An Invitation to p-adic Methods in Number Theory

An Invitation to p-adic Methods in Number Theory

 

Dates

Nine two-hour weekly lectures. The meeting days and time will be decided taking into account the availability of the participants.

Starting the week of the 23rd of March, 2020.

Location

Historical building of the 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.

The course will be organized into nine two-hour weekly lectures, which will take place at the historic building of Universitat de Barcelona. The meeting days and time will be decided taking into account the availability of the participants.

Dates: starting the week of the 23th of March, 2020.

 

Contents
  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

An Invitation to p-adic Methods in Number Theory

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Lecturers

Xavier Guitart (Universitat de Barcelona)

Marc Masdeu (Universitat Autònoma de Barcelona)

Santiago Molina (Universitat Politècnica de Catalunya)

 

Dates

Nine two-hour weekly lectures. The meeting days and time will be decided taking into account the availability of the participants, starting the week of the 23rd of March, 2020.

Location

Historical building of the Universitat de Barcelona (Gran Via de les Corts Catalanes, 585, 08007 Barcelona).

Course Description 

Since their introduction by Kurt Hensel in 1987, 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.

The course will be organized into nine two-hour weekly lectures, which will take place at the historic building of Universitat de Barcelona. The meeting days and time will be decided taking into account the availability of the participants.

Dates: starting the week of the 29th of March, 2020.

Lecturers

TBP

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