Aim: Fix π and let χ vary P -adically. We want to show that, generically,

ord_{s=1/2} L(s, π × π(χ)) = \{ \begin{matrix}0 & if & ϵ = +1 \\ 1 & if & ϵ = −1 \end{matrix}

With:

• π is an irreducible automorphic representation of GL_{2}(A_{F}), where

• F is a totally real number field;

• K is a totally imaginary quadratic extension of F and

• χ is a Grossencharacter of K, inducing an automorphic representation π(χ) of GL_{2}(A_{F});

• P is a prime of F.

Notations and Assumptions

• π: cuspidal, weight (2, ⋯, 2), level N, and central char. ω : F^{×}\A_{F}^{×} → C^{×}.

• χ is a ring class character, i.e.: there exists c ⊂ O_{F} such that χ factors through

χ : K^{×}\A_{K}^{×}/K_{∞}^{×}Ô_{c}^{×} ≃ Pic (O_{c}) → C^{×}. where O_{c} := O_{F} + cO_{K} . The conductor c(χ) of χ is the largest such c.

• By CFT, we may view χ as a character on Gal(K[c]/K), where K[c] is the ring class field of conductor c of K.

• D is the support of disc (K/F). Write

D = (P)D′ and N = P^{δ}N′

with (D′, P) = 1 = (N′, P). We require:

(N′, D′) = 1.

We need:

ω · χ|_{A_{F}^{×}} = 1 on A_{F}^{×}.

Then:

• L(s, π × π(χ)) is entire, and

• L(s, π × π(χ)) has a F.E. s ↔ 1 − s,

• This F.E. has a sign ϵ ∈ {±1}. Moreover, ω is an unramified character of F (because χ is a ring class character of K).

(Thm) For all n ≫ 0, there exists a ring class character χ with c(χ) = P^{n} and ω·χ|_{A_{F}^{×}} = 1 such that

ord_{s=1/2} L(s, π × π(χ)) = \{ \begin{matrix}0 & if & ϵ = +1 \\ 1 & if & ϵ = −1 \end{matrix}

From now on,

Weonlyconsiderthe ϵ = −1

case.

Thanks to the work of S. Zhang (Gross-Zagier formulas for GL_{2}, II), we now have to show:

Thm For all n ≫ 0 and x ∈ CM(P^{n}), there exists a χ with c(χ) = P^{n} and ω · χ|_{A_{F}^{×}} = 1 such that

e_{χ} · α(x) ≠ 0 in A ⊗ C.

Here:

• M is a Shimura curve, A is a quotient of J = Pic^{0}(M) and α : M ⇝ A is an F -morphism.

• CM: Points with CM by K in M and CM(c) ⊂ M(K[c]): those of conductor c.

• G(n) = Gal(K[P^{n}]/K) and

e_{χ} = \frac{1}{|G(n)|} _{σ∈G(n)}^{∑} \overline{χ}(σ) · σ.

Formula for ϵ.

When ω · χ|_{A_{F}^{×}} = 1,

ϵ = ∏ ϵ_{v} = (−1)^{|S|}
and S = {v; ϵ_{v} ≠ η_{v} · ω_{v}(−1)}.

Under our assumption that N and D are relatively prime away from P, and provided that χ is sufficently ramified at P, we find:

S = \{v | \begin{matrix}v | ∞ or v ∤ P ∞ is inert in K \\ and such that v(N) ≡ 1 mod 2 \end{matrix} \}

Therefore

ϵ = −1 ⇔ |S| ≡ 1 mod 2.

The Hodge “embedding”.

There exists δ ∈ Pic (M) ⊗ Q which has degree 1 on each geometrical connected component of M. This δ defines an element ι : M ⇝ J of Mor_{F} (M, J) ⊗ Q given by

x ↦ (x − δ_{c})

where δ_{c} is the restriction of δ to the geometrical connected component c of x ∈ M(C).

We put

α : M ⇝^{ι} J ⟶^{π} A

so that α belongs to Mor_{F} (M, A) ⊗ Q.

A Filtration 1 ⊂ G_{2} ⊂ G_{1} ⊂ G_{0} ⊂ G(∞).

• G(∞) = \underleftarrow{lim} G(n) = Gal(K[P^{∞}]/K) and G_{0} = G(∞)_{torsion} .

• G(∞)^{rat} =< Frob _{Q}; Q ∤ P >⊂ G(∞) and G_{1} = G(∞)_{torsion}^{rat} .

• G_{2} = rec_{K}(F̂^{×}) ⊂ G(∞).

Then G_{2} ⊂ G_{1} ⊂ G_{0} and

• G_{0} is finite, G(∞) ≃ G_{0} × Z_{p}^{[F_{P} ,Q_{p}]}.

• G_{2} ≃ Pic (O_{F}).

• G_{1}/G_{2} is an F_{2}-vector space with basis

{σ_{q} = Frob _{Q}; Q^{2} = qO_{K}, q | D′}.

Fix χ_{0} : G_{0} → C^{×} such that χ_{0} · ω = 1 on A_{F}^{×}.

Thm For all n ≫ 0 and x ∈ CM(P^{n}), there exists a χ with c(χ) = P^{n} and χ |_{G_{0}}= χ_{0} such that

e_{χ} · α(x) ≠ 0 in A ⊗ C.

Proof : Put e_{χ_{0},n} = ∑ e_{χ} where χ runs through the above characters (i.e. those inducing χ_{0} on G_{0} and such that c(χ) = P^{n}). We want:

Thm For all n ≫ 0 and x ∈ CM(P^{n}),

e_{χ_{0},n} · α(x) ≠ 0 in A ⊗ C.

Lemma 1 In the group ring C[G(n)],

e_{χ_{0},n} = \frac{1}{q |G_{0}|} _{σ∈G_{0}}^{∑} \overline{χ}_{0}(σ)σ · (q − Tr_{Z(n)})

with q = |O_{F} /P | and

Z(n) = Gal(K[P^{n}]/K[P^{n−1}]).

Lemma 2 ∃M′, J′ = Pic^{0}M′, π′ : J′ ↠ A and

x ∈ CM(P^{n}) ↦ x′ ∈ CM′(P^{n}) (n ≥ 2) such that if α′ = π′ ∘ ι′ : M′ ⇝ A,

(q − Tr_{Z(n)})α(x) = α′(x′) in A ⊗ C. We thus want: Thm For all n ≫ 0 and x′ ∈ CM′(P^{n}),

_{σ∈G_{0}}^{∑} \overline{χ}_{0}(σ) · σα′(x′) ≠ 0 in A ⊗ C.

Put χ = χ_{0}, M = M′, J = J′, α = α′… Put C_{i} = Z[χ(G_{i})] so that

Z ⊂ C_{2} ⊂ C_{1} ⊂ C_{0} ⊂ C

We want: Thm For all but finitely many x ∈ CM(P^{∞}),

(∗) _{σ∈G_{0}}^{∑} \overline{χ}(σ)σα(x) ≠ tors. in A ⊗_{Z} C_{0}

Proposition 3 (∗) ⇔

⇔ _{σ∈G_{0}/G_{2}}^{∑} \overline{χ}(σ)σα(x) ≠ tors. in A ⊗_{C_{2}} C_{0}
⇔ _{σ∈G_{0}/G_{1}}^{∑} \overline{χ}(σ)σα_{1}(x_{1}) ≠ tors. in A_{1} ⊗_{C_{1}} C_{0}

Action of G_{2}.

The center F̂^{×} of G(A_{f}) acts on M, J, and A through its quotient

F̂^{×}/F^{×}Ô_{F}^{×} ≃ Pic (O_{F}) (≃ G_{2})

For a CM point x = [g] ∈ T (Q)\G(A_{f})/H, Galois action: σ ∈ Gal _{K}^{ab} acts by

σ · x = [λg]

where σ = rec_{K}(λ) with λ ∈ K̂^{×}. Automorphic action: θ ∈ Pic (O_{F}) acts by

θ(x) = [gλ]

where θ = [λ] with λ ∈ F̂^{×}. For σ ∈ G_{2} ↔ θ ∈ Pic (O_{F}), λ ∈ F̂ and

σ · x = [λg] = [gλ] = θ(x).

Action of G_{1}/G_{2}.

Recall that G_{1}/G_{2} = {σ_{D}; D | D′} where σ_{D} = _{q|D}^{∏} σ_{q}, σ_{q} = Frob _{Q}, Q^{2} = qO_{K}.

Lemma 4 ∃{deg _{D} : M_{1} → M; D | D′} and

x ∈ CM(P^{n}) ↦ x_{1} ∈ CM_{1}(P^{n})

such that for all D | D and x ∈ CM(P^{n}), σ_{D} · x = deg _{D}(x_{1}) (in M.)

Take A_{1} = A ⊗_{C_{2}} C_{1} and

α_{1} : M_{1} ⇝^{ι_{1}} J_{1} ↠^{π_{1}} A_{1}

where π_{1} : J_{1} ↠ A_{1} is defined by

\begin{matrix}J_{1} & → & J^{\{D|D\}} & →^{π} & A^{\{D|D\}} & → & A_{1} \\ x & ↦ & (deg _{D}(x)) & & (x_{D}) & ↦ & ∑ \overline{χ}(σ_{D})x_{D} \end{matrix}

Fix a set R ⊂ G_{0} of representatives for G_{0}/G_{1}. We now want:

Thm For all but finitely many x ∈ CM_{1}(P^{∞})

_{σ∈R}^{∑} \overline{χ}(σ) · σα_{1}(x) ≠ torsion in A_{1} ⊗_{C_{1}} C_{0}.

Lemma 5 For any abelian variety B,

B(K[P^{∞}])_{torsion} is finite.

Let E ⊂ CM_{1}(P^{∞}) be an infinite subset of counterexample to our theorem. Then:

x ↦ _{σ∈R}^{∑} \overline{χ}(σ) · σα_{1}(x) ∈ A_{1} ⊗_{C_{1}} C_{0}

takes finitely many values on E, and we want to get a contradiction out of this.

Replace M_{1}, J_{1}, A_{1}… by M, J, A… First Proof: using Andre-Oort (Edixhoven + Yafaev)

Proposition 6 For any infinite subset E of CM(P^{∞}), the Zariski closure of δ(E) in M^{R}/C contains a connected component.

If Φ ∘ δ(E) is finite, Φ is constant on some geometrical connected component of M^{R}. Then α : M → A should be constant on some geometrical connected component of M. Being defined over F, α should then be constant on M (because M is connected as an F -curve), and π would be trivial, a contradiction.

Sketch of the proof of Proposition 6. If some component of the Zariski closure of

{(x, σx); x ∈ E} ⊂ M^{2}

is a curve, then

• This curve is a connected component of some Hecke correspondance T_{M}, and

• σ belongs to the subgroup

< Frob _{Q}; Q | M; Q ∤ P >

of G(∞)^{rat}.

• If also σ ∈ G_{0} = G(∞)_{torsion} , we obtain σ ∈ G(∞)_{torsion}^{rat} = G_{1}.

Second Proof: using Ratner’s Theorem

1- We want: for all n ≫ 0 and x ∈ CM(P^{n}),

_{σ∈R}^{∑} \overline{χ}(σ) · σα(x) = Φ ∘ δ(x)

has a large Galois orbit. This orbit equals

G(∞) · Φ ∘ δ(x) = Φ ∘ δ(G(∞) · x).

2- Reducing everything at some place v of K[P^{∞}] and using red ∘ Φ = Φ ∘ red , we want

RED (G(∞) · x) ⊂ M^{R}(k)

to be large when n ≫ 0, where

RED = red ∘ δ : CM(P^{∞}) → M^{R}(k)

maps x to (red (σx))_{σ∈R} and k is the residue field of v.

3- Choose v above some prime ℓ of F which is inert in K and not in N. Then k is finite and

red (CM) ⊂ M^{ss}(k) = {supersingular points }.

Is it true that

RED : CM(P^{∞}) → M^{ss}(k)^{R}

is surjective? The answer is no. 4- Let c : M → M be the Stein factorisation of M → Spec (O_{F_{v}}), put C = c^{R}

C : M^{ss}(k)^{R} → M(k)^{R}.

Then C ∘ RED : CM(P^{∞}) → M(k)^{R} is not surjective, but: Thm For any x ∈ CM(P^{n}) with n ≫ 0, RED (G(∞) · x) = C^{−1} (C ∘ RED (G(∞) · x) ) .

5- Carayol gives an adelic description of M^{ss}(k), M(k) and c : M^{ss}(k) → M(k). The strong approximation theorem at P then provides a P - adic description of the fibers of c : M^{ss}(k) → M(k). On the other hand, one checks that CM(P^{∞}) may be covered by finitely many “strong P -isogeny classes”, and the latter each have a P -adic parametrisation by B_{P}^{1} ≃ SL_{2}(F_{P}).

Altogether, we find that for a strong P -isogeny class H ⊂ CM(P^{∞}) and z = C ∘ RED (H), the map

H → C^{−1}(z) ⊂ M^{ss}(k)^{R}

looks like

SL_{2}(F_{P}) ⟶^{Δ} _{σ∈R}^{∏} Γ_{σ}\SL_{2}(F_{P})/V

where Δ is the diagonal, V is a compact open subgroup of SL_{2}(F_{P}) and the Γ_{σ}’s are noncommensurable cocompact lattices in SL_{2}(F_{P}).

6- Let us already show that this map is surjective. Put G = SL_{2}(F_{P}) and Γ = ∏_{σ∈R} Γ_{σ}.

Ratner’s theorem (Orbit closure) There exists a closed subgroup Σ of G^{R} such that

\overline{Γ · Δ(G)} = Γ · Σ in G^{R}.

We may furthermore assume that Δ(G) ⊂ Σ: then Σ is essentially a “product of diagonals”. As Γ · Σ is closed, Γ ∩ Σ is a cocompact lattice in Σ. The non commensurability of the Γ_{σ}’s then implies Σ = G^{R}, and Γ · Δ(G) is dense in G^{R}. A fortiori, Γ · Δ(G) · V^{R} = G^{R}.

7- To obtain our theorem, we need a better decomposition of CM(P^{∞}): a finite union of

∪_{n≥0}G(∞) · U(κ_{n})

where t ↦ U(t) is “one parameter unipotent familly of CM points”, κ is a compact open subgroup of O_{F}^{×} and κ_{n} = π^{−n}κ, with π a local uniformiser at P.

Another theorem of M. Ratner tells us that

\frac{1}{λ(κ_{n})} ∫_{κ_{n}} f ∘ Δ(σU(t))dt → ∫ fdμ_{σ}

for any continuous function f on Γ\G^{R}. We show that for almost all σ ∈ G(∞), μ_{σ} = μ is the unique G^{R} invariant measure on Γ\G^{R}. Fubini’s theorem then allows us to analyse the asymptotic behavior of

∫_{G(∞)} f ∘ RED (g · x)dg

for x ∈ CM(P^{n}) with n → ∞ and f any function on the finite set M^{ss}(k)^{R}.