Abstracts for Oberseminar Arithmetic Geometry and Representation Theory

In recent years, many efforts have been made to formalise, in the most efficient way,  Grothendieck’s six operations of sheaves (tensor product and internal hom, inverse and direct image, and exceptional inverse and direct image) and their properties. In addition, one might want to encode natural isomorphisms between the inverse image and exceptional inverse image for a certain class of “étale” morphisms, and between the direct image and the exceptional direct image for a class of “proper morphisms”.

Associated to a modular form f is a two-dimensional Galois representation whose Frobenius eigenvalues can be expressed in terms of the Fourier coefficients of f, using a formula known as the Eichler--Shimura congruence relation. This relation was proved by Eichler--Shimura and Deligne by analyzing the mod p (bad) reduction of the modular curve of level ?0(p). In this talk, I will discuss joint work with Patrick Daniels, Dongryul Kim and Mingjia Zhang, where we give a new proof of this congruence relation that happens "entirely on the rigid generic fibre".