Seminar

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Families of p-adic cusp forms were first introduced by Hida, later leading to the construction of the eigencurve by Coleman and Mazur. Generalizations to reductive groups of higher rank, called eigenvarieties, are rigid analytic spaces providing  the correct setup for the study of  p-adic deformations of automorphic forms. In order to obtain  arithmetic applications, such  as constructing p-adic L-functions or proving explicit reciprocity laws  for Euler systems, one needs to  perform a meaningful limit process requiring to understand the geometry of the eigenvariety at the point corresponding to the p-stabilization of the  automorphic form we are interested in. 

While the geometry of an eigenvariety at  points of cohomological weight is well understood  thanks to  classicality results, the study at  classical  points which are limit of discrete series (such as weight 1 Hilbert modular forms or weight (2,2) Siegel modular forms) is much more involved and the smoothness at such points is a crucial input in the proof of many cases of the Bloch--Kato Conjecture, the Iwasawa Main Conjecture and Perrin-Riou's Conjecture.

Far more fascinating is the study of the geometry at singular points, especially at those arising as intersection between irreducible components of the eigenvariety, as those are related to trivial zeros of adjoint p-adic L-functions.

In this talk we will illustrate some of these ideas using a recent joint work with Adel Betina and Sheng-Chi Shih on the geometry of the Hilbert cuspidal eigenvariety at  weight 1 Eisenstein points. 

ES Program Special Seminar: P-Adic Deformations Of Automorphic Forms And Iwasawa Theory