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 pioneering work of Iwasawa (cf. for example [Iw 64], [Iw 69], or [Co 99]) revealed that much is to be gained by replacing the classical L-series, an analytic function of a complex variable, by a corresponding function of a p-adic variable. Ideally, the definition of the p-adic L-function should closely mimic that of its classical counterpart, while bearing a more direct relation to (p-adic, or eventually -adic) cohomology, so that the resulting analogues of the Birch and Swinnerton-Dyer conjecture becomemore tractable. The first steps in investigating the Birch and Swinnerton–Dyer conjecture along p-adic lines were taken by Manin [Ma 72] and Mazur and SwinnertonDyer [Mz-SD 74], [Mz 71], who attached a p-adic L-function L p(E, s) to a modular elliptic curve E , or, more generally, to a cuspidal eigenform on a congruence subgroup of SL2(Z). The key ingredient in the construction of L p(E, s), recalled in Chapter 2, is the notion of a modular symbol, which relies on the classical modular parametrisation