Uniformization of modular elliptic curves via p-adic periods

p-ADIC PERIODS, p-ADIC L-FUNCTIONS, AND THE p-ADIC UNIFORMIZATION OF SHIMURA CURVES

p-ADIC UNIFORMIZATION OF CURVES

Let us presume that we have at our disposal the fully-formed theory of rigidanalytic spaces., as sketched in my last talk. Why would we care to look for uniformizations of algebraic objects in the rigid analytic category? Let’s look at the analogous situation in the complex-analytic category. We know that, for example, abelian varieties are uniformized by spaces of the form C/Λ, where Λ is a fr...

The p-Adic L-Functions of Modular Elliptic Curves

The arithmetic theory of elliptic curves enters the new century with some of its major secrets intact. Most notably, the Birch and Swinnerton-Dyer conjecture, which relates the arithmetic of an elliptic curve to the analytic behaviour of its associated L-series, is still unproved in spite of important advances in the last decades, some of which are recalled in Chapter 1. In the 1960’s the pione...

Height pairings on Shimura curves and p-adic uniformization

In a recent paper [16] of one of us it was shown that there is a close connection between the value of the height pairing of certain arithmetic 0-cycles on Shimura curves and the values at the center of their symmetry of the derivatives of certain metaplectic Eisenstein series of genus 2. On the one hand, the height pairing can be written as a sum of local height pairings. For example, if the 0...

Elliptic curves and p-adic uniformisation

The Diophantine theory studies the rational solutions (X, Y, Z) ∈ Q of equation (1). It is convenient to ignore the trivial solution (0, 0, 0) and to identify solutions if they differ by multiplication by a non-zero scalar. Solutions to (1) are thus viewed as points in the projective plane P2(Q). Let E(Q) ⊂ P2(Q) denote this solution set. More generally, if F is any field, let E(F ) ⊂ P2(F ) be...