Values of L-functions and p-adic cohomology

Jan Nekovář

In this article we give a survey of two relatively recent developments in number theory: (1) the method of “Euler systems”; (2) new ideas and techniques coming from p-adic cohomology (or, rather, p-adic Hodge theory).

The common thread underlying these two themes is a relationship (still largely conjectural) between

(L) Values of L-functions (A) Arithmetic invariants

Well-known prototypes of such relationships are the Iwasawa Main Conjecture or the conjecture of Birch and Swinnerton-Dyer. In this context, Euler systems have been used to prove theorems about (A) ⟷ (L) relations, while results about p-adic cohomology suggested formulation of precise conjectures about (A) ⟷ (L) relations in a general ‘motivic’ setting.

I would like to thank R. Kučera, K. Rubin and the referee for helpful comments on the first version of this paper.

1. A brief history

What we now call “Euler systems” is a new descent method developed in pioneering works of F. Thaine, K. Rubin and V.A. Kolyvagin. Their most spectacular results are summed up in the following table:

1986 Thaine  – new bounds for ideal class groups of cyclotomic

fields [77]

Rubin  – finiteness of \underline{III} for some elliptic curves with

complex multiplication [57]

1987 Kolyvagin – finiteness of \underline{III} for some modular elliptic curves

[35],[36]

1989 Kolyvagin – determines the structure of \underline{III} in some cases [38]

– gives a general treatment of “Euler systems” used by Rubin and himself [37]

Rubin  – new proof of the Iwasawa Main Conjecture over Q

[58] (much simpler than the proof given by Mazur and Wiles [43])

1990 Rubin  – proof of the Iwasawa Main Conjecture over ima-

ginary quadratic fields [59]

Kolyvagin – determines the structure of Selmer groups in some

cases [39] Since then, the method has been applied in other situations ([6],[9],[16], [34],[40],[41],[42],[45],[61],[69]) and has gradually acquired more cohomological flavour. As for p-adic Hodge theory, it owes its origin to the works of Tate [74] and Grothendieck [26]. Today we view it as a theory of p-adic periods, comparing étale and de Rham comology of algebraic varieties over p-adic numbers. The current state of the art is described in [27] and is a result of a collective effort by many people. In their seminal paper [7], Bloch and Kato used p-adic Hodge theory to formulate a conjecture about precise values of L-functions of (pure) motives, improving thus on previous conjectures, which left an undetermined rational factor:

1978 Deligne  – conjecture about critical values of L-functions,

up to a rational factor [11]

1983 Beilinson  – conjectures about all special values of motivic

L-functions, up to a rational factor [1],[2],[3],[4]

1988 Bloch, Kato – precise conjecture about special values [7]
1991 Kato  – formulation of the Iwasawa Main Conjecture for

motivic L-functions [33]

Fontaine,  – generalization and simplification of the conjecture
Perrin-Riou of Bloch and Kato [18],[19]

One of the central objects of the theory of Bloch–Kato are Selmer groups associated to general p-adic Galois representations. As we shall see below in Sec. 9, that is where cohomological Euler systems lie. In general, an Euler system is formed by elements “explaining” values of a given L-function in a tower of abelian extensions of a base number field. These elements are not independent, but satisfy several compatibility relations. These relations can be plugged into a descent machinery due to Kolyvagin and Rubin, giving a non-trivial information about relevant arithmetic invariants (A). In all known cases the Euler system can be constructed with no apriori reference to L-functions. The relation to L-values must then be established separately. Symbolically, an Euler system (E) is an “incarnation” of L-values (L), and provides a sought-for link between (A) and (L):

(L) ⟵ (E) ⟶ (A)

For a fixed prime number p, the “p-part” of (A) is controlled by the Euler system restricted to (some) abelian extensions of degrees divisible by a sufficiently high power of p, but ramified at various primes l ≠ p – this is a basic difference from Iwasawa theory (cf., however, [69]).

Values of L-functions 3

2. Euler systems – classical examples

(E1) E : “cyclotomic units” (or, rather, cyclotomic numbers) in real abelian extensions of Q ([37],[58]) L : L′(0, χ) for even Dirichlet characters χ over Q A : ideal class groups of real abelian extensions of Q

Fix, for every integer n > 1, a primitive n-th root of unity ζ_{n} ∈ μ_{n} in such a way that ζ_{mn}^{m} = ζ_{n} (e.g. ζ_{n} = exp(2πi/n)). Then, for n > 1,

u_{n} := (1 − ζ_{n})(1 − ζ_{n}^{−1})

lies in Q(μ_{n})^{+} (the maximal real subfield of Q(μ_{n})) and is a unit if n is divisible by two different primes, or a p-unit if n = p^{r}. Basic relations: let l be a prime number not dividing n > 1. Then

\begin{aligned}N_{l} u_{nl} & = (1 − F r_{l})u_{n} \\ u_{nl} & ≡ F r_{l}(u_{n}) (mod λ) \end{aligned}  (2.1)

Here N_{l} denotes the norm map in the extension Q(μ_{nl})^{+}/Q(μ_{n})^{+}, F r_{l} the geometric Frobenius at l (the inverse of which raises the roots of unity in μ_{n} to the l-th power) and λ is any prime in Q(μ_{nl})^{+} above l.

(E2) E : “elliptic units” in abelian extensions of an imaginary quadratic field K = Q(\sqrt{−D}) ([59]) L : L′(0, χ) for abelian Artin characters χ over K A : ideal class groups of abelian extensions of K

There exist various versions of elliptic units, which satisfy relations similar to (2.1) – see [10, II.2],[21],[59].

(E3) E : “Stark units” in abelian extensions of a number field K with at most one complex place L : L′(0, χ) for abelian extensions of K A : ideal class groups of (some) abelian extensions of K

Stark units ([61],[72],[76]) conjecturally appear whenever we are given an abelian extension F/K and a character χ : G(F/K) ⟶ C^{∗} such that the L-function L(s, χ) has zero of order 1 at s = 0. This can happen only if all infinite places of K except for one, say v_{0}, are real and v_{0} splits completely in F/K. Stark [72] conjectures that in this case there is a number ϵ ∈ F^{∗} (in fact a unit, unless K = Q or Q(\sqrt{−D}) and the conductor of F/K is a prime power) such that for each character χ of G(F/K) we have

L′_{S}(0, χ) = −\frac{1}{m} ∑_{σ∈G(F/K)} χ(σ) log |ϵ^{σ}|_{w_{0}}  (2.2)

for a suitable integer m. Here w_{0} is a fixed place of F above v_{0} and L_{S} is the incomplete L-function with Euler factors at primes dividing the conductor of F/K deleted. If χ is faithful, then L_{S}(s, χ) = L(s, χ) is the complete L-function. In this case, L(s, χ) has a simple zero at s = 0 iff all infinite places v ≠ v_{0} ramify in F/K.

(E4) E : “Gauss sums” ([37],[60],[66])

L : L(0, χ) for odd Dirichlet characters over Q

A : odd part of ideal class groups of abelian extensions of Q A typical Gauss sum is obtained as follows: let K = Q(μ_{m}) and let ℘ be a prime of K above p |̸ m. Write ω_{℘} for the character (O_{K}/℘O_{K})^{∗} ⟶ μ_{m} such that

ω_{℘}(a) ≡ a^{(N℘−1)/m} (mod ℘) a ∈ O_{K}

and let ε : O_{K}/℘O_{K} ⟶ μ_{p} be a non-trivial additive character factoring through the trace to Z/pZ. The Gauss sum

S = ∑_{a∈(O_{K}/℘O_{K})^{∗}} ω_{℘}(a)^{−1}ε(a)

lies in K(μ_{p})^{∗} and is a unit outside of primes above p. If we denote by σ_{b} (for an integer b prime to m) the automorphism of K(μ_{p}) which leaves μ_{p} fixed and raises elements of μ_{m} to the b-th power, then S^{b−σ_{b}} lies in K^{∗} (in fact in O_{K}[1/p]^{∗}). Varying m and p, Kolyvagin [37] and Rubin [60] define an Euler system formed by certain products of Gauss sums; their definitions are notationally quite involved, one has to keep track of a lot of different indices, so they will not be reproduced here.

(E5) E : “Heegner points” on Jacobians of modular (or Shimura) curves ([6],[9],[23],[24],[25],[35],[36],[37],[38],[39],[40],[41]) defined over ring class fields of an imaginary quadratic field K = Q(\sqrt{−D}) L : L′(f ⊗ K, 1) for modular forms of weight 2 A : rational points and Tate-Šafarevič groups of the Jacobians of modular (Shimura) curves over ring class fields

For an integer N ≥ 1, the modular curve Y_{0}(N) classifies, in a suitable sense, isogenies λ : E ⟶ E′ between elliptic curves E, E′ with Ker(λ) ⟶^{∼} Z/NZ. For example, its complex points correspond bijectively to isogenies between elliptic curves defined over complex numbers. Y_{0}(N) has a natural compactification X_{0}(N) and both curves are defined over Q. For a fixed order O in an imaginary quadratic field K, a Heegner point corresponding to O is any point on X_{0}(N) represented by an isogeny λ between two curves with complex multiplication by O, i.e. End(E) = End(E′) = O. The conductor of the Heegner point is, by definition, the conductor of O, equal to [O_{K} : O]. Suppose that N is prime to the discriminant of K. Then a Heegner point of conductor n on X_{0}(N) exists iff the following two conditions are satisfied:

Each prime q|N splits completely in K  (2.3a)
N = N_{0}d^{2}, n = n_{0}d, (N_{0}, n_{0}) = 1  (2.3b)

In fact, such a point is defined over K_{n} – the ring class field of K of conductor n. Suppose, for simplicity, that n is prime to N. Choosing an ideal A ⊂ O_{K} with O_{K}/A ≃ Z/NZ (such ideals exist, if (2.3a) is satisfied) and writing O_{n} = Z + nO_{K} for the order of conductor n, then the isogeny

C/O_{n} ⟶ C/(O_{n} ∩ A)^{−1}

represents a Heegner point P_{n} on X_{0}(N), defined over K_{n}. The divisor (P_{n}) − (∞) then defines a point x_{n} on the Jacobian J_{0}(N) of X_{0}(N), also defined over K_{n}. Recall that K_{1} is nothing else than the Hilbert class field of K and that G(K_{n}/K) is canonically isomorphic to Pic(O_{n}). From the exact sequence

\frac{(Z/nZ)^{∗}}{Z^{∗}} ⟶ \frac{(O_{K}/nO_{K})^{∗}}{O_{K}^{∗}} ⟶ Pic(O_{n}) ⟶ Pic(O_{K}) ⟶ 0

we see that

G(K_{n}/K_{1}) ⟶^{∼} \frac{(O_{K}/nO_{K})^{∗}}{O_{K}^{∗}(Z/nZ)^{∗}}

Basic relations: let l be a prime not dividing n and (nl, ND) = 1. Then

u_{l} N_{l}x_{nl} = \{ \begin{matrix}T_{l}x_{n} & if l is inert in K \\ T_{l}x_{n} − σ(x_{n}) − σ^{−1}(x_{n}) & if l splits in K \end{matrix}
x_{nl} ≡ F r_{l}(x_{n}) (mod λ)

(2.4)

Here T_{l} is the l-th Hecke operator, u_{l} the order of the image of O_{K}^{∗}/Z^{∗} in (O_{K}/lO_{K})^{∗}/(Z/lZ)^{∗}, σ ∈ G(K_{n}/K) corresponds by class field theory to one of the factors of l in K and λ is any prime above l in K_{nl}.

3. Relations (E) ⟷ (L)

Let us make explicit how various Euler systems from the previous section “compute” values of L-functions. (E1) Let χ : (Z/NZ)^{∗} ⟶ C^{∗} be an even (χ(−1) = 1) Dirichlet character. Then the (incomplete) L-function

L(s, χ) = ∑_{{n=1 \atop (n,N)=1}}^{∞} χ(n) n^{−s}

has a first order zero at s = 0 and its derivative is given by

L′(0, χ) = −\frac{1}{2} ∑_{σ} χ(σ) log |ϵ^{σ}|  (3.1)

where ϵ = (1 − ζ_{N})(1 − ζ_{N}^{−1}), ζ_{N} = exp(2πi/N) and σ runs through G(Q(μ_{N})^{+}/Q) ⟶^{∼} (Z/NZ)^{∗}/{±1}. This follows from the following elementary fact:

\frac{d}{ds} ∑_{{n∈Z \atop n≡a (mod N)}} |n|^{−s} |_{s=0}= − log |1 − ζ_{N}^{a}|.

(E2) If K is an imaginary quadratic field and χ any ray class character over K, say, of conductor f, then there is a similar formula

L′(0, χ) = −\frac{1}{m} ∑ χ(σ) log |ϵ^{σ}|  (3.2)

for a suitable integer m (see [10, II.5.1]) with σ running through G(K(f)/K) (K(f) denoting the ray class field of conductor f over K) and ϵ a suitable “elliptic unit” in K(f).

(E3) The formula (2.2), which generalizes (3.2), is one of the defining properties of the Stark units.

(E4) The classical factorization formula for the Gauss sums ([78, ch. 6]) relates them to the values of Dirichlet L-functions at s = 0. In the notation of the previous section, fix a non-trivial character χ of G = G(K/Q) with values in a suitable ring R in which [K : Q] is invertible. Then the factorization of the χ-component of S^{b−σ_{b}} is given by the formula

(S^{b−σ_{b}})^{χ} = (χ(b) − b)L(0, \overline{χ})[℘]^{χ}

Here the equality takes place in the free R-module generated by the primes of K above p and

a^{χ} = (\frac{1}{♯(G)} ∑_{σ∈G} χ(σ)^{−1} σ) a

for each element a of an arbitrary R[G]-module.

(E5) In this case, the result is due to Gross–Zagier [25]. Let, as before, K be an imaginary quadratic field of discriminant −D, let f be a newform of weight 2 on Γ_{0}(N). Assume that all primes q|N split completely in K. If n is an integer prime to N, then there is a Heegner point x_{n} ∈ J_{0}(N)(K_{n}) on the Jacobian of X_{0}(N), defined over the ring class field of conductor n. Associated to f, there is a quotient J_{0}(N) ⟶ A of J_{0}(N) (determined up to an isogeny), also defined over Q. Let y_{n} be the image of x_{n} in A(K_{n}). For each ring class character χ : G(K_{n}/K) ⟶ C^{∗} we can take the χ-component of y_{n}:

y_{n}^{χ} ∈ (A(K_{n}) ⊗ C)^{χ}

The main identity expresses the derivative of the L-series of A over K, twisted by the character χ, in terms of y_{n}^{χ}:

\frac{L′(A ⊗_{Q} K, χ, 1)}{Ω_{A,K}} ≐ ⟨y_{n}^{χ}, y_{n}^{χ}⟩  (3.3)

Here Ω_{A,K} denotes certain period associated to A and ⟨ , ⟩ is (the hermitian extension of) the Néron-Tate height pairing on A(K_{n}) ⊗ C (the sign ≐ means equality up to an elementary factor which can be made explicit). The formula (3.3) has been proved so far only for n = 1 ([25]). In a special case when the form f has rational coefficients, A = E is an elliptic curve. If we take n = 1, χ = 1, (3.3) reads as

\frac{L′(E/K, 1)}{Ω_{E/K}} ≐ ⟨y, y⟩  (3.4)

for y = T r_{K_{1}/K}y_{1} ∈ E(K).

4. Classical relations (A) ⟷ (L)

Dedekind’s class number formula relates the behaviour of the zeta function of a number field F at s = 1 (or, equivalently, at s = 0, thank to the functional equation) to the basic arithmetic invariants of F. If, in the standard notation, r_{1} and r_{2} denote the number of real and complex places of F respectively, then

\begin{matrix}r := ord_{s=0} ζ_{F} (s) = r_{1} + r_{2} − 1 \\ ζ_{F}^{∗} (0) := lim_{s→0} ζ_{F} (s) s^{−r} = −\frac{h_{F} · R_{F}}{w_{F}} , \end{matrix}

where w_{F} = ♯(O_{F}^{∗})_{tors} is the number of roots of unity in F, h_{F} = ♯ Pic(O_{F}) the class number and R_{F} the regulator of units – the absolute value of the determinant of the (r × r) matrix (log |σ_{i}(ϵ_{j})|) (here ϵ_{j} is a basis of O_{F}^{∗} modulo torsion and σ_{i} : F ↪ C embeddings given by (some) r distinct archimedean places of F). Combined with relations (3.1-2) of the previous section, one gets purely algebraic statements, involving only (A) and (E). (E1) For F = Q(μ_{N})^{+}, let C_{F} be the intersection of the group of units O_{F}^{∗} with the group generated by ±1, ϵ_{m} = (1 − ζ_{m})(1 − ζ_{m}^{−1}) and their conjugates for all divisors m of N (the group of “cyclotomic units”). Write A_{F} for the ideal class group of O_{F}. Then

♯A_{F} (= h_{F}) = [O_{F}^{∗} : C_{F}],  (4.1)

possibly up to a power of 2 ([68],[78, ch.8]). This result admits a stronger version, known as a conjecture of Gras [20] (proved in [43] as a corollary of the Iwasawa Main Conjecture). For simplicity, we consider only the special case when F = Q(μ_{N})^{+} with p |̸ [F : Q]. For a fixed irreducible Z_{p}-representation χ of G = G(F/Q) and a Z_{p}[G]- module M write

M^{χ} = (\frac{1}{♯(G)} ∑_{σ∈G} Tr(χ(σ^{−1})) σ) M

for the χ-component of M. The conjecture of Gras says that

♯(A_{F} ⊗ Z_{p})^{χ} = ♯((O_{F}^{∗} /C_{F}) ⊗ Z_{p})^{χ}  (4.2)

(E2–3) If F is a ray class field of K = Q(\sqrt{−D}), the formula (4.1) holds true with C_{F} replaced by the group of “elliptic units” [59]. The formula (4.2) is also true whenever p |̸ [F : K] [59]. Both are also expected to hold in the context of Stark units (see [61]). (E4) A special case of the Iwasawa Main Conjecture for the odd part of the class groups of cyclotomic fields is the following ([58],[60],[78, 13.6]): let F = Q(μ_{p}) and let χ : G(F/Q) ⟶ Z_{p}^{∗} be an odd (χ(−1) = −1) character different from the Teichmüller character ω. Then

♯(A_{F} ⊗ Z_{p})^{χ} = [Z_{p} : L(0, \overline{χ})Z_{p}]  (4.3)

(E5) Let us consider an elliptic curve E, which is a factor of J_{0}(N) corresponding to a newform f with rational coefficients. In this context, the conjecture of Birch and Swinnerton Dyer predicts the following ([44, I.7]):

Conjecture (BSD). For any number field F, ord_{s=1} L(E/F, s) = 1 iff E(F)/E(F)_{tors} is isomorphic to Z. If P ∈ E(F) is any element of infinite order, then

\frac{L′(E/F, 1)}{Ω_{E/F}} ≐ \frac{♯\underline{III}(E/F)⟨P, P ⟩}{[E(F) : ZP]^{2}}

In particular, if F = K = Q(\sqrt{−D}) and y ∈ E(K) are as in Sec. 2, then (BSD) together with (3.4) predict that

y ∈ E(K) is of infinite order ⟹ \{ \begin{matrix}[E(K) : Zy] is finite \\ ♯\underline{III}(E/K) ≐ [E(K) : Zy]^{2} \end{matrix}

5. Euler systems and descent

Let us first describe some of the results. The descent machine of Thaine– Rubin–Kolyvagin operates as follows: the given Euler system serves as an imput; and the output is a bound on (A) in terms of (E), or, if we are lucky, a precise structure of (A): (E1) In the situation of Sec. 4, one gets ([37],[58])

♯(A_{F} ⊗ Z_{p})^{χ} | ♯((O_{F}^{∗} /C_{F}) ⊗ Z_{p})^{χ}  (5.1)

It follows from (4.1) that (5.1) is in fact an equality. The method even describes elementary divisors of (A_{F} ⊗Z_{p})^{χ}. Using further tools from Iwasawa theory, Rubin [58] is able to deduce the whole Iwasawa Main Conjecture.

(E2-3) These results are extended over Q(\sqrt{−D}) in [59] and also into the (still conjectural) context of Stark units [61]. (E4) In the notation of Sec. 4, it is proved in [37],[60] that

♯(A_{F} ⊗ Z_{p})^{χ} | [Z_{p} : L(0, \overline{χ})Z_{p}]  (5.2)

for all odd characters of G(F/Q) = G(Q(μ_{p})/Q) different from ω. The class number formula for the odd part of A_{F} ([78, 4.17]) then implies that there must be an equality in (5.2), proving thus (4.3). In his forthcoming work [66], Schoof will give an interpretation of the descent in terms of the Fitting ideals of A_{F} ⊗ Z_{p}. (E5) In [37], improving upon previous works [35],[36], Kolyvagin proves that, if y ∈ E(K) is of infinite order, then

[E(K) : Zy] is finite  (5.3a)
♯\underline{III}(E/K) | [E(K) : Zy]^{2} · controlled factor  (5.3b)

A version of (5.3a) over K_{n} with (n, N) = 1 is proved in [6]. In [38], a precise description of the structure of \underline{III}(E/K)_{p^{∞}} in terms of the Euler system {y_{n}} is given (for almost all p), again assuming that y is not torsion. This result is further generalized in [39]. An interested reader can visualize the descent machine on the picture at the end of this article. The Euler system gets in, elementary divisors of (A) are falling out. The procedure, however, requires some effort on the part of the descenter or descentress. I am greatly indebted to my sister Tereza Nekovářová for drawing the picture.

We now describe, briefly, the main ideas involved (cf. [49]). In the context of ideal class groups, the classical way how to find upper bounds for A_{F} is to produce sufficiently many elements in F^{∗} with an explicit factorization into prime ideals (using Gauss sums, this leads to the Stickelberger theorem). The descent method of Thaine–Rubin–Kolyvagin does the same, but inside F^{∗} ⊗ Z/p^{M}Z, where p^{M} is a fixed power of prime p. The starting point is an observation that we have an isomorphism

F^{∗} ⊗ Z/p^{M}Z ⟶^{∼} (F^{′∗} ⊗ Z/p^{M}Z)^{G(F′/F)},

whenever F′ is a Galois extension of F not containing μ_{p}. The following notation will be useful: let I = ⨁_{v} Z · [v] be the group of (fractional) ideals of F, i.e. the free abelian group on the set of nonarchimedean primes v. For x ∈ F^{∗} ⊗ Z/p^{M}Z and a rational prime l write [x]_{l} ∈ ⨁_{v|l} Z/p^{M}Z · [v] for the factorization of x at primes dividing l.

Suppose we want to prove (5.1). Take F′ = F_{n} = Q(μ_{Nn})^{+} for suitable integers n prime to N. The cyclotomic units

x_{n} = (1 − ζ_{Nn})(1 − ζ_{Nn}^{−1}) ∈ O_{F_{n}}^{∗}

satisfy the relations (2.1). If n = ∏_{l|n} l is square-free, then the Galois group G_{n} = G(F_{n}/F) is the product of G_{l} ⟶^{∼} Z/(l − 1)Z. Fixing a generator σ_{l} of G_{l}, define

N_{l} = ∑_{i=0}^{l−2} σ_{l}^{i}, D_{l} = ∑_{i=1}^{l−2} iσ_{l}^{i} ∈ Z[G_{l}], D_{n} = ∏_{l|n} D_{l} ∈ Z[G_{n}]

Then

(σ_{l} − 1)D_{l} = (l − 1) − N_{l},  (5.4)

i.e., modulo l − 1, N_{l} is “divisible” by σ_{l} − 1. Suppose that n is square-free and such that each of its prime factors l satisfies l ≡ 1 (mod p^{M}), l ≡ ±1 (mod N). It then follows from the relations (2.1) and (5.4) that

(1) The class of D_{n}x_{n} in F_{n}^{∗} ⊗ Z/p^{M}Z is G_{n}-invariant, hence comes from an element c_{M}(n) ∈ F^{∗} ⊗ Z/p^{M}Z.

(2) As F_{n}/F is unramified outside n, c_{M}(n) lies, in fact, in O_{F} [1/n]^{∗} ⊗ Z/p^{M}Z, i.e. [c_{M}(n)]_{l} = 0 for l |̸ n.

(3) If l|n, then [c_{M}(n)]_{l} is explicitly determined by the image of c_{M}(n/l) in (O_{F} /lO_{F})^{∗} ⊗Z/p^{M}Z (this makes sense by the previous remark). More precisely, we have

[c_{M}(n)]_{l} = φ_{l}(c_{M}(n/l))

for a natural G(F/Q)-equivariant surjective homomorphism (see [58])

φ_{l} : (O_{F} /lO_{F})^{∗} ⊗ Z/p^{M}Z ⟶ ⨁_{v|l} Z/p^{M}Z · [v]

We now sketch the proof of (5.1) in the simplest case F = Q(μ_{p})^{+} (see [58] for more details). Note that for χ ≠ 1 (for trivial χ both sides of (5.1) are equal to one) the statement of (5.1) is equivalent to

♯(A_{F} ⊗ Z/p^{M}Z)^{χ} | ♯((O_{F}^{∗} ⊗ Z/p^{M}Z)^{χ}/⟨c_{M}(1)⟩^{χ})

for sufficiently large M, where ⟨c_{M}(1)⟩^{χ} denotes the subgroup of O_{F}^{∗} ⊗ Z/p^{M}Z generated by c_{M}(1)^{χ}. This is true because c_{M}(1) is the image of (1−ζ_{p})(1−ζ_{p}^{−1}) in F^{∗} ⊗Z/p^{M}Z (and its χ-component lies in O_{F}^{∗} ⊗Z/p^{M}Z for non-trivial χ). Fix sufficiently large M and ideal classes A_{1}, …, A_{k} generating (A_{F} ⊗Z_{p})^{χ}. We shall construct by induction square-free integers n_{r} = l_{1} … l_{r} with l_{i} ≡ 1 (mod p^{M}) (for r ≤ k) and interpret the explicit factorization of c_{M}(n_{r})^{χ} ∈ (F^{∗} ⊗ Z/p^{M}Z)^{χ} as a non-trivial relation in (A_{F} ⊗ Z/p^{M}Z)^{χ}. Put n_{0} = 1 and assume that n_{r} has been defined in such a way that, for each i ≤ r, the class of [λ_{i}]^{χ} (for a fixed prime λ_{i} in F over l_{i}) is equal to A_{i}. Let t_{r} be the largest integer t ≤ M such that c_{M}(n_{r})^{χ} is divisible by p^{t} in (F^{∗} ⊗ Z/p^{M}Z)^{χ}. We claim that there exists a prime l ≡ 1 (mod p^{M}) not dividing n_{r} with the following two properties: (A) The class of [λ]^{χ} is equal to A_{r+1}, where λ is one of the primes of F

above l. (B) φ_{l}(c_{M}(n_{r})^{χ}) is divisible by p^{t_{r}}, but not by p^{t_{r}+1} in ⊕_{v|l}Z/p^{M}Z · [v].

This means, in fact, that

φ_{l}(c_{M}(n_{r})^{χ}) = p^{t_{r}} · u · [λ]^{χ}, u ∈ (Z/p^{M}Z)^{∗}

To show this, it is necessary to reformulate both conditions (A),(B) in terms of the splitting of l in a suitable Galois extension of Q and then apply the Čebotarev density theorem (see [58, 3.1]). We put l_{r+1} = l and continue until n_{k} has been defined. According to (3) and (B) above, the divisor of c_{M}(n_{r+1})^{χ} is equal, modulo a linear combination of [λ_{1}]^{χ}, …, [λ_{r}]^{χ}, to u · p^{t_{r}} · [λ_{r+1}]^{χ}. As it is divisible by p^{t_{r+1}}, we must have t_{r+1} ≤ t_{r} (hence t_{r+1} ≤ t_{0}). Dividing by p^{t_{r+1}}, we see that the class of

p^{t_{r}−t_{r+1}}[λ_{r+1}]^{χ}

in (A_{F} ⊗ Z/p^{M−t_{0}}Z)^{χ} (= (A_{F} ⊗ Z_{p})^{χ} if M was chosen large enough) lies in the group generated by A_{1}, …, A_{r}. This shows that

♯(A_{F} ⊗ Z_{p})^{χ} | p^{t_{0}−t_{k}} | p^{t_{0}} = ♯((O_{F}^{∗} ⊗ Z/p^{M}Z)^{χ}/⟨c_{M}(1)⟩^{χ}),

proving thus (5.1).

The descent on modular elliptic curves using Heegner points goes essentially along the same lines, but in the framework of Galois cohomology (the above construction could be interpreted cohomologically as well, as F^{∗} ⊗ Z/p^{M}Z ⟶^{∼} H^{1}(F, μ_{p^{M}}) by Kummer theory). For an elliptic curve E over a field F (say, of characteristic 0) there is the standard exact sequence (cf. [44, I.6])

0 ⟶ E(F) ⊗ Z/p^{M}Z ⟶^{δ} H^{1}(F, E_{p^{M}}) ⟶ H^{1}(F, E)_{p^{M}} ⟶ 0 (5.5)

coming from the short exact sequence of G(\overline{F} /F)-modules

0 ⟶ E_{p^{M}} ⟶ E(\overline{F}) ⟶^{p^{M}} E(\overline{F}) ⟶ 0

If F is a number field, the Selmer group relative to the multiplication by p^{M} on E, defined as

S(E/F, p^{M}) = Ker (H^{1}(F, E_{p^{M}}) ⟶ ⨁_{v} H^{1}(F_{v}, E)_{p^{M}} ) ,

sits in an exact sequence

0 ⟶ E(F) ⊗ Z/p^{M}Z ⟶ S(E/F, p^{M}) ⟶ \underline{III}(E/F)_{p^{M}} ⟶ 0,

where the third term is the p^{M}-torsion of the Tate-Šafarevič group of E over F

\underline{III}(E/F) = Ker (H^{1}(F, E) ⟶ ⨁_{v} H^{1}(F_{v}, E))

Passing to the projective limit gives

0 ⟶ E(F) ⊗ Z_{p} ⟶ S_{p}(E/F) ⟶ T_{p}\underline{III}(E/F) ⟶ 0

Here, for any abelian group A, T_{p}A = \underleftarrow{lim} A_{p^{M}} denotes its (p-)Tate module. The p-primary part of \underline{III} has finite corank and is conjectured to be finite. If this is the case, then S_{p}(E/F) is isomorphic to E(F) ⊗ Z_{p}. If F′/F is a Galois extension, there are maps

(E(F′) ⊗ Z/p^{M}Z)^{G(F′/F)} ↪^{δ} H^{1}(F′, E_{p^{M}})^{G(F′/F)} ⟵^{res} H^{1}(F, E_{p^{M}}) (5.6)

In the situation of (3.4), there are Heegner points y_{n} ∈ E(K_{n}) (for n prime to N) satisfying (if −D ≠ −3, −4)

\begin{aligned}N_{l}y_{nl} & = a_{l}y_{n} \\ y_{nl} & ≡ F r_{l}(y_{n}) (mod λ) \end{aligned}  (5.7)

for each prime l |̸ n which is inert in K (here a_{l} is the l-th coefficient in the L-series L(E/Q, s) and λ any prime above l in K_{nl}). Kolyvagin takes n square-free, divisible only by primes l inert in K and such that a_{l}, l + 1 are both divisible by p^{M} (this is a condition on F r_{l} in the extension Q(E_{p^{M}})/Q), so there is a sufficient supply of such primes by the Čebotarev density theorem). Writing G_{n} = G(K_{n}/K_{1}), then G_{n} = ∏_{l|n} G_{l}. Fixing a generator σ_{l} of G_{l} ⟶^{∼} Z/(l + 1)Z, we have operators D_{n} ∈ Z[G_{n}] as above (with l − 1 replaced by l + 1). Thank to the relations (5.4) and (5.7), the image of D_{n}y_{n} in E(K_{n}) ⊗ Z/p^{M}Z is G_{n}-invariant. The restriction map in (5.6) is for all practical purposes an isomorphism (if F′/F = K_{n}/K_{1}); its “inverse” carries δ([D_{n}y_{n}]) into H^{1}(K_{1}, E_{p^{M}}). Finally, taking the trace cor_{K_{1}/K} we get a cohomology class

c_{M}(n) ∈ H^{1}(K, E_{p^{M}})

Denote by d_{M}(n) its image in H^{1}(K, E)_{p^{M}}. These classes enjoy the following properties:

(1) c_{M}(1) = δ(y) ∈ S(E/K, p^{M})

(2) For each prime v |̸ n of K, the localization d_{M}(n)_{v} vanishes in H^{1}(K_{v}, E)_{p^{M}} (in reality, one has to be careful with primes of bad reduction; multiplying c_{M}(n) by a small power of p takes care of that).

(3) If v = l for l|n, then the localization d_{M}(n)_{l} can be explicitly computed in terms of the localization c_{M}(n/l)_{l}.

These classes give a plenty of annihilators of the Selmer group S(E/K, p^{M}): The Weil pairing E_{p^{M}} × E_{p^{M}} ⟶ μ_{p^{M}} induces local pairings

⟨ , ⟩_{v} : H^{1}(K_{v}, E_{p^{M}}) × H^{1}(K_{v}, E_{p^{M}}) ⟶ H^{2}(K_{v}, μ_{p^{M}}) ⟶^{∼} Z/p^{M}Z

According to the Tate duality theorem ([44, I.3]), these pairings are nondegenerate and δ(E(K_{v})⊗Z/p^{M}Z) is its own orthogonal complement. This implies, in view of (2) above and (5.5) over K_{v}, that for each element s ∈ S(E/K, p^{M}) of the Selmer group, ⟨s, c_{M}(n)⟩_{v} vanishes for v |̸ n. The reciprocity law says that the sum of the local pairings between two global elements x, y ∈ H^{1}(K, E_{p^{M}}) vanishes. In particular,

∑_{v}⟨s, c_{M}(n)⟩_{v} = ∑_{l|n}⟨s, c_{M}(n)⟩_{l} = 0 ∈ Z/p^{M}Z  (5.8)

for each s ∈ S(E/K, p^{M}). Each of the local terms in the last formula involves only s_{l}, c_{M}(n/l)_{l} by (3). Assuming that y is of infinite order, an inductive argument relying on (5.8) and Čebotarev shows that (see [37])

• S(E/K, p^{M})/⟨c_{M}(1)⟩ is killed by a fixed power of p; consequently, \underline{III}(E/K)_{p^{∞}} is finite and S_{p}(E/K) = E(K) ⊗ Z_{p}

• The order of \underline{III}(E/K)_{p^{∞}} divides (up to a controlled factor) [S_{p}(E/K) : Z_{p}c_{∞}(1)]^{2} = [E(K) ⊗ Z_{p} : Z_{p}y]^{2}.

There are variations of this method, which treat factors of J_{0}(N) of higher dimensions ([40]), factors of Jacobians of Shimura curves ([41]), precise structure of \underline{III} ([38]) or S(E/K, p^{∞}) ([39]) (even in the case when y is not a torsion point).

The assertions (5.1-2) involve L-functions of the base field twisted by characters of finite order. Once the Iwasawa Main Conjecture has been established, it is possible to say something about L-functions twisted by more general Hecke characters. Over Q, this amounts to a Tate twist, which merely shifts the variable s (cf. (6.1)); one obtains a relation between the values ζ(1 − 2r) and orders of étale cohomology groups H^{i}((Spec Z[1/p])_{et}, Z_{p}(2r)) ([43]). Over Q(\sqrt{−D}), there are more algebraic Hecke characters, some of them are attached to elliptic curves with complex multiplication. In this way, certain non-trivial versions of the BSD conjecture can be deduced (see [59]).

6. Motives and L-functions

In both of the examples treated in the previous section we could have started from the beginning with cohomology classes, not with units or points on the elliptic curve: no p-adic information is lost if we pass from F_{n}^{∗} to F_{n}^{∗} ˆ⊗Z_{p} = H^{1}(F_{n}, Z_{p}(1)) or from E(K_{n}) to H^{1}(K_{n}, T_{p}E) (here we take continuous Galois cohomology in the sense of [29],[75]).

In order to understand which Galois cohomology groups are related to more general L-functions, we recall the conjectures of Beilinson, which interpret special values of L-functions in terms of motivic extensions (see [1],[2],[3],[4],[5],[12],[14],[30],[47],[54],[65]). The general philosophy of mixed motives (due to Beilinson and Deligne), which underlies these conjectures, is still largely a dream (cf. [31]). Nevertheless, one can produce a lot of examples which fit into the picture predicted by the conjectures ([5],[14],[54],[55]).

Fix a number field F and assume, for the moment, that all dreams have come true and that we know what (mixed) motives over F are. For our purposes, a motive M over F is the most general arithmetic object which possesses an L-function. The latter is defined using various realizations of M:

• the étale realization M_{p} (for each prime number p) — a finite dimensional p-adic (continuous) representation of G(\overline{F} /F).

• the Betti realization M_{B,v} (for each infinite place v of F, where we fix an isomorphism between \overline{F_{v}} and C) — a mixed Q-Hodge structure over F_{v} (see [19],[31],[47]). Recall that a mixed Q-Hodge structure H over C is a Q-vector space with an increasing filtration W_{n}H of H and a decreasing filtration F^{p}H_{C} of its complexification, satisfying

(W_{n}H/W_{n−1}H)_{C} = F^{p}((W_{n}H/W_{n−1}H)_{C}) ⊕ \overline{F}^{n+1−p}((W_{n}H/W_{n−1}H)_{C})

for all integers n, p (here \overline{F}^{p} denotes the filtration obtained from F^{p} by complex conjugation). A mixed Q-Hodge structure over R has, in addition, an involution Φ_{∞} preserving the filtration W_{n}H and such that its anti-linear extension to H_{C} preserves the filtration F^{p}H_{C}.

• the de Rham realization M_{dR} — a finite dimensional F -vector space with a decreasing filtration. There are comparison isomorphisms between different realizations:

• I_{v} : M_{B,v} ⊗_{Q} C ⟶^{∼} M_{dR} ⊗_{F,v} C

• I_{p} : M_{B,v} ⊗_{Q} Q_{p} ⟶^{∼} M_{p}

(I_{p} depending on the choice of an embedding \overline{F} ↪ \overline{F_{v}}). At each place v of F, the motive M has a local L-factor ([67],[11]): for non-archimedean v,

L_{v}(M, s) = det(1 − F r_{v}(Nv)^{−s} | M_{p}^{I_{v}})^{−1} (p |̸ Nv)

(where I_{v} is the inertia group of v), conjecturally independent of p. For an archimedean place v, L_{v}(M, s) is a product of gamma factors depending only on the real Hodge structure M_{(v)} = M_{B,v} ⊗_{Q} R. See [13] for a unified treatment of the local L-factors at all places. For any finite set of (non-archimedean) places S, write

L_{S}(M, s) = ∏_{v∉S} L_{v}(M, s),

where the product is taken over non-archimedean places outside S. If S is empty, we write L(M, s) instead of L_{∅}(M, s). The total L-factor at infinity will be denoted by

L_{∞}(M, s) = ∏_{v|∞} L_{v}(M, s).

A typical motive should be of the form M = h^{m}(X)(n) for a variety X over F (not necessarily smooth or irreducible). Its realizations are

\begin{aligned}M_{p} & = H^{m}((X ×_{F} \overline{F})_{et}, Q_{p})(n) \\ M_{B,v} & = H^{m}((X ×_{F,v} C)(C), Q)(n) \\ M_{dR} & = H_{dR}^{m}(X_{/F})(n) \end{aligned}

Various constructions of linear algebra should make sense for motives. Above M(n) refers to the Tate twist, given by the tensor product with Q(1)^{⊗n}, where Q(1) is the Tate motive, dual to h^{2}(P^{1}), with realizations

Q(1)_{p} = Q_{p}(1) = Z_{p}(1) ⊗_{Z_{p}} Q_{p} = \underleftarrow{lim}(μ_{p^{M}} (\overline{F})) ⊗_{Z_{p}} Q_{p}
Q(1)_{B,v} = (2πi)Q, Hodge type (−1, −1)
Q(1)_{dR} = F = F^{−1}Q(1)_{dR} ⊃ F^{0}Q(1)_{dR} = 0

We have

L_{v}(M(n), s) = L_{v}(M, s + n)  (6.1)

for each place v. Let us mention two standard examples:

(1) The unit motive Q(0) = h^{0}(Spec(F)). Then L(M, s) = ζ_{F} (s) is the Dedekind zeta function of F.

(2) Let X be smooth and projective over F and M = h^{1}(X)(1). Let A = Pic^{o}(X), Â = Alb(X) be the Picard and Albanese variety of X respectively. Then M_{p} = T_{p}(A) ⊗_{Z_{p}} Q_{p} and L(M, s) = L(Â, s) = L(A, s).

If X is smooth and projective over F, then M = h^{m}(X)(n) is a pure motive of weight w = m−2n. In realizations, all M_{B,v} are pure Hodge structures of weight w; there is a finite set S_{bad} of places at which X has bad reduction. If v ∉ S_{bad} is nonarchimedean and prime to p, then M_{p} is unramified at v and all eigenvalues of the geometric Frobenius F r_{v} on M_{p} are algebraic numbers with absolute value (Nv)^{w/2}. According to the monodromy conjecture, a more refined version of the purity should hold also at bad places, see [28]. In general, every motive M should have a canonical (increasing) weight filtration W_{.}M with W_{i}M/W_{i−1}M pure of weight i. For our M = h^{m}(X)(n), the series L_{S_{bad}}(M, s) is absolutely convergent and non-zero for Re(s) > 1 + \frac{w}{2} ; if the monodromy conjecture holds, the same is true for L(M, s). The gamma factors L_{∞}(M, s) have no poles or zeroes in this region either. Conjecturally, the L-function has a meromorphic continuation to C, with only possible pole at 1+ \frac{w}{2} if w is even, and satisfies the functional equation

L_{∞} · L(M, s) = ε(M, s)L_{∞} · L(M^{∗}(1), −s)  (6.2)

with an ε-factor of the form ε(M, s) = a · b^{s}. For our M = h^{m}(X)(n), we have M^{∗}(1) = M(w + 1), so (6.2) becomes

L_{∞} · L(M, s) = ε(M, s)L_{∞} · L(M, w + 1 − s)  (6.3)

Suppose that we are interested in the behaviour of L(M, s) at an integer s = j. Applying a Tate twist, we can assume (by (6.1)) that j = 0. Assuming that the functional equation (6.3) holds, it sufficies to treat only the case when s = 0 lies to the right of the central point (w + 1)/2 (or coincides with it), which happens iff w ≤ −1. In fact,

w = −1 ⟺ s = 0 is the central value (w + 1)/2
w = −2 ⟺ s = 0 is the near central value w/2 + 1
w ≤ −3 ⟺ s = 0 is in the convergence region

The collection of Hodge structures M_{∞} = (M_{(v)})_{v|∞} can be viewed as an object of the category MH_{F ⊗R} of mixed Hodge structures over F ⊗ R. Writing M_{(v)}^{+} for the invariants under G(\overline{F_{v}}/F_{v}), the comparison maps I_{v} induce a map (“the Deligne period map”)

α_{M} : M_{∞}^{+} = ⨁_{v|∞} M_{(v)}^{+} ⟶ ⨁_{v|∞} M_{dR}/F^{0} ⊗_{F} F_{v} = M_{dR}/F^{0} ⊗_{Q} R

Because of our assumption w ≤ −1, α_{M} is injective and we have ([47],[55, ch.1])

− ord_{s=0} L_{∞}(M^{∗}(1), s) = dim_{R} Coker(α_{M})
ord_{s=0} L_{∞}(M, s) = 0

Note that det_{R} Coker(α_{M}) has a natural Q-structure, coming from the Qstructures M_{B,v}^{+} and M_{dR}/F^{0}. The group Coker(α_{M}) has an interpretation in terms of Hodge theory as the group of extensions

Ext_{MH_{F ⊗R}}^{1}(R(0)_{∞}, M_{∞}) = H^{1}(MH_{F ⊗R}, M_{∞})

of the unit Hodge structure R(0)_{∞} = (R(0))_{v|∞} by M_{∞} ([3],[31],[47]). Suppose, for simplicity, that w ≤ −3 (then s = 0 is in the region of absolute convergence), or that w = −2 and M contains no copy of Q(1) (then, according to the Tate conjecture ([73],[31, ch.5],[55, ch.1]), L(M, s) has no pole or zero at s = 0 either). The L-function L(M^{∗}(1), −s) = L(M, w + 1 − s) then has a zero of order

r = dim_{R} H^{1}(MH_{F ⊗R}, M_{∞})

at s = 0.

The fundamental idea of Beilinson and Deligne is that the value L(M, 0) should be given in terms of the group of motivic extensions

Ext_{MM_{F}}^{1} (Q(0), M) = H^{1}(MM_{F} , M)

and the Hodge realization map

r_{∞} : H^{1}(MM_{F} , M) ⟶ H^{1}(MH_{F ⊗R}, M_{∞})

More precisely, they conjecture ([3],[12]) that there is a subgroup

H_{f}^{1}(MM_{F} , M) ⊆ H^{1}(MM_{F} , M)

formed by extension classes ‘with good reduction’ everywhere such that (1) The Hodge realization r_{∞} induces an isomorphism

r : H_{f}^{1}(MM_{F} , M) ⊗_{Q} R ⟶^{∼} H^{1}(MH_{F ⊗R}, M_{∞})  (6.4)

(2) Modulo a non-zero rational factor, the value L(M, 0) is equal to the determinant det(r) ∈ R^{∗}/Q^{∗}. Here the determinant is computed with respect to the Q-structures det_{Q} H_{f}^{1}(MM_{F} , M) and the Q-structure of det_{R} Coker(α_{M}) described above.

In [65], Scholl interprets H_{f}^{1}(MM_{F} , M) as the group of extensions H^{1}(MM_{O}, M) in a certain subcategory MM_{O} (containing all pure motives) of MM_{F}, consisting of “motives over the ring of integers O”. If w = −2 and M contains Q(1), the conjecture needs some modification: for M = Q(1), we have Coker(α_{M}) = R^{r_{1}+r_{2}} and one expects that H^{1}(MM_{F} , Q(1)) = F^{∗} ⊗ Q, H_{f}^{1}(MM_{F} , Q(1)) = O^{∗} ⊗ Q. In this case r is the standard logarithm map and the image of O^{∗} ⊗ R has codimension 1, which is the order of the pole of L(Q(1), s) = ζ(s + 1) at s = 0. There is a conjectural description of the motivic H^{1}, H_{f}^{1} in terms of algebraic K-theory (see [1],[2],[4],[14],[47],[55, ch.1]). In particular, H^{1}(MM_{F} , h^{m}(X)(n)) should be isomorphic to the subspace of K_{2n−m−1}(X) ⊗ Q = K_{−w−1}(X) ⊗ Q on which Adams operations act with weight n. For this K-theoretical version of the motivic cohomology, the map r_{∞} (“the regulator”) was constructed in [1]. If s = 0 is the central value, i.e. if w = −1, the special value should be given by a generalization of the conjecture of Birch and Swinnerton-Dyer. For n ≥ 1, let CH^{n}(X)_{0} be the group of homologically trivial algebraic cycles of codimension n on X (which is supposed to be smooth and projective as before, but also equidimensional), modulo rational equivalence. Under certain conditions on X, there is a height pairing ([4])

⟨ , ⟩ : CH^{n}(X)_{0} × CH_{0}^{dim X+1−n} ⟶ R

The conjecture predicts ([1],[4],[55, ch.1]):

(1) ⟨ , ⟩ is non-degenerate modulo torsion.

(2) r := ord_{s=0} L(h^{2n−1}(X)(n), s) is equal to dim_{Q} CH^{n}(X)_{0} ⊗ Q = dim_{Q} CH^{dim X+1−n}(X)_{0} ⊗ Q.

(3) L^{∗}(h^{2n−1}(X)(n), 0) := lim_{s⟶0} L(h^{2n−1}(X)(n), s)s^{−r} is equal, up to a rational factor, to Ω_{X} · det⟨ , ⟩, where Ω_{X} is a suitable period.

The conjectures predict, in particular, that

ord_{s=0} L(M^{∗}(1), s) = dim_{Q} H_{f}^{1}(MM_{F} , M)

in all three cases, i.e. whenever w ≤ −1. It is also expected that

H^{1}(MM_{F} , M) = 0  if w ≥ 0  (6.5)

Indeed, for w > 0 every extension of Q(0) by M would be split by the weight filtration; there should also be no non-trivial extensions of pure motives of the same weight.

7. Selmer groups

In order to determine the missing rational factors in Beilinson’s conjectures, Bloch and Kato [7] considered also p-adic realization maps (using the Ktheoretic version of the motivic cohomology)

r_{p} : H^{1}(MM_{F} , M) ⟶ H^{1}(F, M_{p})

into (continuous) Galois cohomology (or, in the notation explained below, to its subgroup \underrightarrow{lim}_{S} H^{1}(G_{S}, M_{p})).

One of their main objectives was a definition of Q_{p}-subspaces (we use a slightly different notation, cf. [19])

H_{f}^{1}(F, M_{p}) ⊆ H_{g}^{1}(F, M_{p}) ⊆ H^{1}(F, M_{p})

such that, conjecturally, r_{p} induces isomorphisms

\begin{aligned}H^{1}(MM_{F} , M) ⊗_{Q} Q_{p} & ⟶^{∼} H_{g}^{1}(F, M_{p}) \\ H_{f}^{1}(MM_{F} , M) ⊗_{Q} Q_{p} & ⟶^{∼} H_{f}^{1}(F, M_{p}) \end{aligned}  (7.1)

for all mixed motives M (although Bloch and Kato considered only pure motives). Let us first offer a geometric insight into the cohomology groups H_{f}^{1}, H_{g}^{1}, which is based on a speculation about motivic sheaves. Let S be a finite set of places of F containing all v|p. Write O_{S} for the localization of O at S and G_{S} for the Galois group of the maximal extension of F ramified only at S. Every p-adic representation V of G_{S} determines a p-adic étale sheaf V_{S,et} on (Spec O_{S})_{et} and we have the basic isomorphism between Galois and étale cohomology (see [44, ch.2])

H^{i}(G_{S}, V) = H^{i}((Spec O_{S})_{et}, V_{S,et})

In the context of motives, the pair

Rep_{Q_{p}}(G_{S}) = { p−adic representations of G_{S} }
S((Spec O_{S})_{et}) = { p−adic etale sheaves on Spec(O_{S}) }

should be replaced by

Rep_{S,g,Q_{p}}(F) ={ “geometric” p−adic representations with good

reduction outside S }

S((Spec O_{S})_{mot}) ={ p − adic “motivic” sheaves on Spec(O_{S}) }

(in this case, S is not supposed to contain all places dividing p). Every representation V in Rep_{S,g,Q_{p}}(F) shoud determine a motivic sheaf V_{S,mot} in such a way that (for all i)

H^{i}(Rep_{S,g,Q_{p}}(F), V) = H^{i}((Spec O_{S})_{mot}, V_{S,mot})

This cohomology group will be denoted H_{g}^{i}(O_{S}, V) and we should have

\begin{aligned}H_{g}^{i}(F, V) & = \underrightarrow{lim}_{S} H_{g}^{i}(O_{S}, V) \\ H_{f}^{i}(F, V) & = H^{i}((Spec O)_{mot}, j_{∗}V_{S,mot}), \end{aligned}

where j : Spec(O_{S}) ↪ Spec(O) is the inclusion map. There is an algebraic candidate for Rep_{S,g,Q_{p}}(F) proposed in [19]. The definition is local; suppose that, for each non-archimedean place v of F, we are given two subcategories

Rep_{g,Q_{p}}(O_{v}) ⊆ Rep_{g,Q_{p}}(F_{v})

of the category Rep_{Q_{p}}(F_{v}) of all p-adic representations of G(\overline{F_{v}}/F_{v}), consisting of “geometric p-adic representations of G(\overline{F_{v}}/F_{v}) with good reduction” resp. of “geometric p-adic representations of G(\overline{F_{v}}/F_{v})”. A global p-adic representation of G(\overline{F} /F) then belongs to Rep_{S,g,Q_{p}}(F) if its restrictions to G(\overline{F_{v}}/F_{v}) lie in Rep_{g,Q_{p}}(F_{v}) (for v ∈ S) and in Rep_{g,Q_{p}}(O_{v}) (for v ∉ S). For v |̸ p, one defines ([19])

Rep_{g,Q_{p}}(F_{v}) = Rep_{Q_{p}}(F_{v})
Rep_{g,Q_{p}}(O_{v}) = \{ unramified representations of G(\overline{F_{v}}/F_{v}) \}

For v|p, one has to use Fontaine’s theory of Grothendieck’s ‘mysterious functor’ ([26],[27]), which relates p-adic representations of G(\overline{F_{v}}/F_{v}) to certain objects of p-linear algebra. The definition given in [19] is the following:

Rep_{g,Q_{p}}(F_{v}) = \{ potentially semistable representations of G(\overline{F_{v}}/F_{v}) \}
Rep_{g,Q_{p}}(O_{v}) = \{ crystalline representations of G(\overline{F_{v}}/F_{v}) \}

See [19],[27],[46] for more details about p-adic Galois representations. According to various conjectures in the p-adic Hodge theory, the étale realization M_{p} of every mixed motive M should be in Rep_{S,g,Q_{p}}(F) for suitable S. This is known to be true for M = h^{m}(X)(n) with X proper and smooth over F, provided X has good reduction (or a semistable reduction and p > 2m) at all v|p ([27]). We are now ready to define the groups H_{f}^{1}, H_{g}^{1}. First the local case. Suppose that V is any p-adic representation of G(\overline{F_{v}}/F_{v}). Denote by V_{g} (resp. V_{f}) its largest subrepresentation which belongs to Rep_{g,Q_{p}}(F_{v}) (resp. Rep_{g,Q_{p}}(O_{v})) and put

H_{g}^{i}(F_{v}, V) = H^{i}(Rep_{g,Q_{p}}(F_{v}), V_{g})
H_{f}^{i}(F_{v}, V) = H^{i}(Rep_{g,Q_{p}}(O_{v}), V_{f})

(if v |̸ p, then H_{g}^{i} coincide with the usual cohomology and H_{f}^{i} with the unramified cohomology groups). If T is a submodule (resp. a factor-module) of V, write H_{∗}^{i}(F_{v}, T) for the preimage (resp. the image) of H_{∗}^{i}(F_{v}, V) in H^{i}(F_{v}, T). The most important property of these groups is that under the duality between H^{1}(F_{v}, V) and H^{1}(F_{v}, V^{∗}(1)) given by the cup product ([43, 1.3]), the subgroups H_{f}^{1}(F_{v}, V) and H_{f}^{1}(F_{v}, V^{∗}(1)) are orthogonal complements of each other. For a global p-adic representation V unramified outside S ∪ {v|p} and its sub- or factor-module T, we define

H_{g}^{1}(O_{S}, T) = \{ x ∈ H^{1}(G_{S∪\{v|p\}}, T) | \begin{matrix}x_{v} ∈ H_{g}^{1}(F_{v}, T) v|p, v ∈ S \\ x_{v} ∈ H_{f}^{1}(F_{v}, T) v|p, v ∉ S \end{matrix} \}

H_{g}^{1}(F, T) = \underrightarrow{lim}_{S} H_{g}^{1}(O_{S}, T)

H_{f}^{1}(F, T) = { x ∈ H_{g}^{1}(O_{S}, T) | x_{v} ∈ H_{f}^{1}(F_{v}, T) v|p or v ∈ S }

If T is a Z_{p}-lattice in V, invariant under G(\overline{F} /F), the cohomological Tate-Šafarevič group \underline{III}(F, T) is defined as ([15])

\underline{III}(F, T) = Coker(H_{f}^{1}(F, V) ⟶ H_{f}^{1}(F, V/T))

It is always finite ([7],[15]). Example (1): V = Q_{p}(1) ⊃ T = Z_{p}(1). Then V is in Rep_{g,Q_{p}}(O_{v}) for each finite place v of F and we have

H_{g}^{1}(F_{v}, Q_{p}(1)) = H^{1}(F_{v}, Q_{p}(1)) = F_{v}^{∗} ˆ⊗Q_{p}  (⟶^{∼} Q_{p} if v |̸ p)
H_{f}^{1}(F_{v}, Q_{p}(1)) = O_{v}^{∗} ˆ⊗Q_{p}  (= 0 if v |̸ p)

Here Aˆ⊗Q_{p} = (\underleftarrow{lim} A/p^{M}A)⊗_{Z_{p}} Q_{p} denotes the completed tensor product. It follows that (for any finite set S of non-archimedean places)

\begin{aligned}H_{g}^{1}(O_{S}, Q_{p}(1)) & = O_{S}^{∗} ⊗ Q_{p} \\ H_{g}^{1}(F, Q_{p}(1)) & = F^{∗} ⊗ Q_{p} \\ H_{f}^{1}(F, Q_{p}(1)) & = O^{∗} ⊗ Q_{p} \end{aligned}

The Tate-Šafarevič group

\underline{III}(F, Z_{p}(1)) = \frac{Ker(F^{∗} ⊗ Q_{p}/Z_{p} ⟶ ⊕_{v}Q_{p}/Z_{p} · [v])}{O^{∗} ⊗ Q_{p}/Z_{p}} = (A_{F})_{p^{∞}}

is equal to the p-part of the class group of F. Example (2): Let E be an elliptic curve over F and T = T_{p}E ⊂ V = T ⊗_{Z_{p}} Q_{p}. In this case, V lies in Rep_{g,Q_{p}}(F_{v}) (resp. Rep_{g,Q_{p}}(O_{v})) for all v (resp. for places v of good reduction of E). The sequence (5.5) over F_{v} defines, in the limit, an injection δ : E(F_{v})ˆ⊗Q_{p} ↪ H^{1}(F_{v}, V). Its image is equal to H_{f}^{1}(F_{v}, V) = H_{g}^{1}(F_{v}, V) for all v ([7]). In fact, H^{1}(F_{v}, V) = 0 for all primes v |̸ p. Consequently,

\begin{aligned}H_{f}^{1}(F, V) & = H_{g}^{1}(F, V) = S_{p}(E/F) ⊗_{Z_{p}} Q_{p} \\ H_{f}^{1}(F, T) & = H_{g}^{1}(F, T) = S_{p}(E/F) \end{aligned}

and the Tate-Šafarevič group \underline{III}(F, T) is equal to the factor of \underline{III}(E/F)_{p^{∞}} by its maximal divisible subgroup.

8. Values of L-functions

Let, as before, M = h^{m}(X)(n) (for a smooth projective variety over F) be a pure motive of weight w = m − 2n ≤ −1. Let us consider the composite map

H_{f}^{1}(MM_{F} , M) ⊗_{Q} Q_{p} ⟶ H_{f}^{1}(F, M_{p}) ⟶ H_{f}^{1}(F ⊗ Q_{p}, M_{p})

Bloch and Kato [7] define a canonical exponential map

M_{dR}/F^{0} ⊗_{Q} Q_{p} ⟶ H_{f}^{1}(F ⊗ Q_{p}, M_{p}) = ⨁_{v|p} H_{f}^{1}(F_{v}, M_{p})

which is conjectured to be an isomorphism for our M (this is known to be true if X has good reduction at all v|p). Tate’s global duality ([44, I.4.10]) gives an exact sequence

\begin{matrix}H^{2}(G_{S}, M_{p}^{∗}(1))^{∗} ⟶ H_{f}^{1}(F, M_{p}) ⟶ ⨁_{v∈S} H_{f}^{1}(F_{v}, M_{p}) ⟶ \\ ⟶ & H^{1}(G_{S}, M_{p}^{∗}(1))^{∗} ⟶ H_{f}^{1}(F, M_{p}^{∗}(1))^{∗} ⟶ 0 \end{matrix}

(for any set of places S containing all places of bad reduction and v|p). Suppose now that w ≤ −2. Then the first term should vanish by a conjecture of Jannsen [28], the last term by a combination of (6.5) and (7.1). The local groups H_{f}^{1}(F_{v}, V) should vanish for v |̸ p by the monodromy conjecture. Assuming all this, we get a conjectural exact sequence [33]

0 ⟶ H_{f}^{1}(M) ⊗_{Q} Q_{p} ⟶ M_{dR}/F^{0} ⊗_{Q} Q_{p} ⟶ H^{1}(G_{S}, M_{p}^{∗}(1))^{∗} ⟶ 0

(8.1)

Define a one-dimensional Q-vector space

Θ(M) = det_{Q} H_{f}^{1}(MM_{F} , M) ⊗ det_{Q}(⊕_{v|∞}M_{B,v}^{+}) ⊗ det_{Q}(M_{dR}/F^{0})^{⊗−1} Suppose, for simplicity, that M contains no copy of Q if w = −2. Beilinson’s conjecture (6.4) and (7.1) then predict that r_{∞} and r_{p} induce isomorphisms

s_{∞} : Θ(M) ⊗_{Q} R ⟶^{∼} R

s_{p} : Θ(M) ⊗_{Q} Q_{p} ⟶^{∼} det_{Q_{p}} H^{1}(G_{S}, M_{p}^{∗}(1)) ⊗ det_{Q_{p}}(F ⊗ R, M_{p})

The essential point is that the one-dimensional Q_{p}-vector space on the right hand side has a canonical Z_{p}-submodule ([18],[19]), say, X_{p}. We can now formulate the conjecture specifying the value L(M, 0) ([7],[18],[19],[33]): There is an element z_{M} ∈ Θ(M) (unique up to a sign) such that s_{p}(z_{M} ⊗1) generates X_{p} for all p. Then

s_{∞}(z_{M} ⊗ 1) = ±L(M, 0)  (8.2)

This is, in fact, a simplified version of a general conjecture of Fontaine and Perrin-Riou ([17],[18],[19]), which makes sense for all mixed motives. They work consistently with determinants of all cohomology groups, making no appeal to vanishing conjectures à la Jannsen. Kato [33] considers a fixed abelian extension F′/F with Galois group G and works over F′, taking into account the action of G. His ‘Iwasawa Main Conjecture for motives’ goes essentially along the same lines, one just has to take all determinants over the group ring Q[G]. In practice, one has to replace H_{f}^{1}(MM_{F} , M) by a suitable subspace of the K-theory of X, which satisfies (6.4) and (7.1). There are quite a few instances in which one can construct enough elements in K-theory and compute their image under r_{∞} (see [1],[2],[14],[54],[55]). In order to verify the conjecture (8.2), one must be able to compute also their image under r_{p}. This is far from being easy: for example, it is not known, in general, whether K-theory actually maps into H_{f}^{1}(F, V) ⊆ H^{1}(F, V) (see [22],[46],[63] for some partial results; note that [22] contains a fair amount of misprints). The computation of the map r_{p} can be viewed alternately as a p-adic integration (this is the point of view of [8]), or as an explicit reciprocity law (see [7],[32],[51] for these).

9. Cohomological Euler systems

It is now clear where to look for further Euler systems in Galois cohomology: if we are given a family of extensions F_{n} of the base number field F together with elements in H_{f}^{1}(MM_{F_{n}}, M) (or in some more wordly version of this not quite well-defined group, probably given by algebraic K-theory), consider their p-adic realizations in H^{1}(F_{n}, M_{p}). If we can control the denominators, they will in fact lie in (the image of) H^{1}(F_{n}, T) for a suitable Galois invariant Z_{p}-lattice T in M_{p}. If we are able to prove that these cohomological elements satisfy relations analogous to (2.1) and if we can control their localizations at various places (most notably those dividing p), we can then apply the descent procedure to obtain various finiteness results for Selmer and Tate-Šafarevič groups involved. We now give a list of cases when this has been done.

(E1–3) The Euler system lies in H_{f}^{1}(F_{n}, Z_{p}(1)) = O_{F_{n}}^{∗} ⊗ Z_{p}.

(E4) Gauss sums lie in H_{g}^{1}(F_{n}, Z_{p}(1)), but not in H_{f}^{1}(F_{n}, Z_{p}(1)) for suitable abelian extensions F_{n} of Q(μ_{p}).

(E5) Under the boundary map δ in (5.5), the Heegner points define an Euler system in H_{f}^{1}(K_{n}, T_{p}E).

(E6) E : “Heegner cycles” over ring class fields of an imaginary quadratic field K [45] L : L(f ⊗ K, r) for modular forms of weight 2r > 2 A : Selmer group associated to f over K

Let f = ∑ a_{n}q^{n} be a normalized newform of weight 2r > 2 on Γ_{0}(N) with rational coefficients. There is a ‘motive’ M associated to f such that L(M, s) = L(f, s) = ∑ a_{n}n^{−s}. In fact, M ⊂ h^{2r−1}(X) for a suitable smooth compactification of the (2r−1)-dimensional Kuga-Sato variety over (a suitable cover) of Y_{0}(N) (see [64]). If K is an imaginary quadratic field as in (E5), it is possible to define Heegner cycles x_{n} ∈ CH^{r}(X ⊗_{K} K_{n})_{0} ⊗ Q. Fix a prime number p not dividing N and write y_{n} for the image of x_{n} under the Abel-Jacobi map

ϕ : CH^{r}(X ⊗_{K} K_{n})_{0} ⊗ Q ⟶ H_{f}^{1}(K_{n}, V),

where V = M_{p}(r). They all lie inside H_{f}^{1}(K_{n}, T) for a suitable Z_{p}-lattice in V and satisfy relations analogous to (2.4). It is possible to construct cohomology classes c_{M}(n) ∈ H^{1}(K, T/p^{M}T) following the procedure in Sec. 5 and then apply the descent machine. The result of the descent is the following: if y = cor_{K_{1}/K}(y_{1}) is not torsion in H_{f}^{1}(K, T), then ([45])

the index [H_{f}^{1}(K, T) : Z_{p}y] is finite  (9.1)

In fact, the methods of [37] extend to this case, showing that the order of \underline{III}(T, K) is divisible by the square of this index times a controlled power of p.

The field K has a unique extension K_{∞} with G(K_{∞}/K) ⟶^{∼} Z_{p}^{2}. If p |̸ a_{p}, then there is a p-adic L-function L_{p} defined on continuous characters χ : G(K_{∞}/K) ⟶ C_{p}^{∗} that interpolates algebraic parts of the values of the complex L-function: if χ is a character of finite order, then

L_{p}(f ⊗ K, χ) ≐ i_{p} ∘ i^{−1} (\frac{L(f ⊗ K, χ, r)}{Ω_{f,K,χ}})  (9.2)

Here Ω_{f,K,χ} is a suitable period and i (resp. i_{p}) denotes a fixed embedding of \overline{Q} into C (resp. \overline{Q_{p}}). Restricting to characters of G(KQ_{∞}/K), where Q_{∞}/Q is the cyclotomic Z_{p}-extension of Q, we get a function L_{p}(f ⊗ K, s) of a p-adic variable s ∈ pZ_{p}, if we use the canonical isomorphisms

G(KQ_{∞}/K) ⟶^{∼} G(Q_{∞}/Q) ⟶^{∼} (1 + pZ_{p}) ⟶^{log_{p}} pZ_{p}

Under the same conditions (p |̸ Na_{p}), there exists a canonical (symmetric) p-adic height pairing ([46],[50])

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

It is shown in [48] that (if p splits in K and p |̸ 2Na_{p})

L′_{p}(f ⊗ K, 0) ≐ ⟨y, y⟩_{p}  (9.3)

In particular, it follows from (9.1) and (9.3), that

ord_{s=0} L_{p}(f ⊗ K, s) = 1 ⟹ dim_{Q_{p}} H_{f}^{1}(K, V) = 1

(E7) E : “cyclotomic elements” in H_{f}^{1}(F, Z_{p}(r)) for abelian extensions F of Q for r > 1 odd ([42],[70]) L : L′(1 − r, χ) for Dirichlet characters over Q A : H_{f}^{1}(Q, Q_{p}/Z_{p}(1 − r)) = \underline{III}(Q, Z_{p}(1 − r))

The Euler system is formed by cyclotomic elements of Deligne-Soulé ([70]) y_{n} ∈ H_{f}^{1}(Q(μ_{n}), Z_{p}(r)) satisfying relations similar to (2.1). Out of these elements one manufactures cohomology classes c_{M}(n) ∈ H^{1}(Q, Z/p^{M}Z(r)) which give a bound on H_{f}^{1}(Q, Q_{p}/Z_{p}(1−r)): if s ∈ H_{f}^{1}(Q, Q_{p}/Z_{p}(1−r))_{p^{M}}, the reciprocity law gives a formula analogous to (5.8)

∑_{l}⟨s, c_{M}(n)⟩_{l} = ∑_{l|n}⟨s, c_{M}(n)⟩_{l} = 0 ∈ Z/p^{M}Z

with

⟨ , ⟩_{l} : H^{1}(Q_{l}, Z/p^{M}Z(1 − r)) × H^{1}(Q_{l}, Z/p^{M}Z(r)) ⟶ Z/p^{M}Z

Each term ⟨s, c_{M}(n)⟩_{l} can be again computed inductively using c_{M}(n/l)_{l} ∈ H^{1}(Q_{l}, Z/p^{M}Z(r)). The descent shows that ([42])

♯H_{f}^{1}(Q, Q_{p}/Z_{p}(1 − r)) | [H_{f}^{1}(Q, Z_{p}(r)) : Z_{p} y_{1}]  (9.3)

In [7], Bloch and Kato prove (for p > 2) that the Iwasawa Main Conjecture over Q implies that there is in fact an equality in (9.3), which is a “ppart” of the conjecture (8.2) for M = Q(r). They also need the fact that Jannsen’s vanishing conjecture, formulated in Sec. 8, is true for M = Q(r) (see [28]). (E8) E : Beilinson’s elements in K_{2} of a modular curve [1],[34], [55, ch. 9]

L : L′(f, 0) (or L(f, 2) by the functional equation) for modular forms of weight 2

Kato [34] announced a construction of an Euler system x_{n} ∈ H^{0}(X_{1}(Np^{n})⊗ Q(μ_{p^{n}}), K_{2}) ⊗ Q, using symbols {u, v} formed by suitable modular units. If f is a fixed newform of weight 2 on Γ_{0}(N) (for simplicity, with rational coefficients), E the corresponding quotient of J_{0}(N), which is an elliptic curve, V = T_{p}E, then x_{n} map to y_{n} ∈ H^{1}(Q(μ_{p^{n}}), T (1)) for a suitable lattice T ⊂ V. Applying the ‘Tate twisting trick’ of Soulé [70, 1.4] backwards, Kato uses relations between y_{n} to define a new cohomology class z ∈ H^{1}(Q, T), with the property

• If L(E/Q, 1) ≠ 0, then z_{p} ∉ H_{f}^{1}(Q_{p}, V) Suppose that L(E/Q, 1) ≠ 0. If s ∈ H_{f}^{1}(Q, V), then the localizations s_{l} vanish for l ≠ p, as H^{1}(Q_{l}, V) = 0. The reciprocity law then implies that

⟨s_{p}, z_{p}⟩_{p} = 0 ∈ H^{2}(Q_{p}, Q_{p}(1)) = Q_{p}

However, H_{f}^{1}(Q_{p}, V) is one-dimensional and is equal to its own orthogonal complement, hence s_{p} ∈ H_{f}^{1}(Q_{p}, V) must vanish. This proves that E(Q) is torsion, because the composite map

E(Q) ⊗ Q ⟶ H_{f}^{1}(Q, V) ⟶ H_{f}^{1}(Q_{p}, V)

is injective. This proof of the finiteness of E(Q) provided L(E/Q, 1) ≠ 0 is purely p-adic; if one could define cohomology classes analogous to c_{M}(n) in this context as well, that would probably prove the finiteness of \underline{III}(E/Q)_{p^{∞}}. Hopefully a written version of [34] will be soon available. (E9) E : cohomology classes in H^{1}(Q, M_{p}) for M = Sym^{2}(h^{1}(E)(1)), where

E is a modular elliptic curve over Q ([16]).

L : L(Sym^{2}(h^{1}(E)), 2)

A : H_{f}^{1}(Q, V/T) = \underline{III}(Q, T) for a suitable Z_{p}-lattice T in V = M_{p}.

This construction seems to be of a different nature than (E6–8), as it deals with a critical value, when both L(M, s) and L(M^{∗}(1), −s) are non-zero at s = 0. Fix a modular parametrization ϕ : X_{0}(N) ⟶ E defined over Q. Assuming that E does not have a complex multiplication and p is a fixed prime (outside of a finite set of exceptional primes), Flach constructs, for each prime l |̸ N an element ϵ(l) ∈ H^{1}(X_{0}(N) × X_{0}(N), K_{2}), which is then mapped to c(l) ∈ H^{1}(Q, T) (with T = Sym^{2}(H^{1}((E ⊗_{Q} \overline{Q})_{et}, Z_{p}(1)))) via a sequence of maps

\begin{matrix}H^{1}(X_{0}(N) × X_{0}(N), K_{2}) ⟶ H^{1}(E × E, K_{2}) & ⟶ H^{3}((E × E)_{et}, Z_{p}(2)) \\ ⟶ H^{1}(Q, H^{2}(((E × E) ⊗_{Q} \overline{Q})_{et}, Z_{p}(2))) & ⟶ H^{1}(Q, T) \end{matrix}

It is shown in [16] that (for l |̸ pN) the localizations c(l)_{r} lie in H_{f}^{1}(Q_{r}, T) for all primes r ≠ l, but c(l)_{l} ∉ H_{f}^{1}(Q_{l}, T). An application of Čebotarev gives the main result of [16]:

H_{f}^{1}(Q, V/T) = \underline{III}(Q, T) is killed by deg(ϕ)

10. Concluding remarks

In this survey, we have tried to give examples of Euler systems and their applications. What we are still lacking is a general framework (if only a conjectural one) into which they could be incorporated, although Kato’s conjectural ‘Iwasawa theory for motives’ [33] undoubtedly points in the right direction.

The metaquestion “What is a general motivic Euler system?” seems to be quite important and far from being vacuous; indeed, any answer, however partial, would have to deal with the following topics, which have arithmetic significance in their own right:

(A) p-adic L-functions: Develop a general cohomological theory of padic L-functions, or even a finer theory modulo powers of p, which would explain all congruences between rational parts of special values arising from congruences between Galois representations. Firsts steps in this direction have been taken in [52],[53],[62].

(B) p-adic regulators: Let X be a smooth projective variety over a finite extension F of Q_{p} and X a regular proper flat model of X over the ring of integers O of F. For V = H^{m}((X ⊗_{F} \overline{F})_{et}, Q_{p}(n)) with m − 2n ≤ −2, investigate the Chern character (‘p-adic regulator’)

r_{p} : K_{2n−m−1}(X) ⊗ Q ⟶ H^{1}(F, V)

In particular, show that the image of r_{p} is contained in H_{f}^{1}(F, V) (this is proved - modulo certain unchecked compatibility - in [22] and [63], provided p > n, X is smooth over O and F/Q_{p} is unramified; the same method works even in the ramified case) and give formulas for r_{p}. (C) Explicit reciprocity laws: Formulate and prove general explicit reciprocity laws for p-adic Galois representations (cf. [7],[32]). A conjectural formula for crystalline Galois representations is given in [51],[52]. (D) More examples: Produce new examples; following (E5), (E6), (E8) and (E9) one could try to explore the geometry of higher dimensional Shimura varieties. For example, is it possible to produce Stark units using Hilbert-Blumenthal modular varieties? These problems, while not unrelated to each other, could be pursued separately. Any progress on them, however, would seem to require a more solid foundational background than is currently available, both in p-adic Hodge theory and in our understanding of L-functions of Shimura varieties. Nevertheless, in view of potential arithmetic consequences, each of them is a worthy cause.

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Mathematical Institute, Charles University Sokolovská 83 186 00 Praha 8 Czech Republic

Values of L-functions 35

Department of Mathematics^{1} UCB Berkeley, CA 94720 U.S.A.

1 Miller Fellow