟶ j_{∗}B are supported at the cusps, the map

H^{1}(X(N) ⊗_{Q} \overline{F} , C_{2r−2}) ⟶ H^{1}(X(N) ⊗_{Q} \overline{F} , j_{∗}B)  (2.4.2)

is surjective. This shows that C_{2r−1} = 0 and that (2.4.2) is in fact an isomorphism.

Restricting g to W ⊗_{Q} \overline{F} − U, we get

H^{1}(X(N) ⊗_{Q} \overline{F} − (ỹ ⊗_{F} \overline{F}), C_{2r−2}) ⟶^{∼} εH^{2r−1}(W ⊗_{Q} \overline{F} − U, Q_{p}(r − 1))

An easy argument based on the relative cohomology sequence for X(N) ⊗_{Q} \overline{F} − (ỹ ⊗_{F} \overline{F}) ↪ X(N) ⊗_{Q} \overline{F} then shows that

H^{1}(X(N) ⊗_{Q} \overline{F} − (ỹ ⊗_{F} \overline{F}), C_{2r−2}) ⟶^{∼} εH^{1}(X(N) ⊗_{Q} \overline{F} − (ỹ ⊗_{F} \overline{F}), j_{∗}B))

is an isomorphism as well.

(2.5) The group GL_{2}(Z/NZ)/{±1} acts on X(N), Y (N) and the sheaves B, j_{∗}B. The quotients by the Borel subgroup

B = \{( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} )\} /\{±1\}

are X_{0}(N) = B\X(N), Y_{0}(N) = B\Y (N). The affine curve Y_{0}(N) is the coarse moduli space (over Q) for pairs (E, C), where E is an elliptic curve and C is a subgroup scheme of E isomorphic to Z/NZ. Write j_{0} for the inclusion of Y_{0}(N) to X_{0}(N) and let ε_{B} = (1/♯B) ∑_{g∈B} g be the idempotent corresponding to the trivial representation of B. The sheaf A = ε_{B}π_{∗}B = (π_{∗}B)^{B} (where π denotes the projection Y (N) ⟶ Y_{0}(N)) is smooth outside of the ramification locus Ram of π and π^{∗}A is isomorphic to B outside of π^{−1}(Ram).

Lemma. In the notation of Proposition 2.4, the Abel-Jacobi image

Φ(ε_{B}εY) ∈ H^{1}(F, ε_{B}H^{1}(X(N) ⊗_{Q} \overline{F} , j_{∗}B)(1)) = H^{1}(F, H^{1}(X_{0}(N) ⊗_{Q} \overline{F} , j_{0∗}A)(1))

is represented by the pull-back of the extension

0 ⟶ H^{1}(X_{0}(N) ⊗_{Q} \overline{F} , j_{0∗}A)(1) ⟶ H^{1}(X_{0}(N) ⊗_{Q} \overline{F} − (y ⊗_{F} \overline{F}), j_{0∗}A)(1) ⟶ H^{0}(y ⊗_{F} \overline{F} , A)^{G(\overline{F} /F)} ⟶ 0

by the map Q_{p} ⟶ H^{0}(y ⊗_{F} \overline{F} , A)^{G(\overline{F} /F)} sending 1 to a(Y) := ε_{B}b(Y). Here y = π(ỹ).

Proof. This follows from Prop. 2.4 and the fact that both ε_{B} and π_{∗} are exact functors.

(2.6) Following (and slightly modifying) [Br], we make the following

Definition. If F is an extension of Q and y a closed point of Y_{0}(N) ⊗_{Q} F, a Tate vector at y is an element of H^{0}(y, A) = H^{0}(y ⊗_{F} \overline{F} , A)^{G(\overline{F} /F)}. A Tate cycle on Y_{0}(N) over F is a formal finite sum of Tate vectors at several points y.

Similarly, replacing Y_{0}(N) by Y (N) and A by B, we define Tate vectors and cycles on Y (N). We denote by Z(Y (N), F) (resp. Z(Y_{0}(N), F)) the groups of Tate cycles. In the notation of (2.3)and (2.5), we have

b(Y) ∈ Z(Y (N), F), a(Y) ∈ Z(Y_{0}(N), F)

There is a map

Φ_{T} : Z(Y_{0}(N), F) ⟶ H^{1}(F, H^{1}(X_{0}(N) ⊗_{Q} \overline{F} , j_{0∗}A)(1))

defined as follows: let

a = ∑_{y∈S} a_{y} ∈ Z(Y_{0}(N), F)

be a Tate cycle supported at a finite set S of closed points of Y_{0}(N) ⊗_{Q} F. Then, writing \overline{S} for S ⊗_{F} \overline{F}, Φ_{T} (a) is represented by the pull-back of the exact sequence

0 ⟶ H^{1}(X_{0}(N) ⊗_{Q} \overline{F} , j_{0∗}A)(1) ⟶ H^{1}(X_{0}(N) ⊗_{Q} \overline{F} − \overline{S}, j_{0∗}A)(1) ⟶ H^{0}(\overline{S}, A) ⟶ 0  (2.6.1)

by the map Q_{p} ⟶ H^{0}(\overline{S}, A) = ⊕_{y∈S}H^{0}(y ⊗_{F} \overline{F} , A) sending 1 to a. In this language, Lemma 2.5 says that Φ(ε_{B}εY) = Φ_{T} (a(Y)).

(2.7) According to [ScA 2, 7.5], the relative cohomology sequence

0 ⟶ H^{0}(\overline{S}, A) ⟶ H^{1}(X_{0}(N) ⊗_{Q} \overline{F} rel \overline{S}, j_{0∗}A) ⟶ H^{1}(X_{0}(N) ⊗_{Q} \overline{F} , j_{0∗}A) ⟶ 0

is dual to the sequence (2.6.1). The pairing between them is given by the Poincaré duality

H^{1}(X_{0}(N) ⊗_{Q} \overline{F} rel \overline{S}, j_{0∗}A) × H^{1}(X_{0}(N) ⊗_{Q} \overline{F} − \overline{S}, j_{0∗}A)(1) ⟶ H^{2}(X_{0}(N) ⊗_{Q} \overline{F} , Q_{p}(1)) ⟶^{∼} Q_{p}
↑  ↓  ∥  ∥
H^{1}(X_{0}(N) ⊗_{Q} \overline{F} , j_{0∗}A)  ×  H^{1}(X_{0}(N) ⊗_{Q} \overline{F} , j_{0∗}A)(1)  ⟶ H^{2}(X_{0}(N) ⊗_{Q} \overline{F} , Q_{p}(1)) ⟶^{∼} Q_{p}

and the opposite of the pairing ( , ) : A × A ⟶ Q_{p} that A inherits from B.

(2.8) Let a, b ∈ Z(Y_{0}(N), F) be Tate cycles supported at disjoint finite sets of closed points S, T. The relative cohomology group

H^{1}(X_{0}(N) ⊗_{Q} \overline{F} − \overline{S} rel \overline{T} , j_{0∗}A)(1)

is a mixed extension of exact sequences

0 ⟶ H^{1}(X_{0}(N) ⊗_{Q} \overline{F} , j_{0∗}A)(1) ⟶ H^{1}(X_{0}(N) ⊗_{Q} \overline{F} − \overline{S}, j_{0∗}A)(1) ⟶ H^{0}(\overline{S}, A) ⟶ 0
0 ⟶ H^{0}(\overline{T} , A) ⟶ H^{1}(X_{0}(N) ⊗_{Q} \overline{F} rel \overline{T} , j_{0∗}A) ⟶ H^{1}(X_{0}(N) ⊗_{Q} \overline{F} , j_{0∗}A) ⟶ 0

Take its pull-back by Q_{p} ⟶ H^{0}(\overline{S}, A) (1 ↦ a) and push-out by H^{0}(\overline{T} , A) ⟶ Q_{p} (• ↦ −(•, a)). This will be a mixed extension E of Φ_{T} (a) and (Φ_{T} (b))^{∗}(1). Let y, z be two distinct closed points of Y_{0}(N) ⊗_{Q} F, ỹ (resp. z̃) a point of Y (N) ⊗_{Q} F over y (resp. z) and Y (resp. Z) a cycle of dimension r − 1 supported in the fibre (W ⊗_{Q} F)_{y} (resp. (W ⊗_{Q} F)_{z}). If we apply the previous construction to the Tate cycles a = a(Y), b = a(Z), then the same argument as in the proof of Prop. 2.4 shows that the mixed extension E will be isomorphic to the mixed extension associated to the cycles ε_{B}εY and ε_{B}εZ as in (1.9).

(2.9) We recall a geometric realization of Hecke operators. For an integer m ≥ 1 prime to N, let Y (N, m) be the (reducible) curve over Q parametrizing elliptic curves E with a full level N structure C and an isogeny λ : E ⟶ E′ of degree m. There are two finite maps s, t : Y (N, m) ⟶ Y (N) given by

s(E, C, λ) = (E, C), t(E, C, λ) = (E′, λ_{∗}(C))

In the notation of 2.1, we have a commutative diagram

\begin{matrix}E^{2r−2} & ⟵^{ϕ_{1}} & E^{2r−2} ⊗_{Y (N),s} Y (N, m) & ⟶^{ψ} & E^{2r−2} ⊗_{Y (N),t} Y (N, m) & ⟶^{ϕ_{2}} & E^{2r−2} \\ ↓^{f} & & ↓^{f_{1}} & & ↓^{f_{2}} & & ↓^{f} \\ Y (N) & ⟵^{s} & Y (N, m) & = & Y (N, m) & ⟶^{t} & Y (N) \end{matrix}  (2.9.1)

in which ψ is induced by λ. The m-th Hecke corespondence (α_{1}, α_{2}) : T_{m} ↪ W × W is defined as the Zariski closure of the graph of

(ϕ_{1}, ϕ_{2}ψ) : E^{2r−2} ⊗_{Y (N),s} Y (N, m) ⟶ E^{2r−2} × E^{2r−2}

It acts on Chow groups and cohomology groups of W and commutes with the Scholl projector ε. Sometimes it is necessary to distinguish between the covariant and contravariant actions T_{m}^{∗} = α_{1∗}α_{2}^{∗}, T_{m∗} = α_{2∗}α_{1}^{∗}. In the diagram (2.9.1), the first and third squares are cartesian. By the proper base change theorem, we have

\begin{aligned}s^{∗}B & = εs^{∗}R^{2r−2}f_{∗}Q_{p}(r − 1) = εR^{2r−2}f_{1∗}Q_{p}(r − 1) \\ t^{∗}B & = εt^{∗}R^{2r−2}f_{∗}Q_{p}(r − 1) = εR^{2r−2}f_{2∗}Q_{p}(r − 1) \end{aligned}

and ψ induces homomorphisms

ψ^{∗} : t^{∗}B ⟶ s^{∗}B, ψ_{∗} : s^{∗}B ⟶ t^{∗}B

This defines operators T_{m}^{∗} = s_{∗}ψ^{∗}t^{∗}, T_{m∗} = t_{∗}ψ_{∗}s^{∗} on H^{1}(Y (N) ⊗_{Q} \overline{Q}, B)(1), H_{c}^{1}(Y (N) ⊗_{Q} \overline{Q}, B)(1) and

H^{1}(X(N) ⊗_{Q} \overline{Q}, j_{∗}B)(1) ⟶^{∼} Im (H_{c}^{1}(Y (N) ⊗_{Q} \overline{Q}, B)(1) ⟶ H^{1}(Y (N) ⊗_{Q} \overline{Q}, B)(1))

The isomorphism (2.2.1) is compatible with the action of T_{m}^{∗}, T_{m∗} on both sides. Poincaré duality induces a non-degenerate skew-symmetric pairing

( , ) : H^{1}(X(N) ⊗_{Q} \overline{Q}, j_{∗}B)(1) × H^{1}(X(N) ⊗_{Q} \overline{Q}, j_{∗}B)(1) ⟶ Q_{p}(1)  (2.9.2)

which satisfies

(T_{m∗}x, y) = (x, T_{m}^{∗}y)  (2.9.3)

The action of T_{m} commutes with ε_{B}; induced operators on H^{1}(X_{0}(N) ⊗_{Q} \overline{Q}, j_{∗}B)(1) will be denoted by the same letter. They are symmetric, i.e. satisfy T_{m∗} = T_{m}^{∗} (cf. 2.10 below).

(2.10) Let S be a finite set of closed points on Y (N) ⊗_{Q} F. The operators

\begin{matrix}T_{m}^{∗} : & H^{0}(S ⊗_{F} \overline{F} , B)⟶^{t^{∗}}H^{0}(t^{−1}(S) ⊗_{F} \overline{F} , t^{∗}B)⟶^{ψ^{∗}}H^{0}(t^{−1}(S) ⊗_{F} \overline{F} , s^{∗}B)⟶^{s_{∗}}H^{0}(s(t^{−1}(S)) ⊗_{F} \overline{F} , B) \\ T_{m∗} : & H^{0}(S ⊗_{F} \overline{F} , B)⟶^{s^{∗}}H^{0}(s^{−1}(S) ⊗_{F} \overline{F} , s^{∗}B)⟶^{ψ_{∗}}H^{0}(s^{−1}(S) ⊗_{F} \overline{F} , t^{∗}B)⟶^{t_{∗}}H^{0}(t(s^{−1}(S)) ⊗_{F} \overline{F} , B) \end{matrix}

commute with the Galois action, hence operate on Tate cycles Z(Y (N), F). Applying the projector ε_{B}, we get operators T_{m}^{∗}, T_{m∗} on Z(Y_{0}(N), F). These operators are symmetric, T_{m}^{∗} = T_{m∗}. This can be seen as follows: a Tate vector v on Y_{0}(N) is represented by a triple (E, C, b), where C ⊂ E is a cyclic subgroup scheme of order N and b ∈ Sym^{2r−2} (H^{1}(E ⊗_{F} \overline{F} , Q_{p})) (r − 1). By definition, T_{m∗}(v) is represented by

∑_{{λ:E⟶E′ \atop deg(λ)=m}}(E′, λ_{∗}(C), λ_{∗}(b))

Writing μ := λ̂ : E′ ⟶ E for the dual isogeny, then λ_{∗}(C) = μ^{∗}(C) and λ_{∗}(b) = μ^{∗}(b) (as λ_{∗} = (λ̂)^{∗} on H^{1}(E ⊗_{F} \overline{F} , Q_{p})), hence

T_{m∗}(v) = ∑_{{μ:E′⟶E \atop deg(μ)=m}}(E′, μ^{∗}(C), μ^{∗}(b)) = T_{m}^{∗}(v)

As Abel-Jacobi maps commute with correspondences, we have (in the notation of 2.5)

Φ(T_{m}^{∗}ε_{B}εY) = T_{m}^{∗}Φ(ε_{B}εY) = Φ_{T} (T_{m}^{∗}a(Y)) = Φ_{T} (T_{m∗}a(Y))  (2.10.1)

(2.11) It is well known that T_{m} (= T_{m∗} = T_{m}^{∗}) acts on

V = H^{1}(X_{0}(N) ⊗_{Q} \overline{Q}, j_{0∗}A)(1) = ε_{B}εH^{2r−1}(W ⊗_{Q} \overline{Q}, Q_{p}(r))

as the m-th Hecke operator T_{m} does on S_{2r}(Γ_{0}(N)) ([De 1, 3.19]). Let T be the Q-algebra generated by the operators T_{m} (for m prime to N) on S_{2r}(Γ_{0}(N)). As an abstract algebra, T is a product of finitely many number fields. As in I.2.1, fix embeddings i_{∞} : \overline{Q} ↪ C, i_{p} : \overline{Q} ↪ \overline{Q}_{p}. There is a bijection between Q-algebra homomorphisms β : T ⟶ \overline{Q} and normalized primitive forms f_{β} ∈ S_{2r}(Γ_{0}(N); \overline{Q}), given by

β(T_{m}) = a_{m}(f_{β}), (m, N) = 1

Let N(β) | N be the conductor of f_{β}. Then f_{β} ∈ S_{2r}^{new}(Γ_{0}(N(β)); F (β)), where F (β) = Im(β). The image of the induced map

β_{p} : T ⊗ Q_{p}⟶^{β⊗1}\overline{Q} ⊗_{Q} Q_{p}⟶^{i_{p}}\overline{Q}_{p}

is equal to the closure F̂ (β) of i_{p}(F (β)) in \overline{Q}_{p}. For any T ⊗ Q_{p}-module A, there is a decomposition

A ⊗_{Q_{p}} \overline{Q}_{p} ⟶^{∼} ⨁_{β:T⟶\overline{Q}} (A_{β} ⊗_{F (β)} \overline{Q}_{p})

with

A_{β} = A ⊗_{T⊗Q_{p},β_{p}} F̂ (β)

For A = V, the β-part V_{β} is a vector space over F̂ (β) of dimension 2m(β), where m(β) is the multiplicity of the eigenvalue β in S_{2r}(Γ_{0}(N)), and the action of G(\overline{Q}/Q) is F̂ (β)-linear. Of course, f_{β} is a newform iff N(β) = N iff m(β) = 1. For any prime number ℓ |̸ pN(β), V_{β} is unramified at ℓ and the characteristic polynomial of a geometric Frobenius at ℓ is equal to

det_{F (β)} (1 − F r_{ℓ}t | V_{β}) = (1 − β(T_{ℓ})ℓ^{−r}t + ℓ^{−1}t^{2})^{m(β)}

Both eigenvalues of F r_{ℓ} have absolute value ℓ^{−1/2} ([De 1], [De 2]). For ℓ | N(β) (but ℓ ≠ p), V_{β} is ramified at ℓ and

det_{F (β)} (1 − F r_{ℓ}t | V_{β}^{I_{ℓ}}) = (1 − a_{ℓ}(f_{β})ℓ^{−r}t)^{m(β)}

by [Ca, Thm. A]. The ℓ-th coefficient of f_{β} either vanishes or has absolute value ℓ^{r−1} ([At - Le, Thm. 3]).

(2.12) Lemma. If F is a finite extension of Q_{ℓ} for ℓ ≠ p, then H^{i}(F, V) = 0 for i = 0, 1, 2.

Proof. It follows from what has just been said that H^{0}(F, V_{β}) = 0 for all β. As V_{β} ⟶^{∼} V_{β}^{∗}(1) by 2.9.2, local duality [Se, II.5.2 Thm. 2] gives H^{2}(F, V_{β}) = 0 , hence H^{1}(F, V_{β}) = 0 by the formula for the local Euler characteristic [Se, II.5.7 Thm. 5].

(2.13) Suppose that the field F in 2.8 is a finite extension of Q_{ℓ} with ℓ ≠ p. Since V = H^{1}(X_{0}(N) ⊗_{Q} \overline{Q}, j_{0∗}A)(1) satisfies, by Lemma 2.12,

H^{i}(F, V) = 0  (i = 0, 1, 2),  (2.13.1)

the mixed extension E from 2.8 splits over F into E = V ⊕ U, as in 1.6. We show, following [ScA 2], that the class [U] ∈ H^{1}(F, Q_{p}(1)) can be computed by a formula due to Beilinson [Be]. For any finite set of closed points S ⊂ Y_{0}(N) ⊗_{Q} F, the Hochschild-Serre spectral sequence

H^{a}(F, H_{S⊗_{F} \overline{F}}^{b} (X_{0}(N) ⊗_{Q} \overline{F} , j_{0∗}A(1))) ⟹ H_{S}^{a+b}(X_{0}(N) ⊗_{Q} F, j_{0∗}A(1))  (2.13.2)

degenerates into

H_{S}^{q}(X_{0}(N) ⊗_{Q} F, j_{0∗}A(1)) ⟶^{∼} H^{q−2}(F, H^{0}(S ⊗_{F} \overline{F} , A))  (2.13.3)

We can view, therefore, the Tate classes a, b as elements of

a ∈ H_{S}^{2}(X_{0}(N) ⊗_{Q} F, j_{0∗}A(1)), b ∈ H_{T}^{2} (X_{0}(N) ⊗_{Q} F, j_{0∗}A(1))

The sheaf A is self-dual and has irreducible monodromy. This implies that the analogue of (2.13.2) without supports degenerates into

H^{q}(X_{0}(N) ⊗_{Q} F, j_{0∗}A(1)) ⟶^{∼} H^{q−1}(F, V) = 0

The exact sequence

… ⟶ H^{1}(X_{0}(N) ⊗_{Q} F − T, j_{0∗}A(1)) ⟶ H_{T}^{2} (X_{0}(N) ⊗_{Q} F, j_{0∗}A(1)) ⟶ H^{2}(X_{0}(N) ⊗_{Q} F, j_{0∗}A(1)) = 0

shows that b can be lifted (uniquely) to

b̃ ∈ H^{1}(X_{0}(N) ⊗_{Q} F − T, j_{0∗}A(1))

(2.14) Proposition. The extension class [U] is equal to the image of −a × b̃ under

H_{S}^{2}(X_{0}(N) ⊗_{Q} F, j_{0∗}A(1)) × H^{1}(X_{0}(N) ⊗_{Q} F − T, j_{0∗}A(1))⟶^{∪}H_{S}^{3}(Y_{0}(N) ⊗_{Q} F − T, A^{⊗2}(2))⟶^{( , )}

⟶^{( , )}H_{S}^{3}(Y_{0}(N) ⊗_{Q} F − T, Q_{p}(2)) ⟶^{∼} H^{1}(F, H_{S}^{2}(Y_{0}(N) ⊗_{Q} F − T, Q_{p}(2))(2))⟶^{Tr}H^{1}(F, Q_{p}(1))

Proof. The analogous statement for cohomology with constant coefficients is proved in [ScA 2, Thm. 5.3]. Scholl’s proof works in our case almost word by word.

(2.15) In fact, in [ScA 2] one works over the maximal unramified extension F^{ur} of F. In this case one assumes that a, b ∈ Z(Y_{0}(N), F^{ur}) are Tate cycles with the following properties (which are automatically satisfied if a, b come from Tate cycles over F by a base change): (A) Φ_{T} (a) = Φ_{T} (b) = 0 ∈ H^{1}(F^{ur}, V) (B) The image of b in H^{2}(X_{0}(N) ⊗_{Q} F^{ur}, j_{0∗}A(1)) vanishes. (A) implies that the mixed extension E associated to a, b in 2.8 splits into E = V ⊕ U over F^{ur} and (B) guarantees that b can be lifted to b̃ ∈ H^{1}(X_{0}(N) ⊗_{Q} F^{ur} − T, j_{0∗}A(1)). The argument of [ScA 2, Thm. 5.3] then shows that the class [U] ∈ H^{1}(F^{ur}, Q_{p}(1)) is given by the image of −a × b̃ by a sequence of maps as in the statement of Proposition 2.14, but over F^{ur}.

(2.16) Suppose that X is a proper flat scheme over Λ, the ring of integers in F^{ur}, such that X ⊗_{Λ} F^{ur} ⟶^{∼} X_{0}(N) ⊗_{Q} F^{ur}. Let X_{s} be the special fibre of X and denote by j′ the inclusion j′ : Y_{0}(N) ⊗_{Q} F^{ur} ↪ X. Let a, b be Tate cycles in Z(Y_{0}(N), F^{ur}) with supports at two disjoint finite sets of closed points S, T ⊆ Y_{0}(N) ⊗_{Q} F^{ur}. Write \underline{S} (resp. \underline{T}) for the Zariski closure (with the reduced scheme structure) of S (resp. T) in X. Assume that both \underline{S} and \underline{T} are contained in the regular locus X^{reg} of X and that a (resp. b) extends to a section \underline{a} ∈ H^{0}(\underline{S}, j′_{∗}A) (resp. \underline{b} ∈ H^{0}(\underline{T} , j′_{∗}A)). If y ∈ S, z ∈ T are two points, we define the local intersection index

(\underline{a} · \underline{b})_{x} := (\underline{y} · \underline{z})_{x}(\underline{a}_{x}, \underline{b}_{x})

Here (\underline{y} · \underline{z})_{x} denotes the usual local intersection number on X^{reg} and \underline{a}_{x} (resp. \underline{b}_{x}) the value of \underline{a} (resp. \underline{b}) at x.

Proposition. Assume, furthermore, that H^{2}(X , j′_{∗}A(1)) = 0 and that Φ_{T} (a) = Φ_{T} (b) = 0 ∈ H^{1}(F^{ur}, V). Then the mixed extension E of 2.8 splits as a representation of G(\overline{F} /F^{ur}) into E = V ⊕ U and the image of [U] under the normalized valuation ord : H^{1}(F^{ur}, Q_{p}(1)) ⟶^{∼} Q_{p} is equal to

ord([U]) = ∑_{x∈\underline{S}∩\underline{T}}(\underline{a} · \underline{b})_{x}

Proof. (cf. [ScA 2], [Br]) By [SGA 4\frac{1}{2}, Cycle, Def. 2.1.2], there are cycle classes

cl(\underline{S}) ∈ H2 ; \underline{S}(X , Q_{p}(1)), cl(\underline{T}) ∈ H2 ; \underline{T} (X , Q_{p}(1))

Define α := \underline{a} ∪ cl(\underline{S}) ∈ H2 ; \underline{S}(X , j′_{∗}A(1)). Its restriction to

H_{S}^{2}(X_{0}(N) ⊗_{Q} F^{ur}, j_{∗}A(1)) ⟶^{∼} H^{0}(F^{ur}, H^{0}(S ⊗_{F^{ur}} \overline{F} , A))

(cf. (2.13.3)) is equal to a. Similarly, we get β ∈ H2 ; \underline{T} (X , j′_{∗}A(1)) restricting to b. The exact sequence

H^{1}(X − \underline{T} , j′_{∗}A(1)) ⟶ H2 ; \underline{T} (X , j′_{∗}A(1)) ⟶ H^{2}(X , j′_{∗}A(1)) = 0

shows that β can be lifted to β̃ ∈ H^{1}(X − \underline{T} , j′_{∗}A(1)). Consider now the following commutative diagram:

H_{S}^{2}(X_{0}(N) ⊗_{Q} F^{ur}, j_{∗}A(1)) × H^{1}(X_{0}(N) ⊗_{Q} F^{ur} − T, j_{∗}A(1)) ⟶^{∪} H_{S}^{3}(X_{0}(N) ⊗_{Q} F^{ur} − T, Q_{p}(2)) ⟶^{Tr} H^{1}(F^{ur}, Q_{p}(1))
↑  ↑  |
H2 ; \underline{S}(X , j′_{∗}A(1)) × H^{1}(X − \underline{T} , j′_{∗}A(1))  ⟶^{∪}  H3 ; \underline{S}(X − \underline{T} , Q_{p}(2))  ↓_{δ}
↓^{id⊗δ}  ↓_{δ}  |
H2 ; \underline{S}(X , j′_{∗}A(1)) × H2 ; \underline{T} (X , j′_{∗}A(1))  ⟶^{∪}  H_{X_{s}}^{4}(X , Q_{p}(2))  ⟶^{Tr} H_{•}^{2}(Λ, Q_{p}(1))

in which δ denotes boundary maps in exact cohomology sequences, and the anticommutative diagram ([SGA 4\frac{1}{2}, Cycle, 2.1.3])

H^{1}(F^{ur}, Q_{p}(1)) ⟶^{ord} Q_{p}
↓^{δ}  ∥
H_{•}^{2}(Λ, Q_{p}(1)) ⟵^{cl(•)} Q_{p}

(2.16.1)

Here • denotes the special fibre of Spec(Λ). The statement of the Proposition now follows from 2.15, if we trace down −α × β̃ on the diagram from

H2 ; \underline{S}(X , j′_{∗}A(1)) × H^{1}(X − \underline{T} , j′_{∗}A(1))

to H_{•}^{2}(Λ, Q_{p}(1)) in two different ways.

Our sign disagrees with [Br, Thm. 4.13] (the local height in loc. cit. is defined using a × b̃, not −a × b̃). It seems that the minus sign coming from (2.16.1) has not been taken into account in [Br].

3. Heegner points and cycles

(3.1) We make a digression about cycles on powers of an elliptic curve and their intersection numbers. Consider an elliptic curve E over an algebraically closed field k and we fix a prime p different from the characteristic of k. The simplest case is that of divisors on E × E. The cycle class map

CH^{1}(E × E) ⟶ H^{2}(E × E, Q_{p})(1)

(from now on, we shall drop Q_{p} from the notation) takes a cycle X to its cohomology class [X]. Intersection of divisors is given by a non-degenerate symmetric pairing

( , )_{1} : H^{2}(E × E)(1) × H^{2}(E × E)(1)⟶^{∪}H^{4}(E × E)(2)⟶^{Tr_{E×E}}Q_{p}

The endomorphism ring End_{k}(E) is equipped with a canonical involution α ↦ \overline{α}. We define

t(α) = α + \overline{α}, n(α) = α\overline{α} = \overline{α}α

The degree of the isogeny α : E ⟶ E is equal to n(α). A pair of endomorphisms α, β ∈ End_{k}(E) defines a correspondence

(α, β) : E ⟶ E × E

Let Γ_{α,β} ∈ CH^{1}(E × E) be (the class of) the graph of the correspondence (α, β), i.e. the image of E under

(α, β)_{∗} : CH^{0}(E) ⟶ CH^{1}(E × E)

Its cohomology class is [Γ_{α,β}] = (α, β)_{∗}(1) for

(α, β)_{∗} : H^{0}(E) ⟶ H^{2}(E × E)(1)

We have Γ_{1,β} = Γ_{β}, Γ_{α,1} = (Γ_{α})^{t}, where Γ_{α} denotes the graph of α and ^{t} the transposition. In particular, Γ_{0,1} = {0} × E, Γ_{1,0} = E × {0}.

(3.2) Under the Künneth decomposition of H^{2}(E × E)(1), its two-dimensional subspace

(H^{2}(E)(1) ⊗ H^{0}(E)) ⊕ (H^{0}(E) ⊗ H^{2}(E)(1)) = Q_{p} · [Γ_{0,1}] ⊕ Q_{p} · [Γ_{1,0}]

is orthogonal to H^{1}(E) ⊗ H^{1}(E)(1) under the intersection product ( , )_{1}.

Definition. For α, β ∈ End_{k}(E), let X_{α,β} be the orthogonal projection of [Γ_{α,β}] on H^{1}(E) ⊗ H^{1}(E)(1). Write X_{α} for X_{1,α}.

As ([Γ_{α,β}], [Γ_{0,1}])_{1} = n(α), ([Γ_{α,β}], [Γ_{1,0}])_{1} = n(β), we have

X_{α,β} = [Γ_{α,β}] − n(α)[Γ_{1,0}] − n(β)[Γ_{0,1}]

(3.3) Lemma. (1) X_{α,β} is Z-bilinear in α, β (hence it can be extended Q-linearly to End_{k}(E) ⊗ Q).

(2) (X_{α,β})^{t} = X_{β,α}; X_{αγ,βγ} = n(γ)X_{α,β}; X_{α,β} = X_{β\overline{α}}; (X_{α})^{t} = X\overline{α}

(3) (X_{α}, X_{β})_{1} = −t(α\overline{β})

(4) If λ : E ⟶ E′ is an isogeny, then

\begin{aligned}(λ × λ)_{∗}(X_{α,β}) & = deg(λ)X_{λαλ^{−1},λβλ^{−1}} \\ (λ × λ)^{∗}(X_{α,β}) & = deg(λ)X_{λ^{−1}αλ,λ^{−1}βλ} \end{aligned}

(5) The orthogonal projection of [Γ_{α,β}] on (Sym^{2}(H^{1}(E))) (1) is equal to

\frac{X_{α,β} − X_{β,α}}{2} = X_{(β\overline{α}−α\overline{β})/2}

Proof. (1) For x ∈ H^{1}(E) and y ∈ H^{1}(E)(1), we have

([Γ_{α,β}], p_{1}^{∗}x ⊗ p_{2}^{∗}y)_{1} = Tr_{E×E}((α, β)_{∗}(1) ∪ (p_{1}^{∗}x ⊗ p_{2}^{∗}y)) = Tr_{E×E}((α, β)_{∗}((α, β)^{∗}(p_{1}^{∗}x ⊗ p_{2}^{∗}y))) =
= Tr_{E}((α, β)^{∗}(p_{1}^{∗}x ⊗ p_{2}^{∗}y)) = Tr_{E}(α^{∗}x ⊗ β^{∗}y),

which is Z-bilinear in α, β. The statement follows from the fact that ( , )_{1} is non-degenerate on H^{1}(E) ⊗ H^{1}(E)(1).

(2) The first equality follows from (Γ_{α,β})^{t} = Γ_{β,α}; the second from the fact that γ_{∗} : H^{0}(E) ⟶ H^{0}(E) is the multiplication by n(γ). As n(α) = α\overline{α} ∈ Z, we get n(α)X_{α,β} = X_{α\overline{α},β\overline{α}} = n(α)X_{1,β\overline{α}} = n(α)X_{β\overline{α}}. Finally, (X_{α})^{t} = X_{α,1} = X\overline{α}.

(3) As ([Γ_{α}], [Γ_{β}])_{1} = n(α − β), it follows that (X_{α}, X_{β})_{1} = n(α − β) − n(α) − n(β) = −α\overline{β} − β\overline{α} = −t(α\overline{β}).

(4) We have (λ × λ) ∘ (α, β) = (λαλ^{−1}, λβλ^{−1}) ∘ λ, hence

(λ × λ)_{∗}[Γ_{α,β}] = (λαλ^{−1}, λβλ^{−1})_{∗}deg(λ) = deg(λ)[Γ_{(λαλ^{−1},λβλ^{−1})}]

As (λ̂)^{∗} = λ_{∗} on H^{1}(E), the second equality follows from the first one applied to the dual isogeny.

(5) Follows from (2).

(3.4) Consider now E^{2k} = \underbrace{E × ⋯ × E}_{2k times} for arbitrary k ≥ 1. We have the cycle class map

CH^{k}(E^{2k}) ⟶ H^{2k}(E^{2k})(k)

and the intersection pairing

( , )_{k} : H^{2k}(E^{2k})(k) × H^{2k}(E^{2k})(k)⟶^{∪}H^{4k}(E^{2k})(2k)⟶^{Tr_{E^{2k}}}Q_{p}

Künneth formula and antisymmetrization give a projection (onto a direct summand of H^{2k}(E^{2k})(k))

ε_{k} : H^{2k}(E^{2k})(k) ⟶ H^{1}(E)^{⊗2k}(k) ⟶ (Sym^{2k}(H^{1}(E))) (k)

Note that ε_{k} is given by Scholl’s projector ε (for w = 2k). In this notation, Lemma 3.3.5 says that

ε_{1}([Γ_{α,β}]) = X_{(β\overline{α}−α\overline{β})/2}  (3.4.1)

Proposition. For u, v ∈ (Sym^{2}(H^{1}(E))) (1) (viewed as a subspace of H^{1}(E)^{⊗2}(1)), we have

(ε_{k}(u^{⊗k}), ε_{k}(v^{⊗k}))_{k} = \frac{2^{k}}{\binom{2k}{k}} ((u, u)_{1}(v, v)_{1})^{k/2} P_{k} (\frac{(u, v)_{1}}{((u, u)_{1}(v, v)_{1})^{1/2}} ) ,

where P_{k} is the Legendre polynomial of degree k (see I.1.9). As P_{k}(−t) = (−1)^{k}P_{k}(t), the above expression does not, in fact, contain any square roots.

Proof. This is a question from representation theory of GL_{2}; we omit the calculation but refer the reader to [Br, p.24] and [Ha, pp. 552, 556].

(3.5) Let us recall basic facts about Heegner points on the curve Y_{0}(N) (see [Gr], [Gr - Za] for more details). For a field F of characteristic zero, any F -valued point on Y_{0}(N) can be represented by an isogeny (defined over F) λ : E ⟶ E′ between two elliptic curves with kernel Ker(λ) ≃ Z/NZ. A CM-point (resp. a Heegner point) on Y_{0}(N) is represented by an isogeny between elliptic curves with complex multiplication (resp. by an isogeny between elliptic curves with End(E) = End(E′) ≠ Z). We shall be interested in Heegner points of a special kind, when End(E) = End(E′) = O_{K} is the ring of integers in an imaginary quadratic field K = Q(\sqrt{D}) with discriminant D < 0 prime to N. Such points exist iff each prime number dividing N splits in K. Over the complex numbers, they are classified as follows ([Gr], [Gr - Za, II.1]).

Choose an ideal N ⊂ O_{K} with O_{K}/N ⟶^{∼} Z/NZ. For each ideal A ⊆ O_{K} the isogeny C/A ⟶ C/AN^{−1} represents a Heegner point on Y_{0}(N)(C) that depends only on the ideal class [A] of A. This defines a bijection between Heegner points (in our restricted sense) and pairs ([A], N). All Heegner points are defined over H, the Hilbert class field of K. The Galois group G(H/K) ⟶^{∼} Pic(O_{K}) acts on them by

σ_{𝔭} : ([A], N) ↦ ([A𝔭^{−1}], N)

(𝔭 is a prime ideal of O_{K}, σ_{𝔭} the Artin symbol (H/K, 𝔭)). The complex conjugation acts by

τ : ([A], N) ↦ ([A]^{−1}, \overline{N})

Atkin-Lehner involutions on X_{0}(N) form a group (Z/2Z)^{s}, where s is the number of primes dividing N. The action of this group permutes possible choices of N and (Z/2Z)^{s} × Pic(O_{K}) acts simply transitively on the set of Heegner points.

(3.6) Fix one of the Heegner points y (viewed as a closed point of Y_{0}(N) ⊗_{Q} H) and choose a point ỹ on Y (N) ⊗_{Q} H over y. The fibre E_{y} is an elliptic curve with End(E_{y}) = O_{K}. Let Γ\sqrt{D} ⊂ E_{y}× E_{y} be the graph of \sqrt{D} ∈ O_{K} (we fix one of the square roots) and define

Y = \underbrace{Γ\sqrt{D} × ⋯ × Γ\sqrt{D}}_{r−1 times} ⊂ \underbrace{E_{y} × ⋯ × E_{y}}_{2r−2 times} = (W ⊗_{Q} H)_{y}

The Heegner cycle ε_{B}εY (a cycle with rational coefficients) represents an element of ε_{B}ε(CH^{r}(W ⊗_{Q}H)_{0}⊗Q) and is a higher weight analogue of a Heegner point. Write x = a(Y) ∈ Z(Y_{0}(N), H) for the corresponding Tate cycle. It follows from (3.4.1) that εY and a(Y) remain unchanged if we replace Γ\sqrt{D} by Γ\sqrt{D} − Γ_{1,0} − |D|Γ_{0,1}. Let g : E_{y} ⟶ E′ be an isogeny (defined over some extension of H). It induces a canonical isomorphism (i.e. not depending on g) between End(E′) ⊗ Q and End(E_{y}) ⊗ Q = K, hence \sqrt{D} ∈ End(E′) ⊗ Q is well-defined. Lemma 3.3.4 then gives

\begin{aligned}g_{∗} (X_{√\overline{D}}^{⊗(r−1)}) & = (deg(g))^{r−1}X_{√\overline{D}}^{⊗(r−1)} \\ g^{∗} (X_{√\overline{D}}^{⊗(r−1)}) & = (deg(g))^{r−1}X_{√\overline{D}}^{⊗(r−1)} \end{aligned}  (3.6.1)

(3.7) Up to now, we have been assuming that N ≥ 3. Let us sketch an alternative definition of the sheaf A and the Heegner cycle x ∈ Z(Y_{0}(N), H) which is valid for any integer N ≥ 1. Let M be an arbitrary integer such that NM ≥ 3. The projection π_{M} : Y (NM) ⟶ Y_{0}(N) is the quotient map by the group

B_{M} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(Z/NMZ)/\{±1\} | c ≡ 0 (mod N )\}

The sheaf B_{M} := εR^{2r−2}f_{∗}Q_{p}(r − 1) on Y (NM) is equipped with an action of B_{M}, so it makes sense to define a sheaf on Y_{0}(N)

A_{M} := (π_{M∗}B_{M})^{B_{M}}

Lemma. The sheaf A = A_{M} does not depend on M.

Proof. For M | M′ we have π_{M′} = π_{M} ∘ π_{M′/M}, where π_{M′/M} : Y (NM′) ⟶ Y (NM) is a Galois covering with Galois group B_{M′/M} := Ker(B_{M′} ⟶ B_{M}). As π_{M′/M}^{∗}B_{M} = B_{M′} (proper base change), we have

((π_{M′/M})_{∗}B_{M′})^{B_{M′/M}} = B_{M},  (3.7.1)

hence

A_{M′} = (π_{M′∗}B_{M′})^{B_{M′}} = (π_{M∗}((π_{M′/M})_{∗}B_{M′})^{B_{M′/M}} )^{B_{M}} = (π_{M∗}B_{M})^{B_{M}} = A_{M}

(3.8) In order to define a Heegner cycle x ∈ Z(Y_{0}(N), H), we consider only integers M such that each prime dividing NM splits in K. As in 3.6, we fix a Heegner point y on Y_{0}(N) ⊗_{Q} H and choose a point ỹ on Y (NM) ⊗_{Q} H over y. In the fibre E2r−2 ; _{y} we take the cycle Y_{M} = (Γ\sqrt{D})^{r−1}. The cohomology class of ε_{B_{M}} εY_{M} in the fibre defines a Tate cycle a(Y_{M}) ∈ Z(Y_{0}(N), H). Lemma 2.5 holds true in this case and a(Y_{M}) does not depend on the choice of M, thank to 3.7.1. We put x = a(Y_{M}).

4. Local heights outside of p

(4.1) We are now ready to begin the computation of the p-adic height of a Heegner cycle. Recall our setup: N ≥ 1 is an integer, K = Q(\sqrt{D}) an imaginary quadratic field of discriminant D < 0 in which all primes dividing N split, H is the Hilbert class field of K, O_{K} the ring of integers in K, p a prime number, A the smooth p-adic sheaf on Y_{0}(N) defined in 2.3 (resp. 3.7), V = H^{1}(X_{0}(N) ⊗_{Q} \overline{Q}, j_{∗}A)(1) the corresponding parabolic cohomology group. From now on, we assume that p |̸ N. This implies that X(N), X_{0}(N) and W all have good reduction at p, hence the discussion of 1.4-5 applies over any number field (V satisfies the conditions of 1.4 at places v |̸ p thank to Lemma 2.12). In particular, the Abel-Jacobi map over H factors through

Φ : CH^{r}(W ⊗_{Q} H)_{0} ⟶ H_{f}^{1}(H, V)

The pairing 2.9.2 induces a height pairing

⟨ , ⟩ : H_{f}^{1}(H, V) × H_{f}^{1}(H, V) ⟶ Q_{p},  (4.1.1)

once we specify the data (A), (B) in 1.5. To do that, we fix a continuous homomorphism ℓ_{K} : A_{K}^{∗}/K^{∗} ⟶ Q_{p} and put ℓ_{H} = ℓ_{K} ∘ N_{H/K} : A_{H}^{∗}/H^{∗} ⟶ Q_{p}. We also fix splittings (to be specified later in 5.10)

F^{r}ε_{B}εH_{dR}^{2r−1}(W ⊗_{Q} Q_{p}/Q_{p}) ⟶^{⟵} ε_{B}εH_{dR}^{2r−1}(W ⊗_{Q} Q_{p}/Q_{p})

and define splittings over H ⊗_{Q} Q_{p} by a base change.

(4.2) In the notation of 2.11, let β : T ⟶ \overline{Q} be a Q-algebra homomorphism and f_{β} ∈ S_{2r}^{new}(Γ_{0}(N(β)); F (β)) the corresponding normalized primitive form. We also have the β-component

H_{f}^{1}(H, V)_{β} = H_{f}^{1}(H, V_{β}),

which is a vector space over F̂ (β) (of finite dimension). The height pairing 4.1.1 defines an F̂ (β)-linear pairing

⟨ , ⟩ : H_{f}^{1}(H, V_{β}) × H_{f}^{1}(H, V_{β}) ⟶ F̂ (β)

In 3.6 (resp. 3.8) we defined a Heegner cycle x ∈ Z(Y_{0}(N), H). Let z = Φ_{T} (x) be its Abel-Jacobi image. Of course, we have

z = Φ(ε_{B}εY) (resp. Φ(ε_{B_{M}} εY_{M})) ∈ H_{f}^{1}(H, V)

by Lemma 2.5. Let z_{β} ∈ H_{f}^{1}(H, V_{β}) be the β-component of z. Let C be a character C : G(H/K) ⟶ \overline{Q}^{∗} and L the extension of F (β) generated by the values of C. Then the C-component of z_{β}

z_{β,C} := \frac{1}{[H : K]} ∑_{σ∈G(H/K)} C^{−1}(σ)z_{β}^{σ}

lies in H_{f}^{1}(H, V_{β}) ⊗_{F (β)} L̂. Our aim is to compute ⟨z_{β,C}, z_{β,\overline{C}}⟩ ∈ L̂ (we extend ⟨ , ⟩ L̂-linearly).

(4.3) Proposition. Suppose that f_{β} is a newform. If ℓ_{K} is anticyclotomic, i.e. ℓ_{K} ∘ τ = −ℓ_{K} for the non-trivial element τ ∈ G(K/Q), then

⟨z_{β,C}, z_{β,\overline{C}}⟩ = 0

Proof. Let τ̃ ∈ G(H/Q) be a lift of τ. By [Ne 1, 6.2],

τ̃(z_{β}) = (−1)^{r−1}λ_{N}(f_{β})z_{β}^{σ}

for some σ ∈ G(H/K), hence

τ̃(z_{β,C}) = (−1)^{r−1}λ_{N}(f_{β})C(σ)(z_{β,C})

Since all splittings in 4.1 are invariant under τ̃ by construction, we have

⟨z_{β,C}, z_{β,\overline{C}}⟩_{ℓ_{K}∘N_{H/K}} = ⟨τ̃(z_{β,C}), τ̃(z_{β,\overline{C}})⟩_{ℓ_{K}∘τ∘N_{H/K}} =
= λ_{N}(f_{β})^{2}⟨z_{β,C}, z_{β,\overline{C}}⟩_{−ℓ_{K}∘N_{H/K}} = −⟨z_{β,C}, z_{β,\overline{C}}⟩_{ℓ_{K}∘N_{H/K}}

as λ_{N}(f_{β}) = ±1.

(4.4) Every ℓ_{K} can be written uniquely as a sum ℓ_{K} = ℓ_{K}^{+} + ℓ_{K}^{−} with ℓ_{K}^{±} ∘ τ = ±ℓ_{K}^{±}. In view of Prop. 4.3, the height ⟨z_{β,C}, z_{β,\overline{C}}⟩_{ℓ_{K}} depends only on ℓ_{K}^{+} (recall that heights depend linearly on ℓ_{K}). We can assume, therefore, that ℓ_{K} is cyclotomic, ℓ_{K} = ℓ_{K} ∘ τ. This determines ℓ_{K} up to a multiplicative constant We choose ℓ_{K} = ℓ_{Q} ∘ N, where ℓ_{Q} : A_{Q}^{∗}/Q^{∗} ⟶ Q_{p} is the map

A_{Q}^{∗}/Q^{∗}⟶^{Artin}G(Q^{ab}/Q) ⟶ G(Q(μ_{p^{∞}})/Q)⟶^{χ_{cycl}}Z_{p}^{∗}⟶^{log_{p}}Z_{p} ↪ Q_{p}

Here χ_{cycl} is the cyclotomic character defined by

ζ^{χ_{cycl}(σ)} = σ(ζ), ζ ∈ μ_{p^{∞}}

Another description of ℓ_{Q} is as a composition

A_{Q}^{∗}/Q^{∗} ≃ R_{+}^{∗} × ∏_{ℓ} Z_{ℓ}^{∗} ⟶ Z_{p}^{∗}⟶^{− log_{p}}Z_{p} ↪ Q_{p}

For a prime number ℓ ≠ p, the ℓ-component ℓ_{Q,ℓ} : Q_{ℓ}^{∗} ⟶ Q_{p} is trivial on Z_{ℓ}^{∗} and takes value log_{p}(ℓ) on ℓ.

(4.5) The Heegner cycle x defined in 3.6-7 is supported at one of the Heegner points y. Fix an ideal A ⊂ O_{K} and let σ be the corresponding element of the Galois group G(H/K). According to [Gr - Za, Prop. III.4.3], Tate cycles x and T_{m}x^{σ} have disjoint supports iff

r_{A}(m) = {I ⊂ O_{K} | [I] = [A], N(I) = m} = 0  (4.5.1)

If 4.5.1 is satisfied, we can apply the construction of 2.8 to x and T_{m}x^{σ}, obtaining a mixed extension E of Φ_{T} (x) and (Φ_{T} (T_{m}x^{σ}))^{∗}(1). Fix a (non-archimedean) place v of H of residue characteristic ℓ ≠ p. As a representation of G(\overline{H_{v}}/H_{v}), E splits into E = V ⊕ U_{v} with [U_{v}] ∈ H^{1}(H_{v}, Q_{p}(1)). We shall compute the value of

(x, T_{m}x^{σ})_{v} := ord_{v}([U_{v}]) ∈ Q_{p}  (4.5.2)

using the intersection formula of Proposition 2.16. With our choice of ℓ_{K}, the local height at v is equal to

⟨x, T_{m}x^{σ}⟩_{v} = −(x, T_{m}x^{σ})_{v} log_{p}(Nv)

The calculation will follow quite closely the article of Gross and Zagier [Gr - Za] and the reader is invited to open his or her copy of it.

(4.6) Let \underline{X}_{0}(N) (resp. \underline{Y}_{0}(N)) be the model of X_{0}(N) (resp. Y_{0}(N)) over Z defined in [Ka - Ma]. Over Z[1/N], \underline{X}_{0}(N)⊗Z[1/N] is a proper and smooth curve. If ℓ is a prime dividing N, N = ℓ^{n}M with (ℓ, M) = 1, then \underline{X}_{0}(N) ⊗ Z/ℓZ consists of (n + 1) irreducible components F_{a,b} (a, b ≥ 0, a + b = n). The reduced curve (F_{a,b})_{red} is isomorphic to \underline{X}_{0}(M) ⊗ Z/ℓZ. All curves F_{a,b} meet at supersingular points of \underline{X}_{0}(N) ⊗ Z/ℓZ (i.e. points represented by elliptic curves with supersingular reduction modulo ℓ).

Lemma. For ℓ | N, (\underline{X}_{0}(N)^{reg}) ⊗ Z/ℓZ contains all ordinary (i.e. not supersingular) points of F_{0,n} ∪ F_{n,0}.

Proof. [Li, Prop. 2.7] (note that [Gr - Za, Prop. III.1.4] is incorrect)

(4.7) Recall that v is a place of of H with residue characteristic ℓ ≠ p and y a Heegner point on Y_{0}(N). Let \underline{y} be the Zariski closure of y in \underline{X}_{0}(N) ⊗ O_{H_{v}}. If ℓ |̸ N, then \underline{X}_{0}(N) ⊗ O_{H_{v}} is smooth over O_{H_{v}}.

Lemma. If ℓ | N, then both \underline{y} and \underline{y}^{σ} reduce to ordinary non-cuspidal points on the component

\{ \begin{matrix}F_{0,n} & if v | \overline{N} \\ F_{n,0} & if v | N \end{matrix}

Proof. [Gr - Za, Prop. 3.1]

Corollary. Both \underline{y} and \underline{y}^{σ} are contained in (\underline{X}_{0}(N) ⊗ O_{H_{v}})^{reg}.

(4.8) We want to compute (x, T_{m}x^{σ})_{v} using Proposition 2.16. In the notation of 2.16, we take F = H_{v}, X = \underline{X}_{0}(N) ⊗ Λ ⊃ Y = \underline{Y}_{0}(N) ⊗ Λ, where Λ is the ring of integers in F^{ur}. The point y ⊗_{H} F^{ur} extends to a section \underline{y} of Y ⟶ Spec(Λ). Our aim is to extend the Heegner cycle x ∈ H^{0}(y, A) to \underline{x} ∈ H^{0}(\underline{y}, j′_{∗}A). We imitate the construction of 3.7 on integral models: choose an integer M prime to ℓ such that NM is divisible by at least two odd primes. The moduli problem for elliptic curves with a Drinfeld [Γ(NM)]-structure is then representable by a regular scheme \underline{Y} (NM), which is a model of Y (NM) over Z. There is a universal elliptic curve \underline{f} : \underline{E} ⟶ \underline{Y} (NM) extending f : E ⟶ Y (NM). The p-adic sheaf \underline{B}_{M} = Sym^{2r−2}R^{1}\underline{f}_{∗}Q_{p}(r − 1) is smooth on \underline{Y} (NM) and satisfies k_{M}^{∗} \underline{B}_{M} = B_{M}, k_{M∗}B_{M} = \underline{B}_{M} for k_{M} : Y (NM) ↪ \underline{Y} (NM), by proper and smooth base change theorems. The finite group B_{M} acts on \underline{Y} (NM) and \underline{B}_{M}. The quotient B_{M}\\underline{Y} (NM) is equal to \underline{Y}_{0}(N) ([Ka - Ma, Thm. 7.4.2 & Lemma 8.1.5]). Write \underline{π}_{M} for the projection \underline{Y} (NM) ⟶ \underline{Y}_{0}(N). The sheaf \underline{A} := (\underline{π}_{M∗}\underline{B}_{M})^{B_{M}} on \underline{Y}_{0}(N) extends A = (π_{M}B_{M})^{B_{M}} on Y_{0}(N) and is independent on the choice of M, as in Lemma 3.7. We now assume that all primes dividing M split in K. As in 3.7, we choose a point ỹ on Y (NM) ⊗_{Q} H over y. It extends to a section \underline{ỹ} of \underline{Y} (NM) ⊗ Λ ⟶ Λ. Since

H^{0}(\underline{ỹ}, \underline{B}_{M}) = H^{0}(ỹ, B_{M}),

the Tate cycle b(Y_{M}) ∈ H^{0}(ỹ, B_{M}) from 3.7 (given by the cohomology class of εY_{M} in the fibre) extends uniquely to a section \underline{b}(Y_{M}) of \underline{B}_{M} over \underline{ỹ}. The canonical map \underline{A} ⟶ j′_{∗}A takes ε_{B_{M}} \underline{b}(Y_{M}) ∈ H^{0}(\underline{y}, (\underline{π}_{M∗}\underline{B}_{M})^{B_{M}}) into the sought-for section \underline{x} ∈ H^{0}(\underline{y}, j′_{∗}A) extending x.

It is easy to see that \underline{x} does not depend on the choice of M by looking at its special fibre \underline{x}_{s} ∈ H^{0}(\underline{y}_{s}, j′_{∗}A). According to [Se - Ta], \underline{y} can be represented by an isogeny λ : E ⟶ E′ between elliptic curves defined over Λ. The reduction map defines an inclusion

O_{K} = End_{Λ}(\underline{y}) ↪ End_{k}(\underline{y}_{s})  (4.8.1)

Here k ≃ \overline{F}_{ℓ} is the residue field of Λ. This defines an element \sqrt{D} ∈ End(\underline{y}_{s}) and we have

\underline{x}_{s} = \frac{1}{♯Aut(\underline{y}_{s})} ∑_{α∈Aut(\underline{y}_{s})} α_{∗}ε_{r−1} (X_{√\overline{D}}^{⊗(r−1)}) ∈ H^{2r−2}(E_{s}^{2r−2}, Q_{p})(r − 1)  (4.8.2)

The same construction also applies to x^{σ}, yielding \underline{x}^{σ} ∈ H^{0}(\underline{y}^{σ}, j′_{∗}A).

(4.9) By 4.8.1, we have an inclusion O_{K}^{∗} = Aut(\underline{y}) ↪ Aut(\underline{y}_{s}). We say that we are in the special case if this inclusion is strict.

Lemma. (1) The special case occurs iff N = 1, ℓ ∈ \{2, 3\}, (\frac{D}{ℓ} ) ≠ 1.

(2) If we are not in the special case, then Aut(\underline{y}_{s}) acts trivially in (4.8.2) and \underline{x}_{s} = ε_{r−1} (X_{√\overline{D}}^{⊗(r−1)}).

Proof. (1) Suppose that we are in the special case. The well-known classification of automorphisms of elliptic curves implies that the special fibre E_{s} must be a supersingular elliptic curve in characteristic ℓ = 2 or ℓ = 3 (hence ℓ |̸ N). We have to show that N = 1. The level structure is given by a subgroup scheme Z/NZ ↪ E_{s}, as k is algebraically closed and ℓ |̸ N. The action of A := Aut(\underline{y}_{s}) on this subgroup is given by a homomorphism φ : A ⟶ (Z/NZ)^{∗}. If α lies in the commutant [A, A], then φ(α) = 1, hence deg(α − 1) must be divisible by N. A short inspection of the lattice of subgroups between O_{K}^{∗} and Aut(E_{s}) shows that there is always α ≠ 1 in the commutant [A, A] satisfying α^{ℓ} = 1, hence N | deg(α − 1) = 4 (resp. 3) if ℓ = 2 (resp. if ℓ = 3). As ℓ |̸ N, this implies that N = 1. Conversely, if N = 1, ℓ ∈ \{2, 3\} and (\frac{D}{ℓ} ) ≠ 1, then Aut(\underline{y}_{s}) = Aut(E_{s}) is of order 12 or 24.

(2) This follows from 3.6.1.

(4.10) We now examine T_{m}x^{σ} ∈ H^{0}(T_{m}y^{σ}, A), first in the case when ℓ |̸ m. The section \underline{y}^{σ} can be represented by an elliptic curve \underline{E} over Λ with a Drinfeld [Γ_{0}(N)]-structure. Let E be the generic fibre of \underline{E}. Any isogeny E ⟶ E′ of degree m is automatically defined over H_{v}^{ur} and lifts to an étale isogeny g : \underline{E} ⟶ \underline{E}′ over Λ. The curve \underline{E}′ with the induced [Γ_{0}(N)]-structure represents a section \underline{y}_{g}^{σ} of X ⟶ Spec(Λ). The divisor T_{m}\underline{y}^{σ} (Zariski closure of T_{m}y^{σ}) decomposes into

T_{m}\underline{y}^{σ} = ∑_{deg(g)=m} \underline{y}_{g}^{σ}

According to 3.6.1, the Tate cycle T_{m}x^{σ} is equal to

T_{m}x^{σ} = m^{r−1} ∑_{deg(g)=m} x_{g}^{σ},

where x_{g}^{σ} = ε_{r−1} (X_{√\overline{D}}^{⊗(r−1)}) is a Tate vector supported at y_{g}^{σ}. We extend every x_{g}^{σ} to a section \underline{x}_{g}^{σ} ∈

H^{0}(\underline{y}_{g}^{σ}, j′_{∗}A) as in 4.8 and define

T_{m}\underline{x}^{σ} = m^{r−1} ∑_{deg(g)=m} \underline{x}_{g}^{σ}

If we are not in the special case of 4.9, then the special fibre (\underline{x}_{g}^{σ})_{s} is equal to ε_{r−1} (X_{√\overline{D}}^{⊗(r−1)}), where

\sqrt{D} ∈ End(\underline{y}_{g}^{σ}) ⊗ Q ⊂ End((\underline{y}_{g}^{σ})_{s}).

(4.11) There is another description of Hecke operators on integral models which works also for for m divisible by ℓ (but prime to N). Choose an auxiliary integer M prime to pm, such that NM is divisible by at least two odd primes. The moduli problem for elliptic curves with a combination of (Drinfeld) [Γ(NM)] and [Γ_{0}(m)] structures is representable by a curve \underline{Y} (NM, m), flat over Z, which is an integral model of Y (NM, m). As in 2.9, there are two finite maps \underline{s}, \underline{t} : \underline{Y} (NM, m) ⟶ \underline{Y} (NM) and a universal elliptic curve \underline{f} : \underline{E} ⟶ \underline{Y} (NM). They define an integral version of the diagram 2.9.1, which gives rise to homomorphisms

\underline{ψ}^{∗} : \underline{t}^{∗}\underline{B}_{M} ⟶ \underline{s}^{∗}\underline{B}_{M}, \underline{ψ}_{∗} : \underline{s}^{∗}\underline{B}_{M} ⟶ \underline{t}^{∗}\underline{B}_{M}

Assume that all primes dividing M split in K. As in 4.8, we have a section \underline{ỹ}^{σ} of \underline{Y} (NM) ⊗ Λ ⟶ Λ and an extended Tate cycle \underline{b}(Y_{M}^{σ}) ∈ H^{0}(\underline{ỹ}^{σ}, \underline{B}_{M}). Applying to \underline{b}(Y_{M}^{σ}) operators T_{m∗} = \underline{t}_{∗}\underline{ψ}_{∗}\underline{s}^{∗} (resp. T_{m}^{∗} = \underline{s}_{∗}\underline{ψ}_{∗}\underline{t}^{∗}; cf. 2.10) and ε_{B_{M}}, we obtain elements

\begin{aligned}T_{m∗}(ε_{B_{M}} \underline{b}(Y_{M}^{σ})) & ∈ H^{0}(T_{m∗}\underline{y}, \underline{A}) \\ T_{m}^{∗}(ε_{B_{M}} \underline{b}(Y_{M}^{σ})) & ∈ H^{0}(T_{m}^{∗}\underline{y}, \underline{A}) \end{aligned}

which are equal by the same argument as in 2.10. Denote by T_{m}\underline{x}^{σ} their common image in H^{0}(T_{m}\underline{y}^{σ}, j′_{∗}A). It extends T_{m}x^{σ} and is equal to T_{m}\underline{x}^{σ} constructed in 4.10 if ℓ |̸ m.

(4.12) Proposition. Let m ≥ 1 be prime to N such that r_{A}(m) = 0. If N = 1, assume that both 2 and 3 split in K. Then

(x, T_{m}x^{σ})_{v} = \frac{1}{2}m^{r−1} ∑_{n≥1} ∑_{g} (ε_{r−1} (X_{g\sqrt{D}g^{−1}}^{⊗(r−1)}) , ε_{r−1} (X_{√\overline{D}}^{⊗(r−1)}))_{r−1} ,

where the sum extends over g ∈ Hom_{Λ/π^{n}}(\underline{y}^{σ}, \underline{y}) of degree deg(g) = m.

Corollary. If ℓ splits in K, then (x, T_{m}x^{σ})_{v} = 0.

Proof of the Corollary. According to [Gr - Za, III.7.1], the set of g in the Proposition is empty. Proof of the Proposition. It should be noted that the symbol \sqrt{D} in the above formula has a dual meaning: in the first term, it is an element of End(\underline{y}^{σ}) ↪ End((\underline{y}^{σ})_{s}) (hence g\sqrt{D}g^{−1} lies in End(\underline{y}_{s})); in the second term, \sqrt{D} ∈ End(\underline{y}) ↪ End(\underline{y}_{s}). The intersection pairing ( , )_{r−1} is computed on E_{s}^{2r−2} (where E with a cyclic subgroup of order N represents y). We shall apply Prop. 2.16 with F = H_{v}, X = \underline{X}_{0}(N)⊗Λ. By [Ka - Ma, 14.5.5.1], we have H^{2}(X , j′_{∗}A(1)) = 0. Let m = m_{0}ℓ^{t} with ℓ |̸ m_{0}. As in 4.10, we have

T_{m_{0}} \underline{x}^{σ} = m_{0}^{r−1} ∑_{g} \underline{x}_{g}^{σ}

Fix one of g ∈ Hom_{Λ}(\underline{y}^{σ}, \underline{y}_{g}^{σ}) (deg(g) = m_{0}) and decompose the divisor T_{ℓ^{t}} \underline{y}_{g}^{σ} on X into irreducible components \underline{y}_{g}^{σ}(k). Then

T_{ℓ^{t}} \underline{x}_{g}^{σ} = ℓ^{t(r−1)} ∑_{k} \underline{x}_{g}^{σ}(k)

with \underline{x}_{g}^{σ}(k) supported at \underline{y}_{g}^{σ}(k). Assumptions of Prop. 2.16 are satisfied for \underline{a} = \underline{x}, \underline{b} = \underline{x}_{g}^{σ}(k) (Lemma 4.6).

The special fibre of \underline{x} is represented by (\underline{y}_{s}, ε_{r−1} (X_{√\overline{D}}^{⊗(r−1)})). It follows from Lemma 4.13 below that the special fibre of \underline{b} does not depend on k, provided the supports of \underline{a} and \underline{b} intersect. It follows from Prop. 2.16 that

(x, T_{ℓ^{t}}x_{g}^{σ})_{v} = ℓ^{t(r−1)}(\underline{y} · T_{ℓ^{t}} \underline{y}_{g}^{σ})(\underline{x}_{s}, \underline{x}_{g}^{σ}(k)_{s})_{r−1}

Taking the sum over g, using [Gr - Za, III.6.1] and Lemma 4.13, we obtain

(x, T_{m}x^{σ})_{v} = \frac{1}{2}m^{r−1} ∑_{n≥1} ∑_{hg} (ε_{r−1} (X_{√\overline{D}}^{⊗(r−1)}) , ε_{r−1} (X_{hg\sqrt{D}g^{−1}h^{−1}}^{⊗(r−1)}))_{r−1}

with hg ∈ Hom_{Λ/π^{n}}(\underline{y}^{σ}, \underline{y}), deg(hg) = m as claimed. It remains to prove the following Lemma.

(4.13) Lemma. Suppose that \underline{y} and \underline{y}_{g}^{σ}(k) intersect. Then the special fibre of \underline{x}_{g}^{σ}(k) is represented by

(\underline{y}_{s}, ε_{r−1} (X_{hg\sqrt{D}g^{−1}h^{−1}}^{⊗(r−1)}))

for arbitrary isogeny h ∈ Hom_{Λ/π^{n}}(\underline{y}^{σ}, \underline{y}) with deg(h) = ℓ^{t}, n ≥ 1.

Proof. If t = 0, then \underline{y}_{g}^{σ} is ireducible and the special fibre of \underline{x}_{g}^{σ} is represented by (use 3.6.1 and 4.9.2)

((\underline{y}_{g}^{σ})_{s}, ε_{r−1} (X_{√\overline{D}}^{⊗(r−1)})) = ((\underline{y}_{g}^{σ})_{s}, ε_{r−1} (X_{g\sqrt{D}g^{−1}}^{⊗(r−1)})) ,

hence also by (\underline{y}_{s}, ε_{r−1} (X_{hg\sqrt{D}g^{−1}h^{−1}}^{⊗(r−1)})), as h induces an isomorphism between (\underline{y}_{g}^{σ})_{s} and \underline{y}_{s}.

Suppose now that t ≥ 1 (thus ℓ |̸ N and X is smooth over Λ). If ℓ splits in K, then \underline{y} and \underline{y}_{g}^{σ}(k) never intersect by [Gr - Za, p. 260], so there is nothing to prove.

Assume now that ℓ is inert in K. If t is even, then the components \underline{y}_{g}^{σ}(k) are given by \underline{y}(2k) for 0 ≤ k ≤ t/2, in the notation of [Gr - Za, p. 259]. For each k, the special fibre of \underline{y}(2k) is isomorphic to the special fibre of \underline{y}_{g}^{σ} (cf. the discussion after [Gr - Za, III.5.3]) and the special fibre of \underline{x}_{g}^{σ}(k) is represented by

((\underline{y}_{g}^{σ})_{s}, ε_{r−1} (X_{g\sqrt{D}g^{−1}}^{⊗(r−1)}))

Every isogeny h ∈ Hom_{Λ/π^{n}}(\underline{y}_{g}^{σ}, \underline{y}) of degree deg(h) = ℓ^{t} is of the form h = ℓ^{t/2}h_{0}, where h_{0} is defined over Λ/π^{m} for some m ≥ 1 (possibly smaller than n) and deg(h_{0}) = 1. This implies that \underline{x}_{g}^{σ}(k)_{s} is represented by

(\underline{y}_{s}, ε_{r−1} (X_{h_{0}g\sqrt{D}g^{−1}h_{0}^{−1}}^{⊗(r−1)})) = (\underline{y}_{s}, ε_{r−1} (X_{hg\sqrt{D}g^{−1}h^{−1}}^{⊗(r−1)}))

as claimed.

If t is odd, then the components \underline{y}_{g}^{σ}(k) are \underline{y}(2k + 1) for 0 ≤ k ≤ (t − 1)/2. Represent \underline{y}_{g}^{σ} by an elliptic curve E′ over Λ with a cyclic subgroup of order N. Every isogeny E′ ⟶ E′′ of degree ℓ is defined over a totally ramified extension Λ′ of Λ of degree ℓ + 1. It lifts the Frobenius morphism h_{1} : E′_{s} ⟶ E_{s}^{′(ℓ)}. The special fibre of \underline{y}(2k + 1) is isomorphic to (\underline{y}^{σ})_{s}^{(ℓ)} ([Gr - Za, III.5.3]) and it follows from the previous discussion and 3.6.1 that the special fibre of \underline{x}_{g}^{σ}(k) is represented by

((\underline{y}_{g}^{σ})_{s}^{(ℓ)}, ε_{r−1} (X_{h_{1}g\sqrt{D}g^{−1}h_{1}^{−1}}^{⊗(r−1)}))

Every isogeny h ∈ Hom_{Λ/π^{n}}(\underline{y}_{g}^{σ}, \underline{y}) of degree deg(h) = ℓ^{t} is of the form h = ℓ^{(t−1)/2}h_{0}h_{1}, where h_{0} is an

isomorphism between (\underline{y}^{σ})_{s}^{(ℓ)} and \underline{y}_{s}. Consequently, \underline{x}_{g}^{σ}(k)_{s} is represented by

(\underline{y}_{s}, ε_{r−1} (X_{h_{0}h_{1}g\sqrt{D}g^{−1}h_{1}^{−1}h_{0}^{−1}}^{⊗(r−1)})) = (\underline{y}_{s}, ε_{r−1} (X_{hg\sqrt{D}g^{−1}h^{−1}}^{⊗(r−1)}))

as claimed.

For ramified ℓ the proof of the Lemma is goes along the same lines, using the description of the components and isogenies of degree ℓ^{t} in [Gr - Za, p. 259, 261]. We omit the details.

(4.14) We rewrite the expression in Proposition 4.12 using quaternions, as in [Gr - Za, III.7]. It suffices to treat only the case when ℓ is not split in K (⟹ ℓ |̸ N). If this is the case, then End_{Λ/π}(\underline{y}) = End(\underline{y}_{s}) = R is an order in the quaternion algebra B over Q ramified at ℓ and ∞. Let 𝔩 be the unique prime dividing ℓ in K. The embedding O_{K} = End(\underline{y}) ↪ R = End(\underline{y}_{s}) extends to K ↪ B. There is a decomposition B = B_{+} + B_{−} with B_{+} = K, B_{−} = Kj, where j satisfies jaj^{−1} = a^{τ} for all a ∈ K. Each element b ∈ B has a unique decomposition b = b_{+} + b_{−} with b_{±} ∈ B_{±}. Write N for the reduced norm on B.

Recall from [Gr - Za, III.7.3] that

End_{Λ/π^{n}}(\underline{y}) = \{b ∈ R | |D|N(b_{−}) ≡ 0 (mod ℓ(N(𝔩))^{n−1} )\}

Hom_{Λ/π^{n}}(\underline{y}^{σ}, \underline{y}) ⟶^{∼} End_{Λ/π^{n}}(\underline{y}) · A

In the latter isomorphism, the isogeny corresponding to b ∈ B has degree N(b)/N(A). We also assume that ℓ |̸ N(A); we can always find A satisfying this condition without changing σ = σ_{A}. Under this assumption, we have (cf. [Li])

Hom_{Λ/π^{n}}(\underline{y}^{σ}, \underline{y}) = \{b ∈ RA | |D|N(b_{−}) ≡ 0 (mod ℓ(N(𝔩))^{n−1} )\}  (4.14.1)

(4.15) Proposition. Under the assumptions of Proposition 4.12,

(x, T_{m}x^{σ})_{v} = \frac{(4|D|m)^{r−1}}{\binom{2r−2}{r−1}} ∑_{{b∈RA/\{±1\} \atop N(b)=mN(A)}} U(b_{−})P_{r−1} (1 − \frac{2N(b_{−})}{N(b)}) ,

where

U(b_{−}) = \{ \begin{matrix}0 & if ℓ splits in K \\ \frac{1}{2}(1 + ord_{ℓ}(N(b_{−})) & if ℓ is inert in K \\ ord_{ℓ}(|D|N(b_{−})) & if ℓ ramifies in K \end{matrix}

Proof. We combine Prop. 3.4, Prop. 4.12 and (4.14.1), observing that, for u = \sqrt{D} and v = b\sqrt{D}b^{−1}, we have ⟨X_{u}, X_{u}⟩_{1} = ⟨X_{v}, X_{v}⟩_{1} = 2D, ⟨X_{u}, X_{v}⟩_{1} = t(uv) = 2D(N(b_{+}) − N(b_{−}))/N(b).

(4.16) We recall some notation from [Gr - Za]: for an ideal B ⊂ O_{K}, denote by {B} the genus of [B]. For an integer n ≥ 1, set

R_{{B}}(n) = ∑_{[A]∈{B}} r_{A}(n)
δ(n) = 2^{s},  s = number of prime factors of (n, |D|)

We shall evaluate the sum

⟨x, T_{m}x^{σ}⟩_{ℓ} = ∑_{v|ℓ}⟨x, T_{m}x^{σ}⟩_{v} = − ∑_{v|ℓ}(x, T_{m}x^{σ})_{v} log_{p}(Nv)

Proposition. Under the assumptions of Prop. 4.12,

(1) If ℓ splits in K, then ⟨x, T_{m}x^{σ}⟩_{ℓ} = 0.

(2) If ℓ is inert in K, then

⟨x, T_{m}x^{σ}⟩_{ℓ} = −u^{2} \frac{(4|D|m)^{r−1}}{\binom{2r−2}{r−1}} log_{p}(ℓ) ∑_{{0<n< \frac{m|D|}{N} \atop ℓ |̸n}} ord_{ℓ}(ℓn)r_{A}(m|D| − nN)δ(n)R_{\{A𝔮N \}}(n/ℓ)P_{r−1} (1 − \frac{2nN}{m|D|})

(3) If ℓ is ramified in K, then

⟨x, T_{m}x^{σ}⟩_{ℓ} = −u^{2} \frac{(4|D|m)^{r−1}}{\binom{2r−2}{r−1}} log_{p}(ℓ) ∑_{{0<n< \frac{m|D|}{N} \atop ℓ |̸n}} ord_{ℓ}(n)r_{A}(m|D| − nN)δ(n)R_{\{A𝔮𝔩N \}}(n/ℓ)P_{r−1} (1 − \frac{2nN}{m|D|})

Here u = ♯(O_{K}^{∗})/2 and 𝔮 is any ideal in O_{K} with N(𝔮) ≡ −ℓ (mod |D|).

Proof. This follows from Prop. 4.15 by the the same calculation as in [Gr - Za, III.9]; the only difference is that we use p-adic logarithms and have an extra term with the Legendre polynomial.

(4.17) Proposition. Assume that (m, N) = 1, r_{A}(m) = 0 and that we are not in the special case of 4.9. Then

∑_{ℓ≠p}⟨x, T_{m}x^{σ}⟩_{ℓ} = −u^{2} \frac{(4|D|m)^{r−1}}{\binom{2r−2}{r−1}} ∑_{0<n< \frac{m|D|}{N}} σ_{A}(n)r_{A}(m|D| − nN)P_{r−1} (1 − \frac{2nN}{m|D|}) ,

where σ_{A}(n) was defined in I.6.6.

Proof. This follows from Prop. 4.16 and the equality ([Gr - Za, IV.4.6.b])

∑_{{ℓ|n prime \atop ℓ≠p}} log_{p}(ℓ) \{ \begin{matrix}ord_{ℓ}(ℓn)r_{A}(m|D| − nN)δ(n)R_{\{A𝔮N \}}(n/ℓ) & ℓ inert in K \\ ord_{ℓ}(n)r_{A}(m|D| − nN)δ(n)R_{\{A𝔮𝔩N \}}(n/ℓ) & ℓ ramified in K \\ 0 & ℓ splits in K \end{matrix} \} =

= ∑_{{k|n \atop k>0}} ε_{A}(n, k) log_{p} (\frac{n}{k^{2}} ) = σ_{A}(n)

5. Local heights at p

(5.1) We now turn our attention to local heights at primes dividing p. The assumptions of 4.1 are in force. We first redo [PR 2, 5.2] in our setting. Let Y_{0}(N)^{ord} be the subset of Y_{0}(N)(\overline{Q}_{p}) represented by pairs (y, C_{N}), where y is an elliptic curve over \overline{Q}_{p} with good ordinary reduction and C_{N} a cyclic subgroup of y of order N (recall that p |̸ N). The p-torsion y_{p} contains a canonical subgroup can(y) of order p, namely the kernel of the reduction on y_{p}. The map

c : Y_{0}(N)^{ord} ⟶ Y_{0}(pN)(\overline{Q}_{p})

given by

c(y, C_{N}) = (y, C_{N} ⊕ can(y))

is a section of the projection Y_{0}(pN) ⟶ Y_{0}(N) over Y_{0}(N)^{ord}. The absolute Frobenius on Y_{0}(N) ⊗ Z/pZ has a canonical lifting

φ : Y_{0}(N)^{ord} ⟶ Y_{0}(N)^{ord},

given by

φ(y, C_{N}) = (y/can(y), C_{N} + can(y)/can(y))

We shall be interested in Tate cycles Z(Y_{0}(N)^{ord}, \overline{Q}_{p}) supported in Y_{0}(N)^{ord}. Any such Tate cycle is a sum of Tate vectors, which are represented by triples \underline{y} = (y, C_{N}, a) with (y, C_{N}) ∈ Y_{0}(N)^{ord}, a ∈ Sym^{2r−2}(H^{1}(y, Q_{p})). For m ≥ 1 prime to N, the Hecke correspondence T_{m} = T_{m∗} = T_{m}^{∗} acts on Tate cycles as in 2.10. There is also a correspondence T′_{m} acting on Z(Y_{0}(pN), \overline{Q}_{p}) by the following formula:

(T′_{m})_{∗}(y, C_{pN}, a) = ∑_{λ:y⟶y′ ; {deg(λ)=m \atop Ker(λ)∩C_{pN} =0}} (y′, λ_{∗}(C_{pN}), λ_{∗}(a))

If p |̸ m, then

c(T_{m}(\underline{y})) = (T′_{m})_{∗}(c(\underline{y}))  (5.1.1)

For k ≥ 0, denote by S(y, k) the set of all subgroups of Y of order p^{k}. Every C ∈ S(y, k) defines an isogeny λ : y ⟶ y/C of degree p^{k}; we define

\underline{y}_{C} := (y/C, C + C_{N}/C, λ_{∗}(a))

(5.2) Lemma. For all m ≥ 1 prime to N and k ≥ 0, we have

(1) (T′_{mp^{k+1}})_{∗} ∘ c(\underline{y}) = c ∘ T_{mp^{k+1}}(\underline{y}) − c ∘ T_{mp^{k}} ∘ φ(\underline{y})

(2) c∘[T_{mp^{k+2}} ∘φ−T_{mp^{k+1}} ∘(p^{2r−2} +φ^{2})+p^{2r−2}T_{mp^{k}} ∘φ](\underline{y}) = (T′_{m})_{∗} ∘[(T′_{p^{k+2}})_{∗} ∘c∘φ−p^{2r−2}(T′_{p^{k+1}})_{∗} ∘c](\underline{y})

Proof. (1) Thank to (5.1.1) and the fact that

T_{mp^{k}} = T_{m} ∘ T_{p^{k}}, (T′_{mp^{k}})_{∗} = (T′_{m})_{∗} ∘ (T′_{p^{k}})_{∗}

for p |̸ m, we can assume that m = 1. Then

T_{p^{k+1}}(\underline{y}) = ∑_{{C∈S(y,k+1) \atop can(y)⊆C}} \underline{y}_{C} + ∑_{{C∈S(y,k+1) \atop can(y)⊈C}} \underline{y}_{C}

If can(y) ⊆ C, then C′ := C/can(y) runs through S(can(y), k) and \underline{y}_{C} = (φ(\underline{y}))_{C′}, so the first sum is equal to T_{p^{k}}(φ(\underline{y})), thus

c ∘ T_{p^{k+1}}(\underline{y}) = c ∘ T_{p^{k}}(φ(\underline{y})) + (T′_{p^{k+1}})_{∗} ∘ c(\underline{y})

(2) This follows from (1) applied to k and k + 1.

(5.3) Lemma. For (y, C_{N}) ∈ Y_{0}(N)^{ord},

(T′_{p^{k+2}})_{∗} ∘ c ∘ φ(\underline{y}) − p^{2r−2}(T′_{p^{k+1}})_{∗} ∘ c(\underline{y}) = ∑_{C}(c ∘ φ(\underline{y}))_{C},

where C runs through elements C ∈ S(φ(y), k + 2) such that can(φ(y)), y_{p}/can(y) ⊄ C.

Proof. By definition,

(T′_{p^{k+1}})_{∗} ∘ c(\underline{y}) = ∑_{{C∈S(y,k+1) \atop can(y)⊄C}} c ∘ \underline{y}_{C}

The set of the C’s is in a bijection with the set of C′ := φ^{−1}(C) – they run through cyclic subgroups C′ ∈ S(φ(y), k + 2) such that C′ ⊃ y_{p}/can(y) = Ker(φ̂). As the diagram

y ⟶^{φ} φ(y) ⟶^{λ_{C′}} φ(y)_{C′}
∥  ↓_{φ}  ∥
y ⟶^{p} y ⟶^{λ_{C}} y_{C}

commutes, we have φ_{∗}(a)_{C′} = [p]_{∗}a_{C} = p^{2r−2}a_{C}. Subtracting from the sum defining (T′_{p^{k+2}})_{∗} ∘ c ∘ φ(\underline{y}) we get the statement of the Lemma.

Corollary. If m = m_{0}p^{n} with p |̸ m_{0}, then

c ∘ [T_{mp^{k+2}} ∘ φ − T_{mp^{k+1}} ∘ (p^{2r−2} + φ^{2}) + p^{2r−2}T_{mp^{k}} ∘ φ](\underline{y}) = (T′_{m_{0}})_{∗} [∑_{C} c ∘ (φ(\underline{y}))_{C}] ,

where C runs through cyclic subgroups of φ(y) of order p^{n+k+2} satisfying can(φ(y)) ⊄ C and y_{p}/can(y) ⊄ C.

(5.4) We now specialize the previous Corollary to CM points, assuming that y is an elliptic curve with a complex multiplication by an order of conductor prime to p in K = Q(\sqrt{D}). Since y is ordinary, p splits in K, (p) = 𝔭\overline{𝔭} with 𝔭 ≠ \overline{𝔭}. Let v be a place of H over 𝔭 and w a place of H_{p^{n+k+2}} (the ring class field of K of conductor p^{n+k+2}) over v.

Lemma. If m = m_{0}p^{n} with p |̸ m_{0}, then

T_{mp^{k+2}}(φ(\underline{y})) − T_{mp^{k+1}}(p^{2r−2}\underline{y} + φ^{2}(\underline{y})) + p^{2r−2}T_{mp^{k}}(φ(\underline{y})) = N_{H_{p^{n+k+2},w}/H_{v}}(T_{m_{0}}(φ(\underline{y})_{C})),

where C is any subgroup of y of order p^{n+k+2} that does not contain can(φ(y)) (= φ(y)_{𝔭}) or y_{p}/can(y) (= φ(y)\overline{𝔭}).

Proof. This follows from Cor. 5.3 and the fact that the subgroups C form a G(H_{p^{n+k+2},w}/H_{v})-torsor.

(5.5) Let F be a number field and v a prime of F dividing p. We denote by Z_{f}(Y_{0}(N), F_{v}) the preimage of H_{f}^{1}(F_{v}, V) under

Φ_{T} : Z(Y_{0}(N), F_{v}) ⟶ H^{1}(F_{v}, V)

Suppose that a, b ∈ Z_{f}(Y_{0}(N), F_{v}) are Tate cycles with disjoint supports. The mixed extension E from 2.8 can be plugged in to the height machinery of 1.8 (provided the relevant splittings and ℓ_{v} : F_{v}^{∗} ⟶ Q_{p} are fixed), defining thus the local height ⟨a, b⟩_{v}. If, in the notation of 2.3-5, a = a(Y ⊗_{F} F_{v}), b = a(Z ⊗_{F} F_{v}), then a, b ∈ Z_{f}(Y_{0}(N), F_{v}) by Lemma 1.4 and ⟨a, b⟩_{v} = ⟨Y, Z⟩_{v}. Denote by A_{Z_{p}} the image of R^{2r−2}f_{∗}Z_{p}(r − 1) in A. One defines integral Tate cycles Z(Y_{0}(N), F)_{Z_{p}} by replacing A by A_{Z_{p}} in 2.6. The discussion of integrality properties in 1.10 carries over to integral Tate cycles. For example,

Φ_{T} (Z_{f}(Y_{0}(N), F_{v})_{Z_{p}}) ⊆ p^{−c}Im(H_{f}^{1}(F_{v}, T) ⟶ H_{f}^{1}(F_{v}, V)),  (5.5.1)

where T is the image of H^{1}(X_{0}(N)⊗_{Q} \overline{Q}, j_{∗}A_{Z_{p}})(1) in V and p^{c} is the order of the finite group H^{2}(X_{0}(N)⊗_{Q} \overline{Q}, j_{∗}A_{Z_{p}}). Similarly, if a, b ∈ Z_{f}(Y_{0}(N), F_{v})_{Z_{p}} and Φ_{T} (a) = 0, then

⟨a, b⟩_{v} ∈ p^{−2c}ℓ_{v}(F_{v}^{∗})  (5.5.2)

(5.6) We continue with the notation and assumptions of 4.1, but from now on we also assume that p splits in K, (p) = 𝔭\overline{𝔭} with 𝔭 ≠ \overline{𝔭}. Let σ_{𝔭} ∈ G(H/K) (resp. σ\overline{𝔭}) be the Frobenius at 𝔭 (resp. \overline{𝔭}). If A_{m}^{σ} is a function of m ∈ Z and σ ∈ G(H/K), we put

A_{m}^{σ}|U_{p} = A_{mp}^{σ}, A_{m}^{σ}|σ′ = A_{m}^{σσ′}

Using this notation, we define, for any m ≥ 1 prime to N and σ ∈ G(H/K),

b_{m}^{σ} := T_{m}x^{σ}|(U_{p} − σ_{𝔭})(U_{p}σ_{𝔭} − p^{2r−2}) = T_{mp^{2}}x^{σσ_{𝔭}} − T_{mp}(p^{2r−2}x^{σ} + x^{σσ_{𝔭}^{2}}) + p^{2r−2}T_{m}x^{σσ_{𝔭}}

This is an element of Z_{f}(Y_{0}(N), H)_{Z_{p}}, as the action of T preserves the f-subspace.

Proposition. Let v be a place of H over 𝔭 and a ∈ Z(Y_{0}(N), H_{v}) a Tate cycle with Φ_{T} (a) = 0. Then there is an integer k_{0} (depending on a) such that

⟨a, b_{mp^{k}}^{σ}⟩_{v} ∈ p^{k−k_{0}}Z_{p}

for all m ≥ 1 prime to N and k ≥ 0, provided a and b_{mp^{k}}^{σ} have disjoint supports.

Proof. Changing k_{0}, we may assume that a is an integral Tate cycle and p |̸ m. According to Lemma 5.4, we have

b_{mp^{k}}^{σ} = N_{H_{p^{k+2},w}/H_{v}}(h_{m,k}^{σ})

for suitable h_{m,k}^{σ} ∈ Z_{f}(Y_{0}(N), H_{p^{k+2},w})_{Z_{p}}. By 1.9.1 and 5.5.2, this implies that

⟨a, b_{mp^{k}}^{σ}⟩_{v,ℓ_{v}} = ⟨a, h_{m,k}^{σ}⟩_{w,ℓ_{w}} ∈ p^{−2c}ℓ_{w}(H_{p^{k+2},w}^{∗}),

where ℓ_{w} = ℓ_{v} ∘ N_{H_{p^{k+2},w}/H_{v}}. However, H_{p^{k+2},w} contains the (k + 2)-th layer Q_{p,k+2} of the cyclotomic Z_{p}-extension Q_{p,∞}/Q_{p} (as p splits in K), hence

ℓ_{w}(H_{p^{k+2},w}^{∗}) ⊆ ℓ_{Q,p}(N_{Q_{p,k+2}/Q_{p}}(Q_{p,k+2}^{∗})) ⊆ p^{k+2}ℓ_{Q,p}(Q_{p}^{∗}) = p^{k+3}Z_{p}

(5.7) Lemma. Let M ≥ 1 be an integer and g = ∑_{n≥1} a_{n}q^{n} ∈ S_{2r}(Γ_{0}(N)) such that a_{m} = 0 for all integers m prime to NM satisfying r(m) = ∑_{[A]∈Pic(O_{K})} r_{A}(m) = 0. Then a_{n} = 0 for all n ≥ 1 prime to N.

Proof. We can write

g = ∑_{β:T⟶\overline{Q}} c_{β} · f_{β} + h,

where a_{n}(h) = 0 for all n prime to N. Let ℓ |̸ NM be a prime number inert in K. Then r(ℓn) = 0 for all n prime to ℓ, hence

0 = a_{ℓn}(g) = ∑_{β} c_{β}a_{ℓn}(f_{β}) = ∑_{β} c_{β}a_{ℓ}(f_{β})a_{n}(f_{β})

whenever (n, NMℓ) = 1. This implies that c_{β}a_{ℓ}(f_{β}) = 0 for all β. Assume that, for some β, a_{ℓ}(f_{β}) = 0 for all primes ℓ |̸ NM inert in K. It follows that

Tr(F r_{ℓ}|V_{β}) = m(β)a_{ℓ}(f_{β})ℓ^{−r} = 0

for such ℓ. Let T_{β} ⊂ V_{β} be a lattice invariant under G(\overline{Q}/Q). The action of the Galois group on T_{β}/p^{n}T_{β} factors through G(F_{n}/Q), where F_{n} is an extension of Q unramified outside pN. As (|D|, pN) = 1, it follows from Čebotarev density theorem for KF_{n}/Q that for any prime number q |̸ NM|D|p there are infinitely many primes ℓ_{j} |̸ NM|D|p inert in K such that

a_{q}(f_{β}) ≡ a_{ℓ_{j}}(f_{β}) (mod p^{n}O_{F (β)} )

Consequently, a_{q}(f_{β}) = 0 for all such q, which is impossible. This contradiction shows that for each β, one can choose ℓ in such a way that a_{ℓ}(f_{β}) ≠ 0. This implies that c_{β} = 0 for all β, hence g = h. Lemma is proved.

(5.8) It follows from Lemma 5.7 that there is a finite subset S ⊂ {m | (m, pN) = 1, r(m) = 0} and constants c_{β,m} ∈ L (β : T ⟶ \overline{Q}, m ∈ S, L a finite extension of Q containing all F (β)) such that

∑_{m∈S} c_{β,m}β′(T_{m}) = \{ \begin{matrix}1 & if β′ = β \\ 0 & if β′ ≠ β \end{matrix}

Put, for each β : T ⟶ \overline{Q},

x_{β} := ∑_{m∈S} c_{β,m}T_{m}x ∈ Z_{f}(Y_{0}(N), H) ⊗_{Q_{p}} L̂

Of course,

Φ_{T} (x_{β}) = z_{β} ⊗ 1 ∈ H_{f}^{1}(H, V_{β}) ⊗_{F (β)} \overline{Q}_{p} ⊂ H_{f}^{1}(H, V) ⊗_{Q_{p}} L̂

Let F be the operator (U_{p} − p^{r−1}σ_{𝔭})^{2}(U_{p} − p^{r−1}σ\overline{𝔭})^{2}. If r_{A}(m) = 0 (for [A] corresponding to σ^{−1}), put

⟨x, T_{m}x^{σ}⟩_{p} = ∑_{v|p}⟨x, T_{m}x^{σ}⟩_{v}

The following statement will be proved in 6.2 (the reader is invited to check that our reasoning is not circular).

Lemma. There is a modular form H_{σ} ∈ \overline{M}_{2r}(Γ_{0}(Np^{∞}); Q_{p}) with the property that

a_{m}(H_{σ}) = ⟨x, T_{m}x^{σ}⟩_{p}|F

for all integers m ≥ 1 such that p | m, (m, N) and r_{A}(m) = 0.

Recall from I.2.3 the operator

e : \overline{M}_{2r}(Γ_{0}(Np^{∞}); Â) ⟶ M_{2r}(Γ_{0}(pN); Â)

Its restriction to S_{2r}(Γ_{0}(Np^{n}); Â) is equal to

e_{n} = lim_{k⟶∞} U_{p}^{cp^{k}}

for some integer c = c(n) ≥ 1. This formula is still valid if c is replaced by its arbitrary mutiple. This implies that

e = lim_{k⟶∞} U_{p}^{b(k)p^{k}}

on \overline{M}_{2r}(Γ_{0}(Np^{∞}); Â) with b(k) = ∏_{n≤k} c(k).

(5.9) For an integer n ≥ 1 and β : T ⟶ \overline{Q}, we define

λ_{n}^{σ}(β) = ∑_{m∈S} c_{β,m}a_{mn}(H_{σ}|e)

Proposition. Let F′ = (U_{p} − σ_{𝔭})(U_{p}σ_{𝔭} − p^{2r−2})(U_{p} − σ\overline{𝔭})(U_{p}σ\overline{𝔭} − p^{2r−2}). If n ≥ 1 is prime to pNM, where M = ∏_{m∈S} m, then

(1) λ_{n}^{σ}(β)|F′ = β(T_{n})λ_{1}^{σ}(β)|F′

(2) If r(n) = 0, then ∑_{β} λ_{n}^{σ}(β)|F′ = a_{n}(H_{σ}|eF′)

Proof. (1) Put

a = T_{n}x_{β} − β(T_{n})x_{β} = ∑_{β} c_{β,m}(T_{m}T_{n}x − β(T_{n})T_{m}x) ∈ H_{f}^{1}(H, V) ⊗_{Q_{p}} L̂

As Φ_{T} (a) = 0, we can apply Prop. 5.6. Our assumptions imply that a and b_{p^{k}}^{σ} have disjoint supports for all k ≥ 0, so we obtain

∑_{m∈S} c_{β,m} lim_{k⟶∞}⟨T_{m}T_{n}x, x^{σ}⟩_{p}|U_{p}^{b(k)p^{k}}F′ = β(T_{n}) ∑_{m∈S} c_{β,m} lim_{k⟶∞}⟨T_{m}x, x^{σ}⟩_{p}|U_{p}^{b(k)p^{k}}F′

Applying F to both sides and using ⟨T_{n∗}x, y⟩_{p} = ⟨x, T_{n}^{∗}y⟩_{p}, we obtain from Lemma 5.8

∑_{m∈S} c_{β,m}a_{mn}(H_{σ}|eF′) = β(T_{n}) ∑_{m∈S} c_{β,m}a_{m}(H_{σ}|eF′),

as was to be shown.

(2) This follows from Prop. 5.6 for a = T_{n}(x − ∑_{β} x_{β}) by the same calculation, once we observe that T_{n}x and b_{p^{k}}^{σ} have disjoint supports for r(n) = 0.

(5.10) Assume that f = f_{β_{0}} (for a fixed β_{0}) is a newform, ordinary at p (i.e. a_{p}(f) = β_{0}(T_{p}) is a p-adic unit). This implies ([Wi, Thm 2]) that the Galois representation V_{β_{0}} is ordinary in the sense of [Gre]. In this case we have a canonical F̂ (β_{0})-linear splitting

F^{r}ε_{B}εH_{dR}^{2r−1}(W ⊗_{Q} Q_{p}/Q_{p})_{β_{0}} ⟶^{⟵} ε_{B}εH_{dR}^{2r−1}(W ⊗_{Q} Q_{p}/Q_{p})_{β_{0}}

We assume that the splittings in 4.1 have been chosen in such a way that they induce this canonical splitting on the β_{0}-component.

Proposition. Under the above assumptions,

(1) λ_{n}^{σ}(β_{0}) = 0 for all n prime to pNM.

(2) L_{f_{0}}(H_{σ}|eF′) = 0

Proof. We first show that (1) implies (2). Indeed, Lemma 5.7 and Prop. 5.9 (for pN instead of N) tell us that

H_{σ}|eF′ = ∑_{β}(λ_{1}^{σ}(β)|F′)f_{β} + h_{σ},

where h_{σ} ∈ S_{2r}(Γ_{0}(pN); Q_{p}) and a_{m}(h_{σ}) = 0 for all m prime to pN. It follows that

L_{f_{0}}(H_{σ}|eF′) = λ_{1}^{σ}(β_{0})L_{f_{0}}(f),

which vanishes iff λ_{1}^{σ}(β_{0}) = 0. Let us now prove (1). We have F′ = F′_{𝔭}F ′ ; \overline{𝔭} with

F′_{𝔭} = (U_{p} − σ_{𝔭})(U_{p}σ_{𝔭} − p^{2r−2}), F ′ ; \overline{𝔭} = (U_{p} − σ\overline{𝔭})(U_{p}σ\overline{𝔭} − p^{2r−2})

Let v be a prime of H dividing 𝔭. For n ≥ 0, define

t_{n}^{σ} = N_{H_{p^{k+2},w}/H_{v,n}}(h_{1,n}^{σ}),

where h_{m,n}^{σ} was defined in 5.6 and H_{v,n} = H_{v}Q_{p,n} is the n-th layer in the cyclotomic Z_{p}-extension of H_{v}. As in 5.6, it follows from Prop. 5.4 that

⟨T_{m}x, x^{σ}⟩_{v}|F′_{𝔭}U_{p}^{n} = ⟨T_{m}x, t_{n}^{σ}⟩_{H_{v,n}}

whenever r(m) = 0, which implies that

⟨x_{β_{0}}, x^{σ}⟩_{v}|F′_{𝔭}U_{p}^{n} = ⟨x_{β_{0}}, t_{n}^{σ}⟩_{H_{v,n}}

The representation V ⊗_{Q_{p}} L̂ of G(\overline{Q}/Q) is a direct sum V′ ⊕ V′′ with

V′ = V_{β_{0}} ⊗_{F (β_{0})} L̂, V′′ = ⨁_{β≠β_{0}} (V_{β} ⊗_{F (β)} L̂)^{⊕m(β)}

The lattice T ⊂ V, defined in 5.5, determines lattices T′ ⊂ V′, T′′ ⊂ V′′. We now want to apply Prop. 1.11 for a_{1} = Φ_{T} (x_{β_{0}}), a_{2} = t_{n}^{σ} over H_{v,n}. The assumptions (A)–(C) of 1.11. are all satisfied, but it is the V′′-component of a_{1,v}, not of a_{2,v}, that vanishes. We have to pass, therefore, to the dual mixed extension E^{∗}(1) and use [Ne 2, 4.10.3]. As V ⟶^{∼} V^{∗}(1) is self-dual (and the same is true also for V′, V′′), we can apply Prop. 1.11 as it stands. Writing

\begin{aligned}p^{d_{n,1}} & = ♯H^{1}(H_{v,n}, T^{′′∗}(1))_{tors} \\ p^{d_{n,2}} & = ♯ (H_{f}^{1}(H_{v,n}, T′)/N_{∞}H_{f}^{1}(H_{v,n}, T′)) , \end{aligned}

we obtain from 5.5.1 and Prop. 1.11

p^{2c+d_{0}+d_{n,1}+d_{n,2}}⟨x_{β_{0}}, t_{n}^{σ}⟩_{H_{v,n}} ∈ ℓ_{v} ∘ N_{H_{v,n}/H_{v}}(H_{v,n}^{∗} ˆ⊗Z_{p}) = p^{n}ℓ_{v}(H_{v}^{∗} ˆ⊗Z_{p})

We shall prove below that d_{n,1}, d_{n,2} are bounded by some integer d. Assuming this, we have

∑_{m∈S} c_{β_{0},m}⟨T_{m}x, x^{σ}⟩_{v}|F′_{𝔭}U_{p}^{n} = ⟨x_{β_{0}}, x^{σ}⟩_{v}|F′_{𝔭}U_{p}^{n} = ⟨x_{β_{0}}, t_{n}^{σ}⟩_{H_{v,n}} ∈ p^{n−2c−d_{0}−2d}ℓ_{v}(H_{v}^{∗} ˆ⊗Z_{p}),

hence

λ_{1}^{σ}(β_{0})|F′_{𝔭} = ∑_{m∈S} c_{β_{0},m}⟨T_{m}x, x^{σ}⟩_{p}|F′_{𝔭}e = lim_{k⟶∞}⟨x_{β_{0}}, x^{σ}⟩_{p}|F′_{𝔭}U_{p}^{b(k)p^{k}} = 0

Combined with Prop. 5.9.1, this implies the statement (1) of the Proposition. It remains to show that d_{n,1}, d_{n,2} are bounded. The Pančiškin condition from 1.11 means that there is an exact sequence of representations of G(\overline{Q}_{p}/Q_{p})

0 ⟶ X ⟶ V′ ⟶ Y ⟶ 0

with suitable properties. Put T_{X} = T′ ∩ X, T_{Y} = T′/T_{X}. According to [Ne 2, 6.9],

p^{d_{n,2}} | ♯H^{0}(H_{v,n}, T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p}) · ♯H^{0}(H_{v,n}, T_{Y} ⊗ Q_{p}/Z_{p})

We also have

p^{d_{n,1}} | ♯H^{0}(H_{v,n}, T^{′′∗}(1) ⊗ Q_{p}/Z_{p})

The boundedness of d_{n,1}, d_{n,2} then follows from the following Lemma.

(5.11) Lemma. H^{0}(H_{v,∞}, Z) = 0 for Z = V^{′′∗}(1), X^{∗}(1), Y.

Proof. If Z is a Hodge-Tate representation of G(\overline{H}_{v}/H_{v}), then

H^{0}(H_{v,∞}, Z) ⊆ ⨁_{j∈Z} H^{0}(H_{v}, Z(j))(−j),

as H_{v,∞} = H_{v}Q_{p,∞}. Assume, moreover, that Z is crystalline (which is true for V by [Fo - Me], as p > 2r − 1 by assumption) and write

D_{v}(Z) = H^{0}(H_{v}, Z ⊗_{Q_{p}} B_{cris})

for the corresponding filtered module. It is equipped with the action of the crystalline Frobenius f and we have

H^{0}(H_{v}, Z(j))(−j) ⊆ D_{v}(Z)^{f=p^{−j}}

By crystalline Weil conjectures [KaN - Me],

D_{v}(V)^{f=p^{−j}} = 0 ∀j ∈ Z

This implies the vanishing of D_{v}(Z)^{f=p^{−j}} for all subquotients Z of V ⊗_{Q_{p}} L̂, which includes V′′, X, Y, hence also V^{′′∗}(1), X^{∗}(1) (as V ⟶^{∼} V^{∗}(1)). Lemma is proved.

6. Proof of Theorem A

(6.1) The assumptions and notation of 0.2 are in place. The newform f corresponds to a homomorphism β_{0} : T ⟶ \overline{Q}. Fix σ ∈ G(H/K) and define F_{σ} ∈ S_{2r}(Γ_{0}(N); Q_{p}) by

F_{σ} = ∑_{β}⟨z_{β}, z_{β}^{σ}⟩f_{β}

For m ≥ 1 prime to N, the m-th Fourier coefficient of F_{σ} is equal to

a_{m}(F_{σ}) = ∑_{β}⟨z_{β}, z_{β}^{σ}⟩β(T_{m}) = ⟨z, T_{m}z^{σ}⟩ = ⟨x, T_{m}x^{σ}⟩ ∈ Q_{p}  (6.1.1)

If, furthermore, r_{A}(m) = 0 (where [A] ∈ Pic(O_{K}) corresponds to σ^{−1}), then we have a decomposition into local heights

a_{m}(F_{σ}) = c_{m}^{σ} + d_{m}^{σ}

with

c_{m}^{σ} = ∑_{v |̸p}⟨x, T_{m}x^{σ}⟩_{v}, d_{m}^{σ} = ∑_{v|p}⟨x, T_{m}x^{σ}⟩_{v}

(6.2) If C is a character of G(H/K), the interpolation property of the p-adic L-function L_{p}(f⊗K, C, W) stated in 0.5 follows from Thm. I.5.10. If ℓ_{K} is anticyclotomic, then the statement of Theorem A becomes 0 = 0, by Cor. I.5.12 and Prop. 4.3. It is sufficient, therefore, to prove Theorem A for a single non-trivial cyclotomic ℓ_{K}; we take ℓ_{K} = ℓ_{Q} ∘ N as in 4.4. By definition, we have ℓ_{K} = log_{p} ∘λ, where λ : G(Q_{∞}/Q) ⟶ 1 + pZ_{p} is given by the cyclotomic character. The corresponding homomorphism

λ̃ : Z_{p}^{∗} ⟶ 1 + pZ_{p},

defined in I.6.2, is equal to λ̃(x) = ⟨x⟩^{−1}, where ⟨x⟩ = xω^{−1}(x) for the Teichmüller character ω : Z_{p}^{∗} ⟶ μ_{p−1}(Q_{p}). According to I.6.2.1, we have

L_{p}(f ⊗ K, C)(λ^{s}) = (−1)^{r}H_{p}(f)(1 − C(\frac{D}{C} )⟨C⟩^{2s})^{−1} ∑_{[A]∈Pic(O_{K})} C(A)^{−1}L_{f_{0}} [∫_{Z_{p}^{∗}} ⟨x⟩^{−s} dΨ̃_{A}^{C}]  (6.2.1)

If σ^{−1} ∈ G(H/K) corresponds to the ideal class [A], put

G_{σ} = (−1)^{r} ∫_{Z_{p}^{∗}} log_{p} dΨ̃_{A} ∈ \overline{M}_{2r}(Γ_{0}(Np^{∞}); Q_{p})

It follows from 6.2.1 that

L′_{p}(f ⊗ K, C, 11) = −H_{p}(f) ∑_{σ∈G(H/K)} C(σ)L_{f_{0}}(G_{σ})  (6.2.2)

Combining Cor. I.6.6, I.6.7.1 and Prop. 4.17, we obtain the main identity

c_{m}^{σ}|(U_{p} − p^{r−1}σ_{𝔭})^{2}(U_{p} − p^{r−1}σ\overline{𝔭})^{2} = u^{2}(4|D|)^{r−1}a_{m}(G_{σ})|(U_{p}^{4} − p^{2r−2}U_{p}^{2})  (6.2.3)

valid for all m ≥ 1 satisfying p | m, (m, N) = 1, r_{A}(m) = 0. The modular form

H_{σ} = F_{σ}|(U_{p} − p^{r−1}σ_{𝔭})^{2}(U_{p} − p^{r−1}σ\overline{𝔭})^{2} − u^{2}(4|D|)^{r−1}G_{σ}|(U_{p}^{4} − p^{2r−2}U_{p}^{2})

is an element of \overline{M}_{2r}(Γ_{0}(Np^{∞}); Q_{p}). According to (6.2.3),

d_{m}^{σ}|F = ∑_{v|p}⟨x, T_{m}x^{σ}⟩_{v}|F = a_{m}(H_{σ})

provided p | m, (m, N) = 1, r_{A}(m) = 0. This proves Lemma 5.8. (6.3) According to Prop. 5.10,

L_{f_{0}}(H_{σ}|F′) = ℓ_{f_{0}}(H_{σ}|F′e) = 0

Consequently,

L_{f_{0}}(F_{σ}|FF′) = u^{2}(4|D|)^{r−1}L_{f_{0}}(G_{σ}|(U_{p}^{4} − p^{2r−2}U_{p}^{2})F′)  (6.3.1)

Multiplying 6.3.1 by C(σ) and summing over σ ∈ G(H/K), we obtain (using 6.2.2 and the fact that L_{f_{0}} ∘U_{p} = α_{p}(f)L_{f_{0}})

(fñ) · L_{f_{0}}(f)h⟨z_{β,C}, z_{β,\overline{C}}⟩ ∏_{𝔭|p} (1 − \frac{C(𝔭)p^{r−1}}{α_{p}(f)} ) (1 − \frac{C(\overline{𝔭})p^{r−1}}{α_{p}(f)} ) = −(fñ) · u^{2}(4|D|)^{r−1}H_{p}(f)^{−1}×

× (1 − \frac{p^{2r−2}}{α_{p}(f)^{2}} ) L′_{p}(f ⊗ K, C, 11) (6.3.2)

with

(fñ) = (α_{p}(f) − C(𝔭))(α_{p}(f)C(𝔭) − p^{2r−2})(α_{p}(f) − C(\overline{𝔭}))(α_{p}(f)C(\overline{𝔭}) − p^{2r−2})

As neither α_{p}(f) nor p^{2r−2}α_{p}(f)^{−1} are roots of unity (by Weil’s conjectures, [De 1], [De 2]), we can divide 6.3.2 by (fñ). Using I.2.4.2 and the definition of H_{p}(f), we finally obtain

L′_{p}(f ⊗ K, C, 11) = − ∏_{𝔭|p} (1 − \frac{C(𝔭)p^{r−1}}{α_{p}(f)} ) (1 − \frac{C(\overline{𝔭})p^{r−1}}{α_{p}(f)} ) \frac{h⟨z_{β,C}, z_{β,\overline{C}}⟩}{u^{2}(4|D|)^{r−1}}

Theorem A is proved. (6.4) It seems that Thm. 1.3 of [PR 2] should also contain a minus sign, as our Thm. A does. Apparently the sign is missed in [PR 2, 2.3], where one should pass from λ to λ̃ through the reciprocity map

Z_{p}^{∗} ↪ Q_{p}^{∗} ⟶ A_{Q}^{∗}/Q^{∗} ⟶ G(Q(μ_{p^{∞}})/Q),

not by the cyclotomic character

χ_{cycl} : G(Q(μ_{p^{∞}})/Q) ⟶ Z_{p}^{∗}

There is also a sign error in the formulas for the local maps ℓ_{ν,v}(x) in [PR 2, p. 501], which disappears on the next page. If we are unwilling to renormalize the p-adic L-function, the only way to get rid of the sign in Thm. A seems to be a sign change in the definition of the p-adic height in [Ne 2]. This would also require a sign change in [ScP 1], which perhaps would not be such a bad idea, given the main comparison result of [ScP 2].

(6.5) The proof of Theorem 0.8 in [Ne 1] works under the assumptions p > 2r − 1, (|D|, N) = 1 and p |̸ 2N. In [Ne 1], it was required that p |̸ 2N(2r − 2)!φ(N), but a better choice of the lattice J makes the factor (2r − 2)!φ(N) superfluous. One should take for J the image of

ε_{B}εH_{et}^{1}(\overline{M}_{N} ⊗_{Q} \overline{Q}, j_{∗}F)

in

ε_{B}εH_{et}^{1}(\overline{M}_{N} ⊗_{Q} \overline{Q}, j_{∗}F ⊗_{Z_{p}} Q_{p})

The assumption p > 2r − 1 is explained in 0.13 and should be eliminated by a further progress in p-adic Hodge theory. Note that Theorem 13.1 of [Ne 1] says that (in our language)

F̂ (f) · Im(Φ_{K,f}) = F̂ (f) · z_{K,f} = H_{f}^{1}(K, V_{f})

However, Φ_{K,f} commutes with the action of T and T ⊗ Q_{p} ⟶ F̂ (f) is surjective, hence

F̂ (f) · Im(Φ_{K,f}) = Q_{p} · Im(Φ_{K,f})

(6.6) The formula I.5.13.4 reads in our case as follows

L_{p}(f ⊗ K, ⟨ ⟩^{s} ∘ N) = \frac{⟨N⟩^{2s}(−1)^{r−1}|D|^{1/2}}{2^{2r−1}(2π)^{2}⟨f, f⟩_{N}} L_{p}^{MTT}(f ⊗ K, x^{r−1}⟨ ⟩^{s}),

which proves 0.9.1.

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