# p-adic methods in the theory of classical automorphic forms

## March 9-14, 2015

**Organizers ** : Henri Darmon (McGill), Matthew Greenberg (Calgary), Adrian Iovita (Concordia), Payman Kassaei (McGill)

This 6 day workshop will divided into two parts of three days each, with slightly different focus around the common unifying theme of *p*-adic methods in number theory and the theory of automorphic forms. The first half will focus on eigenvarieties and their arithmetic ramifications, and the second will focus on the theory of Euler systems.

**Part I: Eigenvarieties, L-functions, and Galois representations, March 9-11.**

There has been recent progress on the one hand on classical problems related to *p*-adic automorphic forms which let to the construction of the cuspidal eigenvarieties for a large variety of PEL Shimura varieties and on the other hand in the theory of automorphic Galois representations. As a consequence it seems natural to concentrate on the following themes:

1) The local and global geometry of the various eigenvarieties and of their morphisms to the respective weight spaces.

2) Properties of various objects parameterized by the points of the eigenvarieties like
Galois representations and *p*-adic *L*-functions. It would be very interesting to study these variations relative to the geometry of the eigenvarieties.

This part of the workshop will contain a combination of background talks aimed at graduate students and postdoctoral fellows explaining the main themes, the main questions and the current state of our knowledge and research reports on the latest developments.

**Part II: Euler systems and reciprocity laws, March 12-14.**

The second part of the workshop will be devoted to the theme of special algebraic cycles and classes (in Chow groups of null-homologous cycles, or in higher Chow groups) and the Bloch-Beilinson conjectures relating these objects to special values of *L*-functions. It will also center on the *Euler systems* arising when these objects are made to vary in *p-adic families* and compared to the *p*-adic *L*-functions attached to the corresponding *p*-adic families of Galois representations. A sample of the questions that are currently being actively pursued include the construction of Perrin-Riou style regulator maps for *p*-adic families of Galois representations attached to *p*-adic families of finite slope eigenforms, the ongoing study of new Euler systems arising from *p*-adic families of generalised Gross-Schoen diagonal cycles and Beilinson-Flach elements and their application to the Birch and Swinnerton-Dyer conjecture, comparisons between Euler systems, the theory of "Stark-Heegner points", and the recent remarkable progress towards converses of Kolyvagin's theorem and conjectures of "Mazur-Tate type" arising from the work of Chris Skinner and Wei Zhang.