An Invitation to p-adic Methods in Number Theory

Since their introduction by Kurt Hensel in 1897, p-adic numbers have become ubiquitous in number theory, as they provide a way to use analytic techniques in arithmetic problems. These play a key role in most of modern results in number theory, such as Fermat’s Last Theorem, the known cases of the Birch and Swinnerton-Dyer conjecture, or the proof of the Sato-Tate conjecture. These lectures, aimed at a broad audience, aim to introduce several techniques that illustrate their power.

The first main goal, after introducing the basic notions, will be the construction of a p-adic analogue of Riemann’s zeta function. We will next introduce modular forms, which are central objects in number theory, and their p-adic avatars. The final part of the course will be devoted to the L-series of modular forms (complex-analytic functions which generalize Riemann’s zeta) and how to p-adically interpolate them.

No specialized background will be assumed.