### __ The infinite fern and families of quaternionic modular forms __

This page collects some informations about my course for the Galois
trimester.

There will be eight lectures, on Tuesdays, between January 12 and
March 9 (13h30-16h30) : see here.

Note that there will be no lecture the weeks 7 and 8 and two lectures on week
9.

This course is intended to beginners. The main aim is to explain the
construction of the "infinite fern" of Gouvea & Mazur via the theory of p-adic families of
quaternionic modular forms. The covered topics should include:

- Galois groups of number fields and Galois representations,

- Geometric and Modular Galois representations,

- Mazur's deformation functor, pseudo-deformation functors,

- The p-adic character variety of a profinite group,

- Definite quaternion algebras and their modular forms,

- p-adic and families of quaternionic modular forms,

- p-adic spectral theory and eigenvarieties,

- The infinite fern and the "density of modular points".

No specific background is required besides the basics of algebraic number
theory and algebraic geometry. It may be useful to study independently
a bit of rigid analytic geometry (e.g. as in Tate's inventiones
paper, or in the book of B.G.R. cited below). We shall sometimes use some
concepts introduced in the lectures of Berger and Henniart, and there will be
alsâo relations with the lectures of Clozel and Colmez.

Lecture Notes :

Lecture 0 (Brief
introduction),

Lecture 1 (Global
Galois groups, Galois representations, Tate modules of elliptic curves I),

Lecture 2 (Tate modules of
elliptic curves II, Geometric and modular Galois representations) + some notes on
Eichler-Shimura's theorem
(note quite complete yet),

Lecture 3 (Deformation
functors, Mazur's universal deformation ring, Pseudocharacters),

Lecture 4 (Pseudodeformation rings, Global Galois cohomology,
the regular G_{Q,S}-case) + some explicit computations with Pari,

Lecture 5 (Rigid analytic
spaces, Generic fibers of deformation rings),

Lecture 6 (Definite quaternion
algebras and their modular forms),

Lecture 7 (p-adic modular
forms for definite quaternion algebras),

Lecture 8 (The eigencurve).