# Overview

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## Higher Coleman theory and applications

This workshop is dedicated to the theme of p-adic variation of automorphic representations. The automorphic representations we are thinking about can be realized in the coherent cohomology of certain PEL Shimura varieties, in fact we will limit ourselves to the Siegel moduli spaces of polarized abelian varieties with level structure.

An old theorem of Langlands and Li-Schwermer states that on a toroidal compactification of a Siegel modular variety of genus g and some level, for generic classical weights, the interior (or parabolic) cohomology with coefficients in the modular sheaf of that weight is concentrated in one degree i, greater or equal to 0.

When that one degree is zero, the p-adic variation of the automorphic sheaves is known, and so also the p-adic variation of the automorphic forms and representations. We refer to this body of knowledge as "Coleman theory."

The p-adic variation of automorphic representations which are realized in the one degree i larger then 0 and less then g, a new result of V. Pilloni and G. Boxer-V. Pilloni, known as "higher Coleman theory," is the main object of study of our workshop, together with its applications to p-adic L-functions and Eichler-Shimura isomorphisms.

The workshop shall be preceeded by a preparatory discussion group that will also meet in the afternoons during the workshop: https://kundudeb.github.io/p-adic-seminar.html