Date 
Title 
Speaker 
Reference 
Notes 
February 08 
Introduction to Eigenalgebras and Eigenvarieties
As motivation for the rest of the semester, we begin the study of eigenalgebras, focusing on the case of Hecke algebras for modular forms. Then, we will initiate the construction of Eigenvarieties. We first set up the required algebraic notions (Fredholm & Riesz Theory, Adapted Pairs) and then briefly discuss some geometric consequences.

David Marcil 
[Bel21, (2.7)(3.3)] 

February 15 
Properties and Machinery of Eigenvarieties
Using the various algebraic concepts introduced last week, we finally define Eigenvariety data and proceed to construct eigenvarieties. We will prove some of their properties and discuss a generalization of this object when working with complexes.

David Marcil 
[Bel21, (3.3)(3.9)] 

February 22 
Modular Symbols and Cohomology
We will introduce abstract modular symbols with values in a system of coefficients V and their relation to cohomology with local coefficients. These will provide examples of Banach modules (over reduced affinoid Banach Q_palgebras) with an Hecke action required for eigenvariety datum. We will later specify to the cases where V is a symmetric power of the GL(2)standard representation or various spaces of distributions. These respectively have relations to classical and padic Lfunctions.

David Marcil 
[Bel21, Ch. 4] 

March 01 
Classical Modular Symbols and Modular Forms
Abstract : TBD

David Marcil 


March 08 
The Fréchet module of Overconvergent Distributions
Abstract : We will construct various modules of sequences and functions until we can define modules of overconvergent distributions. The motivation is to understand tempered distributions as well as measures. These modules will be the ones associated to affinoid neighborhoods of some weight space in the formation of eigenvarieties.

Jiahe Shen 
[Bel21, (6.1)(6.3)] 

March 15 
Spring break  No talk




March 22 
Hecke action on Distributions and Overconvergent Modular Symbols
Abstract : We study the action of GL(2) on the space of overconvergent distributions to transform the latter into a Hecke module. In particular, we will obtain the fundamental exact sequence for overconvergent distributions and the compactness of the usual U_p operator on overconvergent modular symbols. We will conclude with Stevens' Control Theorem and various other results useful in the study of padic Lfunctions on overconvergent modular forms.

Jiahe Shen 
[Bel21, (6.3)(6.5)] 

March 29 
Applications to padic Lfunctions of NonCritical Slope Modular Forms
Abstract : We finish our discussion on overconvergent modular symbols and use them to show how to padic interpolate special Lvalues of classical cuspidal eigenforms whose slope is not too large in comparison to its weight. We will later use the eigencurve to discuss this variation in families.

David Marcil 
[Bel21, (6.5)(6.7)] 
PDF 
April 05 
Comparison between the Eigencurve of Modular Symbols and the ColemanMazur Eigencurve
Abstract : TBD

David Marcil 
[Bel21, (7.1)(7.2)] 

April 12 
Local Geometry of the Eigencurve of Modular Symbols (I)
Abstract : TBD

David Marcil 
[Bel21, (7.3)(7.6)] 

April 19 
Local Geometry of the Eigencurve of Modular Symbols (II)
Abstract : TBD

David Marcil 
[Bel21, (7.6)(7.7)] 

April 26 
padic Lfunctions on the Eigencurve
Abstract : TBD

David Marcil 
[Bel21, Ch. 8] 

May 03 
Abstract : TBD

David Marcil 

