GENERALIZED KUMMER CONGRUENCES AND p-ADIC FAMILIES OF MOTIVES

ALEXEI A. PANCHISHKIN

Abstract. We describe some new general constructions of p-adic Lfunctions attached to certain arithmetically defined complex L-functions coming from motives over Q with coefficiens in a number field T, [T : Q] < ∞. These constructions are equivalent to proving some generalized Kummer congruences for critical special values of these complex L-functions.

The paper is based on some talks given by the auther during his visit to MSRI in February and March 1994. The purpose of this paper is to describe some new general constructions of p-adic L-functions attached to certain arithmetically defined complex Lfunctions coming from motives over Q with coefficiens in a number field T, [T : Q] < ∞. These constructions are equivalent to proving some generalized Kummer congruences for critical special values of these complex L-functions.

The starting point in the theory of L-functions is the expansion of the Riemann zeta-function ζ(s) into the Euler product:

ζ(s) = ∏_{p}(1 − p^{−s})^{−1} = ∑_{n=1}^{∞} n^{−s} (Re(s) > 1).

The set of arguments s for which ζ(s) is defined can be extended to all s ∈ C, s ≠ 1, and we may regard C as the group of all continuous quasicharacters

C = Hom(R_{+}^{×}, C^{×}), y ↦ y^{s}

of R_{+}^{×}. The special values ζ(1 − k) at negative integers are rational numbers:

ζ(1 − k) = −\frac{B_{k}}{k} ,

where B_{k} are Bernoulli numbers, which are defined by the formal power series equality

e^{Bt} = ∑_{n=0}^{∞} \frac{B_{n}t^{n}}{n!} = \frac{t e^{t}}{e^{t} − 1} ,

Research at MSRI is suported in part by NSF grant DMS-9022140. and we know (by the classical Sylvester–Lipschitz theorem) that

c ∈ Z implies c^{k}(c^{k} − 1)\frac{B_{k}}{k} ∈ Z.

The theory of non-Archimedean zeta-functions originates in the work of Kubota and Leopoldt containing p-adic interpolation of these special values. Their construction turns out to be equivalent to classical Kummer congruences for the Bernoulli numbers, which we recall here in the following form. Let p be a fixed prime number, c > 1 an integer prime to p. Put

ζ_{(p)}^{(c)}(−k) = (1 − p^{k})(1 − c^{k+1})ζ(−k)

and let h(x) = ∑_{i=0}^{n} α_{i} x^{i} ∈ Z_{p}[x] be a polynomial over the ring Z_{p} of p-adic integers such that s

x ∈ Z_{p} ⟹ h(x) ∈ p^{m}Z_{p}.

Then we have that

∑_{i=0}^{n} α_{i} ζ_{(p)}^{(c)}(−i) ∈ p^{m}Z_{p}.

This property expresses the fact that the numbers ζ_{(p)}^{(c)}(−k) depend continuously on k in the p-adic sense; it can be deduced from the known formula for the sum of k-th powers:

S_{k}(N) = ∑_{n=1}^{N−1} n^{k} = \frac{1}{k + 1}[B_{k+1}(N) − B_{k+1}]

in which B_{k}(x) = (x+B)^{k} = ∑_{i=0}^{k} \binom{k}{i}B_{i}x^{k−i} denotes the Bernoulli polynomial. Indeed, all summands in S_{k}(N) depend p-adic analytically on k, if we restrict ourselves to numbers n, prime to p, so that the desired congruence follows if we express the numbers ζ_{(p)}^{(c)}(−k) in terms of Bernoulli numbers.

The domain of definition of p-adic zeta functions is the p-adic analytic Lie group

X_{p} = Hom_{contin}(Z_{p}^{×}, C_{p}^{×})

of all continuous p-adic characters of the profinite group Z_{p}^{×}, where C_{p} = ˆ \overline{Q}_{p} denotes the Tate field (completion of an algebraic closure of the p-adic field Q_{p}), so that all integers k can be regarded as the characters x_{p}^{k} : y ↦ y^{k}. The construction of Kubota and Leopoldt is equivalent to existence a p-adic analytic function ζ_{p} : X_{p} → C_{p} with a single pole at the point x = x_{p}^{−1}, which becomes a bounded holomorphic function on X_{p} after multiplication by the elementary factor (x_{p} x − 1) (x ∈ X_{p}), and is uniquely determined by the condition

ζ_{p}(x_{p}^{k}) = (1 − p^{k})ζ(−k) (k ≥ 1).

This result has a very natural interpretation in framework of the theory of non-Archimedean integration (due to Mazur): there exists a p-adic measure μ^{(c)} on Z_{p}^{×} with values in Z_{p} such that ∫_{Z_{p}^{×}} x_{p}^{k} μ^{(c)} = ζ_{(p)}^{(c)}(−k). Indeed, if we integrate h(x) over Z_{p}^{×} we exactly get the above congruence. On the other hand, in order to define a measure μ^{(c)} satisfying the above condition it suffices for any continuous function ϕ : Z_{p}^{×} → Z_{p} to define its integral ∫_{Z_{p}^{×}} ϕ(x) μ^{(c)}. For this purpose we approximate ϕ(x) by a polynomial (for which the integral is already defined), and then pass to the limit.

The important feature of the construction is that it equally works for primitive Dirichlet characters χ modulo a power of p: if we fix an embedding i_{p} : \overline{Q} ↪ C_{p} then the character χ : (Z/Z_{p^{N}})^{×} → (\overline{Q})^{×} becomes an element of the torsion subgroup X_{p}^{tors} ⊂ X_{p} and the above equality also holds for the special values L(−k, χ) of the Dirichlet L-series

L(s, χ) = ∑_{n=1}^{∞} χ(n) n^{−s} = ∏_{p}(1 − χ(p)p^{−s})^{−1},

so that we have

ζ_{p}(χx_{p}^{k}) = i_{p}[(1 − χ(p)p^{k})L(−k, χ)] (k ≥ 1, k ∈ Z, χ ∈ X_{p}^{tors}).

The original construction of Kubota and Leopoldt was successesfully used by Iwasawa for the description of the class groups of cyclotomic fields. Since then the class of functions admitting p-adic analogues has gradually extended.

L-functions (of complex variable) can be attached as certain Euler products to various objects such as diophantine equations, representations of Galois groups, modular forms etc., and they play a crucial role in modern number theory. Deep interrelations between these objects discovered in last decades are based on identities for the corresponding L-functions which presumably all fit into a general concept of the Langlands of L-functions associated with automorphic representations of a reductive group G over a number field K. ¿From this point of view the study of arithmetic properties of these zeta function is becoming especially important.

The major sources of such L-functions are:

1) Galois representations of G_{K} = Gal(\overline{K}/K) for algebraic number fields K, r : G_{K} → GL(V), V a finite dimensional vector space, and one can attach to r an Euler product due to Artin.

2) Algebraic varieties X defned over an algebraic number field K. In this case one can attach to X/K its Hasse–Weil zeta function.

3) Automorphic forms and automorphic representations. In the classical case one associates to a modular form f(z) = ∑_{n=0}^{∞} a_{n} exp(2πinz) its Mellin transform L(s, f) = ∑_{n=1}^{∞} a_{n}n^{−s}. In general an automorphic form generates an automorphic representations in the space of smooth functions over an adelic reductive group, and one can attach an Euler product to it using a decomposition of such a representation into a tensor product indexed by prime numbers p and ∞.

Conjecturally, all the three type of L-functions can be related to each other using a general theory of motives over Q with coefficiens in a number field T, [T : Q] < ∞ (this field coincides with the field Q({a_{n}}_{n≥1}) generated by the coefficients of the corresponding L-function L(M, s) = ∑_{n=1}^{∞} a_{n}n^{−s}.) For a fixed prime number p one can also attach in many cases to the above complex L-function a p-adic L-function. These p-adic L-functions are certain analytic functions in a p-adic domain obtained by an interpolation procedure of certain special values of the corresponding complex analytic L-functions. Their existence is equivalent to certain generalized Kummer congruences for the special values.

We describe a general conjecture on the existence of a p-adic family M_{P} of motives, parametrised by some dense subset of algebraic characters P of a p-adic commutative algebraic group (which we call group of Hida). This group can be regarded as (a maximal torus of) the p-adic part G_{M,p} of the motivic Galois group G_{M} of M (the Tannakian group for the tensor category generated by M). The important condition of motives M_{P} of the above family is that they have the same fixed p-invariant h = h_{P} (generalized slope), which is defined as the difference between the Newton polygon and the Hodge polygon of a motive at certain point d^{+} (the dimension of the subspace M^{+} of the Betti realization M_{B} of M). The corresponding p-adic L-functions of this family can be unbounded in the ”cyclotomic direction” (of Amice–Vélu type [Am-Ve]) but they form a family which is conjectually bounded in the ”weight direction”, that is for P parametrized by algebraic characters of G_{M,p}.

More precisely, the values of the function P ↦ L(M_{P} , 0) satisfy generalised Kummer congruences in the followinfg sense: for any finite linear combination ∑_{P} b_{P} · P with b_{P} ∈ C_{p} which has the property ∑_{P} b_{P} · P ≡ 0(modp^{N}) we have that for some constant C ≠ 0 the corresponding linear combination of the normalized L-values

C ∑_{P} b_{P} c_{p}(M_{P}) · \frac{L_{(p,∞)}(M_{P} , 0)}{c_{∞}(M_{P})} ≡ 0(modp^{N}).

Here c_{p}(M_{P}) and c_{∞}(M_{P}) denote a p-adic and a complex period of M_{P} so that the

ratio ”\frac{c_{p}(M_{P})}{c_{∞}(M_{P})}” is uniquely defined, and L_{(p,∞)}(M_{P} , s) denotes the above L-function

L(M_{P} , s) normalized by multiplying by a certain canonicaly defined Deligne’s p-factor corresponding to a choice of inverse roots α^{(1)}(p), …, α^{(d^{+})}(p) ∈ C_{p} of p-local polynomial of M such that

ord_{p}(α^{(1)}(p)) ≤ ord_{p}(α^{(2)}(p)) ≤ … ord_{p}(α^{(d)}(p)),

d being the common rank of the family M_{P}, d^{+} the T -dimension of the Deligne’a subspace M^{+} of M_{B} (the fixed subspaces of the canonical involution ρ of M over T).

Recent examples of such families related to modular forms were constructed by R.Coleman [CoPBa] who proved the following

Theorem. Suppose α ∈ Q and ε : (Z/pZ)^{×} → C_{p}^{×} is a character. Then there exists a number n_{0} which depends on p, N and ε, and α with the following property: If k ∈ Z, k > α + 1 and there is a unique normalized cusp form F on X_{1}(Np) of weight k, character εω^{−k} and slope α and if k′ > α + 1 is an integer congruent tok modulo p^{n+n_{0}}, for any positive integer n, then there exists a unique normalized cusp form F′ on X_{1}(Np) of weight k′, character εω^{−k′} and slope α (ω denotes the Teichmü ller character). Moreover his form satisfies the congruence

F′(q) ≡ F′(q)(modp^{n+1}).

This result can be regarded as a generalization of the work of Hida [HiGal] who considered the case α = 0 and constructed interesting families of Galois representations of the type

ρ_{p} : G_{Q} → GL_{2}(Z_{p}[[T]]), G_{Q} = Gal(\overline{Q}/Q),

which are non ramified outside p. These representations have the following property: if we consider the homomorphisms

Z_{p}[[T]] ⟶^{s_{k}}Z_{p}, 1 + T ↦ (1 + p)^{k−1},

then we obtain a family of Galois representations

ρ_{p}^{(k)} : G_{Q} → GL_{2}(Z_{p}),

which is parametrized by k ∈ Z, and for k = 2, 3, ⋯, these representations are equivalent over Q_{p} to the p-adic representations of Deligne, attached to modular forms of weight k. This means that the representations of Hida are obtained by the p-adic interpolation of Deligne’s representations. A geometric interpretation of Hida’s representations was given by Mazur and Wiles [Maz–W], cf. [Maz]. For example, for the modular form Δ of weight 12 Hida has constructed a representation

ρ_{p,Δ} : G_{Q} → GL_{2}(Z_{p}[[T]]),

as an example of his general theory, where the prime number p have the property τ(p) ≢ 0(modp) (e.g. p < 2041, p ≠ 2, 3, 5 and 7).

The boundednes property is the subject of a research by G.Stevens and of a forthcoming paper of B.Mazur and F.Q.Gouvêa.

Note that other examples may include Rankin products, Garrett triple products of elliptic and Hilbert modular forms and standard L-functions of Siegel modular forms.

To describe this conjecture more precisely, let M is a motive over Q of with coefficients in T i.e.

M_{B}, M_{DR}, M_{λ}, I_{∞}, I_{λ},

where M_{B} is the Betti realization of M which is a vector space over T of dimension d endowed with a T -rational involution ρ ; M_{B} = M^{+} ⊕ M^{−} denotes the corresponding decomposition into the sum of (+1)-eigenspaces and (−1)-eigenspaces of ρ.

M_{DR} is the de Rham realization of M, a free T -module of rank d, endowed with a decreasing filtration {F_{DR}^{i}(M) ⊂ M_{DR} | i ∈ Z} of T -modules; M_{λ} is the λ-adic realization of M at a finite place λ of the coefficient field T (a T_{λ}- vector space of degree d over T_{λ}, a completion of T at λ) which is a Galois module over G_{Q} = Gal(\overline{Q}/Q) so that we a have a compatible system of λ-adic representations denoted by

r_{M,λ} = r_{λ} : G_{Q} → GL(M_{λ}).

Also,

I_{∞} : M_{B} ⊗_{T} C → M_{DR} ⊗_{T} C

is the complex comparison isomorhism of complex vector spaces

I_{λ} : M_{B} ⊗_{T} T_{λ} → M_{λ}

is the λ-adic comparison isomorphism of T_{λ}-vector spaces. It is assumed in the notation that the complex vector space M_{B} ⊗_{Q} C is decomposed in the Hodge bigraduation

M_{B} ⊗_{T} C = ⊕_{i,j}M^{i,j}

in which ρ(M^{i,j}) ⊂ M^{j,i} and

h(i, j) = h(i, j, M) = dim_{C} M^{i,j}

are the Hodge numbers. Moreover,

I_{∞}(⊕_{i′≥i}M^{i′,j}) = F_{DR}^{i}(M) ⊗ C.

Also, I_{λ} takes ρ to the r_{λ}-image of the Galois automorphism which corresponds to the complex conjugation of C. We assume that M is pure of weight w (i.e. i + j = w).

The L-function L(M, s) of M is defined as the following Euler product:

L(M, s) = ∏_{p} L_{p}(M, p^{−s}),

extended over all primes p and where

L_{p}(M, X)^{−1} = det (1 − X · r_{λ}(F r_{p}^{−1}) | M_{λ}^{I_{p}}) =
(1 − α^{(1)}(p)X) · (1 − α^{(2)}(p)X) · . . . · (1 − α^{(d)}(p)X) =
1 + A_{1}(p)X + … + A_{d}(p)X^{d};

here F r_{p} ∈ G_{Q} is the Frobenius element at p, defined modulo conjugation and modulo the inertia subgroup I_{p} ⊂ G_{p} ⊂ G_{Q} of the decomposition group G_{p} (of any extension of p to \overline{Q}). We make the standard hypothesis that the coefficients of L_{p}(M, X)^{−1} belong to T, and that they are independent of λ coprime to p. Therefore we can and we shall regard this polynomial both over C and over C_{p}. We shall need the following twist operation: for an arbitrary motive M over Q with coefficients in T an integer m and a Hecke character χ of finite order one can define the twist N = M(m)(χ) which is again a motive over Q with the coefficient field T (χ) of the same rank d and weight w so that we have

L(N, s) = ∏_{p} L_{p}(M, χ(p)p^{−s−n}).

The group of Hida and the algebra of Iwasawa–Hida. Now let us fix a motive M with coefficients in T = Q(⟨a(n)_{n}⟩) of rang d and of weight w, and let End_{T} M denote the endomorphism algebra of M (i.e. the algebra of T -linear endomorphisms of any M_{B}, which commute with the Galois action under the comparison isomorphisms). Let

Gal_{p} = Gal(Q_{p,∞}^{ab}/Q)

denotes the Galois group of the maximal abelian extension Q_{p,∞}^{ab} of Q unramified outside p and ∞. Define O_{T,p} = O_{T} ⊗ Z_{p}.

Definition. The group of Hida GH_{M} = GH_{M,p} is the following product

GH_{M} = (End_{T} M)^{×}(O_{T,p}) × Gal_{p},

where (End_{T} M)^{×} denotes (a maximal torus of) the algebraic T -group of invertible elements of End_{T} M (it is implicitely supposed that the group End_{T} M^{×} posesses an O_{T} -integral structure given by an appropriate choice of an O_{T} -lattice).

Consider next the C_{p}-analytic Lie group

X_{M,p} = Hom_{contin}(GH_{M}, C_{p}^{×})

consisting of all continuous characters of the Hida group GH_{M}, which contains the C_{p}-analytic Lie group

X_{p} = Hom_{contin}(Gal_{p}, C_{p}^{×})

consisting of all continuous characters of the Galois group Gal_{p} (via the projection of GH_{M} onto Gal_{p}.

The group X_{M,p} contains the discrete subgroup A of arithmetical characters of the type

χ · η · x_{p}^{m} = (χ, η, m),

where

χ ∈ X_{M,p}^{tors}

is a character of finite order of GH_{M}, η is a T -algebraic character of (End_{T} M)^{×}(O_{T,p}), m ∈ Z, and x_{p} denotes the following natural homomorphism

x_{p} : Gal_{p} = Gal(Q_{p,∞}^{ab}/Q) ≅ Z_{p}^{×} → C_{p}^{×}, x_{p} ∈ X_{p}.

Definition. The algebra of Iwasawa–Hida I_{M} = I_{M,p} of M at p is the completed group ring O_{p}[[GH_{M}]], where O_{p} denotes the ring of integers of the Tate field C_{p}.

Note that this definition is completely analogous to the usual definition of the Iwasawa algebra Λ as the completed group ring Z_{p}[[Z_{p}]] if we take into account that Z_{p} coincides with the factor group of Z_{p}^{×} modulo its torsion subgroup.

Now for each arithmetic point P = (χ, η, m) ∈ A we have a homomorphism

ν_{P} : I_{M,p} → O_{p}

which is defined by the corresponding group homomorphism

P : GH_{M} → O_{p}^{×} ⊂ C_{p}^{×}.

For a I_{M}-module N and P ∈ A we put

N_{P} = N ⊗_{I_{M} ,ν_{P}} O_{p}

(”reduction of N modulo P ”, or a fiber of N at P).

Therefore, for a Galois representation

r_{N} : G_{Q} → GL(N)

of the above type its reduction r_{N_{P}} = r mod P is defined as the natural composition:

G_{Q} → GL(N) → GL(N_{P}).

Remark. In his very recent work [HiGen] Hida gives another version of the above definition, but he starts from a Galois representation φ : Gal(\overline{F} /F) ⟶ GL_{n}(I), where I = O_{K}[[T_{n}(Z_{p}]] and T_{n} the maximal split torus of Res_{O_{F} /Z}GL(n) for the integer ring O_{F} of F, and for the integer ring O_{K} of a suffficiently large finite extension K of Q_{p}. He is interested in representations φ satisfying the following condition:

There are arithmetic points P ”densly populated” in Spec(I(K) such that the Galois representation φ_{P} = P ∘ φ is the p-adic étale realization of a rank n pure motive M_{P} of weight w defined over F with coefficients in a number field E_{P} in \overline{Q}.

We are trying to resolve an inverse problem and to include a given motive M in a maximal possible p-adic family M_{P} parametrized by arithmetic characters of a certain group which we suppose to consist of an ”algebraic part” (End_{T} M)^{×}(O_{T,p}) and of a ”Galois part” Gal_{p}.

A conjecture on the existence of p-adic families of Galois representations attached to motives. Note first that the fixed embeddings T ↪ C,

i_{∞} : \overline{Q} → C, i_{p} : \overline{Q} → C_{p}

define a place λ(p) of T attached to the corresponding composition

T ↪ \overline{Q} →^{i_{p}} C_{p}.

Conjecture I. For every M of rang d with coefficients in T there esists a free I_{M}- module M_{I} of the same rang d, a Galois representation

r_{I} : G_{F} → GL(M_{I}),

a dense subset A′ ⊂ A of characters, and a distinguished point P_{0} ∈ A such that

(a) the reduced Galois representation

r_{I,P_{0}} : G_{F} → GL(M_{I,P_{0}})

is equivalent over C_{p} to the λ(p)-adic representation r_{M,λ(p)} of M at the distinguished place λ(p); (b) for every P ∈ A′ there exists a motive M_{P} over Q of the same rang d such that its λ(p)-adic Galois representation is equivalent over C_{p} to the reduction

r_{I,P} : G_{Q} → GL(M_{I,P}).

We call the module M_{I} the realization of Iwasawa of M.

A generalization of the Hasse invariant for a motive. We define the generalized Hasse invariant of a motive in terms of the Newton polygons and the Hodge polygons of a motive. Properties of these polygons are closely related to the notions of a p-ordinary and a p-admissible motive.

Now we are going to define the Newton polygon P_{Newton}(u) = P_{Newton}(u, M) and the Hodge polygon P_{Hodge}(u) = P_{Hodge}(u, M) attached to M. First we consider (using i_{∞}) the local p-polynomial

L_{p}(M, X)^{−1} = 1 + A_{1}(p)X + ⋯ + A_{d}(p)X^{d}
= (1 − α^{(1)}(p)X) · (1 − α^{(2)}(p)X) · . . . · (1 − α^{(d)}(p)X),

and we assume that its inverse roots are indexed in such a way that

ord_{p}α^{(1)}(p) ≤ ord_{p}α^{(2)}(p) ≤ ⋯ ≤ ord_{p}α^{(d)}(p)

Definition. The Newton polygon P_{Newton}(u)(0 ≤ u ≤ d) of M at p is the convex hull of the points (i, ord_{p}A_{i}(p)) (i = 0, 1, ⋯, d).

The important property of the Newton polygon is that the length the horizontal segment of slope i ∈ Q is equal to the number of the inverse roots α^{(j)}(p) such that ord_{p}α^{(j)}(p) = i (note that i may not necessarily be integer but this will be the case for the p-ordinary motives below).

The Hodge polygon P_{Hodge}(u) (0 ≤ u ≤ d) of M is defined using the Hodge decomposition of the d-dimensional C-vector space

M_{B} = M_{B} ⊗_{T} C = ⊕_{i,j}M^{i,j}

where M^{i,j} as a C-subspace.

Definition. The Hodge polygon P_{Hodge}(u) is a function [0, d] → R whose graph consists of segments passing through the points

(0, 0), …, (∑_{i′≤i} h(i′, j), ∑_{i′≤i} i′h(i′, j)),

so that the length of the horizontal segment of the slope i ∈ Z is equal to the dimension h(i, j).

Now we recall the definition of a p-ordinary motive (see [Co], [Co–PeRi]). We assume that M is pure of weight w and of rank d. Let G_{p} be the decomposition group (of the place λ(p) in T over p) and

ψ_{p} : G_{p} → Z_{p}^{×}

be the cyclotomic character of G_{p}. Then M is called p ordinary at p if the following conditions are satisfied:

(i) The inertia group I_{p} ⊂ G_{p} acts trivially on each of the l-adic realizations M_{l} for l ≠ p; (ii) There exists a decreasing filtration F_{p}^{i}V on V = M_{p} = M_{B} ⊗ Q_{p} of Q_{p}-subspaces which are stable under the action of G_{p} such that for all i ∈ Z the group G_{p} acts on F_{p}^{i}V/F_{p}^{i+1}V via some power of the cyclotomic character, say ψ_{p}^{−e_{i}}. Then

e_{1}(M) ≥ ⋯ ≥ e_{t}(M)

and the following properties take place:

(a)

dim_{Q_{p}} F_{p}^{i}V/F_{p}^{i+1}V = h(e_{i}, w − e_{i});

(b) The Hodge polygon and the Newton polygon of M coincide:

P_{Newton}(u) = P_{Hodge}(u).

If furthermore M is critical at s = 0 then it is easy to verify that the number d_{p} of the inverse roots α^{(j)}(p) with

ord_{p}α^{(j)}(p) < 0 is equal to d^{+} = d^{+}(M) of M_{B}^{+}.

However, it turns out that the notion of a p-ordinary motive is too restrictive, and we have introduced the following weaker version of it.

Definition. The motive M over F with coefficients in T is called admissible at p if

P_{Newton}(d^{+}) = P_{Hodge}(d^{+})

here d^{+} = d^{+}(M) is the dimension of the subspace M^{+} ⊂ M_{B}. In the general case we use the following quantity (a ”generalized slope”) h = h_{p} which is defined as the difference between the Newton polygon and the Hodge polygon of M:

h_{p} = P_{Newton}(d^{+}) − P_{Hodge}(d^{+}).

of M at p. Note the following important properties of h:

(i) h = h(M) does not change if we replace M by its Tate twist. (ii) h = h(M) does not change if we replace M by its twist M = M(χ) with a Dirichlet character χ of finite order whose conductor is prime to p. (iii) h = h(M) does not change if we replace M by its dual M^{∨}

In the next section we state in terms of this quantity a general conjecture on p-adic L-functions.

A conjecture on the existence of certain families of p-adic L-functions. We are going to describe families of p-adic L-functions as certain analytic functions on the total analytic space, the C_{p}-analytic Lie group

X_{M,p} = Hom_{contin}(GH_{M}, C_{p}^{×}),

which contain the C_{p}-analytic Lie subgroup (the cyclotomic line) X_{p} ⊂ X_{M,p}:

X_{p} = Hom_{contin}(Gal_{p}, C_{p}^{×}).

In order to do this we need a modified L-function of a motive. Following J. Coates this modified L-function has a form appropriate for further use in the p-adic construction. First we multiply L(M, s) by an appropriate factor at infinity and define

Λ_{(∞)}(M, s) = E_{∞}(M, s)L(M, s)

where E_{∞}(M, s) = E_{∞}(τ, R_{F/Q}M, ρ, s) is the modified Γ-factor at infinity which actually does not depend on the fixed embedding τ of T into C. Also we put

c^{ν}(M) = (c^{ν}(M)^{(τ)}) = c^{ν}(RM)(2πi)^{r(RM)} ∈ (T ⊗ C)^{×}

where

ν = (−1)^{m}, r(M) = ∑_{j<0} jh(i, j) = ∑_{j<0} jh(i, j),

c^{ν}(M) is the period of M. Note that the quantity r(M) has a natural geometric interpretation as the minimum of the Hodge polygon P_{Hodge}(M).

We define

\begin{matrix}Λ_{(p,∞)}(M(m)(χ), s) = \\ G(χ)^{−d^{ε_{0}} (M(m)(χ))} ∏_{p|p} A_{p}(M(m)(χ), s) · Λ_{(∞)}(M(m)(χ), s) \end{matrix} ,

where

A_{p}(M(χ), s) = \{ \begin{matrix}∏_{i=d^{+}+1}^{d}(1 − χ(p)α^{(i)}(p)p^{−s}) ∏_{i=1}^{d^{+}}(1 − χ^{−1}(p)α^{(i)}(p)^{−1}p^{s−1}) \\ for p |̸ 𝔠(χ) \\ ∏_{i=1}^{d^{+}} (\frac{p^{s}}{α^{(i)}(p)})^{ord_{p}𝔠(χ)}, otherwise. \end{matrix}

Let A be the discrete subgroup A of arithmetical characters,

χ · η · x_{p}^{m} = (χ, η, m) ∈ A,

A′ ⊂ A a certain ”dense” subset of characters, P_{0} ∈ A a distinguished point of conjecture I. Let A′′ ⊂ A′ be the subset of critical elements, which consists of those P, for which the corresponding motives M_{P} are critical (at s = 0). Now we are ready to formulate the following

Conjecture II. There exists a certain choice of complex periods Ω_{∞}(P) ∈ C^{×} and p-adic periods Ω_{p}(P) ∈ C_{p}^{×} for all P ∈ A′′ such that “the ratio” Ω_{p}(P)/Ω_{∞}(P) is canonicaly defined, and there exists a C_{p}-meromorphic function

L_{M} : X_{M,p} → C_{p}

with the properties:

(i)

L_{M}(P) = Ω_{p}(M)\frac{Λ_{p,(∞)}(M(m)(χ), 0)}{Ω_{∞}(P)}

for almost all P ∈ A′′;

(ii) For arithmetic points of type

P = (χ, η, m) ∈ A′′

with η fixed there exists a finite set Ξ ⊂ X_{M,p} of p-adic characters and positive integers n(ξ) (for ξ ∈ Ξ) such that for any g_{0} ∈ Gal_{p} we have that the function

∏_{ξ∈Ξ}(x(g_{0}) − ξ(g_{0}))^{n(ξ)}L_{M}(x · P)

is holomorphic on X_{p};

(iii) For arithmetic points of type

P = (χ, η, m) ∈ A′′

with η fixed the function in (ii) is bounded if and only if the invariant h(P) = h(M_{P}) vanishes;

(iv) In the general case the function L_{M}(P · x) of x ∈ X_{p} is of logarithmic growth type o(log(·)^{h}_{0}) with

h_{0} = [h] + 1.

(v) For arithmetic points of type

P = (χ, η, m) ∈ A′′

with χ and m fixed the function in (ii) is always bounded if the Hasse invariant h(P) = h(M_{P}) does not depend on η.

Note that the assertion (v) means in particulary that the values of the function

P ↦ Ω_{p}(M)\frac{Λ_{(p,∞)}(M_{P} , 0)}{Ω_{∞}(P)}

satisfy generalised Kummer congruences in the followinfg sense: for any finite linear combination ∑_{P} b_{P} · P with b_{P} ∈ C_{p} which has the property ∑_{P} b_{P} · P ≡ 0(modp^{N}) we have that for some constant C ≠ 0 the corresponding linear combination of the normalized L-values

C ∑_{P} b_{P} Ω_{p}(M_{P}) · \frac{Λ_{(p,∞)}(M_{P} , 0)}{Ω_{∞}(P)} ≡ 0(modp^{N}).

In the case of families of supersingular modular forms studied by R.Coleman [CoPBa] the invariant h(P) reduces to the slope of a modular form in such a family.

References

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[PaLNM] Panchishkin A.A.: Non-Archimedean L-functions of Siegel and Hilbert modular forms, Lecture Notes in Math., 1471, Springer, 1991.

[PaAdm] Panchishkin A.A.: Admissible Non-Archimedean standard zeta functions of Siegel modular forms, Proceedings of the Joint AMS Summer Conference on Motives, Seattle 1991, AMS, Providence, R.I., 1993, vol.2, 251–292.

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Mathematical Science Research Institute, Berkeley, CA 94720, USA

and

Institut Fourier, B.P.74, 38402 St.-Martin d’Hères, France