Number Theory Seminar - Spring 2023

Abstract : 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.

Abstract : 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.

Abstract : 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.

Abstract : TBD

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.

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.

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.

Abstract : Using our Hecke modules of overconvergent modular symbols, we construct the corresponding eigencurve by following the machinery we developed a few weeks ago. We will use various properties of the modules to prove certain compatibility, restriction and specialization theorems. Finally, we will compare this construction with the more classical one of Coleman and Mazur.

Abstract : We will define classical points on the eigencurve of modular symbols. Then, by studying the family of Galois representations carried by this eigencurve, we will show that the only classical points are the "very classical points" and the "Hida classical points". If time permits, we will also discuss the ordinary locus of the eigencurve and its local geometry.

Abstract : Using the classification of classical points that we established last week, we will show that the eigencurve is étale at non-critical slope normal classical points. Moreover, we will establish smoothness and étaleness criterions for critical slope very classical points. Finally, we will give a rough overview of the geometry near the other types of classical points as well as global properties of the eigencurve.

Abstract : Previously, we established the construction of a p-adic L-function for a fixed non-critical slope modular forms. We now use the eigencurve to allow ourselves to vary this modular form inside a Coleman family. This way, we will obtain a 2-variable p-adic L-function for overconvergent eigenforms.

Abstract : Using the cohomological variant of the eigenvariety machine, we will construct slightly more general eigencurves for GL_2. This allows us to define an "adjoint L-ideal sheaf" over the latter as well as a p-adic adjoint L-function locally. If time permits, we will discuss its relattion with the classical adjoint L-function of a modular form as well as Hida's formula.