Number Theory Seminar

Seminar information archive ~03/29Next seminarFuture seminars 03/30~

Date, time & place Wednesday 17:00 - 18:00 056Room #056 (Graduate School of Math. Sci. Bldg.)
Organizer(s) Naoki Imai, Shane Kelly


17:00-18:00   Online
Stefano Morra (Paris 8 University)
Some properties of the Hecke eigenclasses of the mod p-cohomology of Shimura curves (English)
[ Abstract ]
The mod p local Langlands program, foreseen by Serre and proposed in precise terms by C. Breuil after his p-divisible groups computations in the Breuil-Conrad-Diamond-Taylor proof of the Shimura-Taiyama-Weil conjecture, was realized in the particular case of GL_2(\mathbf{Q}_p) thanks to a vast convergence of new tools: classification of mod p-representations of GL_2(\mathbf{Q}_p), local Galois deformation techniques, local-global compatibility arguments.
When trying to extend these conjectures to more general groups, multiple problems arise (lack of classification results for smooth mod p-representations of p-adic groups, absence of explicit integral models for Galois representations with the relevant p-adic Hodge theory conditions), and the only way to formulate, and test, conjectures on a mod p local Langlands correspondence relies on its expected realization in Hecke eigenclasses of Shimura varieties (or, in other words, the expectation of a local-global compatibility of the Langlands correspondence).
In this talk we describe some properties of Hecke isotypical spaces of the mod p-cohomology of Shimura curves with infinite level at p, when the reflex field F is unramified at p and the Shimura curve arises from a quaternion algebra which is split at p. These Hecke isotypical spaces are expected to be the “good” smooth mod p-representations of GL_2(F_{\mathfrak{p}}) attached to mod p Galois representations of Gal(\overline{\mathbf{Q}_p}/F_{\mathfrak{p}}) via the expected local Langlands correspondence mod p. We will in particular comment on their Gelfand-Kirillov dimension, and their irreducibility (in particular, the finite length of these Hecke eigenspaces as GL_2(F_{\mathfrak{p}})-representations).
This is a report on a series of work joint with C. Breuil, F. Herzig, Y. Hu et B. Schraen.