| 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_p-algebras) 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 p-adic L-functions.
|
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 p-adic L-functions on overconvergent modular forms.
|
Jiahe Shen |
[Bel21, (6.3)-(6.5)] |
|
| March 29 |
Applications to p-adic L-functions of Non-Critical Slope Modular Forms
Abstract : We finish our discussion on overconvergent modular symbols and use them to show how to p-adic interpolate special L-values 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 Coleman-Mazur 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 |
p-adic L-functions on the Eigencurve
Abstract : TBD
|
David Marcil |
[Bel21, Ch. 8] |
|
| May 03 |
Abstract : TBD
|
David Marcil |
|
|
