An Invitation to p-adic Methods in Number Theory


from 16:00 to 18:00

From March 22nd (9 lectures)

Classroom T2
Facultat de Matemàtiques i Informàtica.
Universitat de Barcelona
(Gran Via de les Corts Catalanes, 585, 08007 Barcelona).

Course Description 

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.

  1. Basics on p-adic numbers.
  2. p-adic measures.
  3. p-adic interpolation of Riemmann’s zeta function.
  4. Modular forms and their L-functions.
  5. Serre’s approach to p-adic modular forms.
  6. Construction of the p-adic families.
  7. Modular symbols.
  8. Admissible p-adic distributions.
  9. p-adic interpolation of L-functions


Xavier Guitart (Universitat de Barcelona)

Marc Masdeu (Universitat Autònoma de Barcelona)

Santiago Molina (Universitat Politècnica de Catalunya)

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