Tag - Cohomology theory

Lue Pan: On the locally analytic vectors of the completed cohomology of modular curves

22nd October, 2020

A classical result identifies holomorphic modular forms with highest weight vectors of certain representations of SL2(ℝ). We study locally analytic vectors of the (p-adically) completed cohomology of modular curves and prove a p-adic analogue of this result. As applications, we are able to prove a classicality result for overconvergent eigenforms and give a new proof of Fontaine-Mazur conjecture in the irregular case under some mild hypothesis. One technical tool is relative Sen theory which allows us to relate infinitesimal group action with Hodge(-Tate) structure.

Tiago Fonseca: A crash course on modular forms and cohomology

11th September, 2020

An LMS online lecture course in modular forms.

This is a geometrically flavoured introduction to the theory of modular forms. We will start with a standard introduction to some basic analytic aspects concerning modular forms and to their interpretation as sections of line bundles on modular curves.

Then, our main goal will be to explain how one can attach certain 2-dimensional cohomology groups to Hecke eigenforms. In this course, we will only deal with algebraic de Rham and Betti cohomology, but this can also serve to build geometric intuition on the l-adic setting, which gives rise to the famous l-adic representations attached to modular forms.

We will finish with a discussion on the Eichler-Shimura isomorphism, periods of modular forms, and, depending on time, Manin's theorem on the critical values of L-functions of modular forms.

Ana Caraiani: Local-global compatibility in the crystalline case

16th April, 2020

Let F be a CM field. Scholze constructed Galois representations associated to classes in the cohomology of locally symmetric spaces for GLn over F with p-torsion coefficients. These Galois representations are expected to satisfy local-global compatibility at primes above p. Even the precise formulation of this property is subtle in general, and uses Kisin’s potentially semistable deformation rings. However, this property is crucial for proving modularity lifting theorems.

I will discuss joint work with J. Newton, where we establish local-global compatibility in the crystalline case under mild technical assumptions. This relies on a new idea of using P-ordinary parts, and improves on earlier results obtained in joint work with P. Allen, F. Calegari, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. Thorne in certain Fontaine-Laffaille cases.

Will Sawin: The Lucky Logarithmic Derivative

29th November, 2018

We study the function field analogue of a classical problem in analytic number theory on the sums of the generalized divisor function in short intervals, in the limit as the degrees of the polynomials go to infinity. As a corollary, we calculate arbitrarily many moments of a certain family of L-functions, in the limit as the conductor goes to infinity. This is done by showing a cohomology vanishing result using a general bound due to Katz and some elementary calculations with polynomials. This method is based on work of Hast and Matei, except that thanks to a trick involving the logarithmic derivative, we are able to achieve a much smaller error term than is possible by this method for a "typical" problem of this type.

Jun Su: Automorphy for coherent cohomology of Shimura varieties

5th December, 2017

We consider the coherent cohomology of toroidal compactifications of Shimura varieties with coefficients in the canonical extensions of automorphic vector bundles and show that they can be computed as relative Lie algebra cohomology of automorphic representations. Consequently, any Galois representation attached to these coherent cohomology should be automorphic. Our proof is based on Franke’s work on singular cohomology of locally symmteric spaces and via Faltings' BGG spectral sequence we’ve also strengthened Franke’s result in the Shimura variety case.