Glenn Stevens

p-adicL-functions andp-adic periods of modular forms

Let E be an elliptic curve which is defined over Q and has stable reduction modulo a given prime p. Assuming that E is modular, one can associate to E a p-adic L-function Lp(E, s). (See [-Mz-SwD, A-V, Vi, Mz-T-T] for its construction in various cases.) This function is defined by a certain interpolation property and is analytic for seZp. In this paper, we will assume that E has split multiplicative reduction at p. Under this assumption the interpolation property implies that Lp(E, 1)=0. We will prove a formula for Ep(E, 1) which was discovered experimentally by Mazur, Tate, and Teitelbaum [Mz-T-T]. By Tates p-adic urfiformization theory, there is a p-adic integer q~:epZp (which we refer to as the Tate period for E) and a p-adic analytic isomorphism

Stickelberger elements and modular parametrizations of elliptic curves

In the present paper we shall give evidence to support the claim (Conjecture I below and (1.3)) that every elliptic curve A/o which can be parametrized by modular functions admits a canonical modular parametrization whose properties can be related to intrinsic properties of A. In particular, we will see how such a parametrizat ion can be used to prove some rather pleasant integrality properties of Stickelberger elements ad p-adic L-functions attached to A. In addition, if Conjecture I is true then we can give an intrinsic characterization of the isomorphism class of a special elliptic curve in the Q-isogeny class of A distinguished by modular considerations. For most of the paper we have opted for the concrete approach and defined modular parametrizations in terms of X I (N) (Definition 1.1). However, to justify our view of these parametrizations as being canonical, we begin here with a more intrinsic definition. Recall that Shimura ([19], Chap. 6; see w 1 of this paper) has defined a compatible system of canonical models of modular curves {Xs, SeS~}, where 5 p is a certain collection of open subgroups of the group GL(2, Az) over the finite adeles A I of Q. We define the adelic upper half-plane to be the pro-variety )~=lL_m Xs and give )~ the Q-structure induced by the s field of modular functions whose q-expansions at the 0-cusp have coefficients in Q. A modular parametrization of A is a Q-morphism ~: ) ( ~ A which sends

Critical slope p-adic L-functions

Let g be an eigenform of weight k+2 on Γ0(p)∩Γ1(N) with p N . If g is non-critical (i.e. of slope less than k + 1), using the methods of [1, 20], one can attach a p-adic L-function to g which is uniquely determined by its interpolation property together with a bound on its growth. However, in the critical slope case, the corresponding growth bound is too large to uniquely determine the p-adic L-function with its standard interpolation property. In this paper, using the theory of overconvergent modular symbols, we give a natural definition of p-adic L-functions in this critical slope case.

Overconvergent Eichler–Shimura isomorphisms

We provide a geometric Hodge-Tate map giving generic description of the overconvergent modular symbols of some p-adic (accessible) weight k, base-changed to C_p, in terms of overconvergent modular forms of weight k+2.

Periods of Modular Forms

In this chapter we develop the tools needed to describe the subgroup of H1(X(Γ);ℚ/ℤ) corresponding to the cuspidal group C(Γ).

An Overview of The Proof of Fermat’s Last Theorem

The principal aim of this article is to sketch the proof of the following famous assertion.

P-adic L-functions and Congruences

Mazur and Swinnerton-Dyer [30] have shown how to pass from a TD-eigenfunction on (ℚ/ℤ)p,Δ to a p-adic distribution on ℤ p,Δ * . Applying this procedure to the universal modular symbol on a modular curve, X, Mazur [26] defines a p-adic L-function LP (X0(N), χ, s) associated to an Eisenstein prime P for Г0(N), N prime.

The Special Values Associated to Cuspidal Groups

Let Γ be a group of type (N1, N2) and E ∈ E 2 (Γ) be a J -eigenfunction. This chapter is devoted to describing the subgroup CE of the cuspidal group and the speical values of the associated cohomology class φE ∈ H1(X;A(E)).

Tables of Special Values

In the remaining pages we display three sets of tables of algebraic parts of special values of L-functions,