Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo

Multiplicity one for the mod p cohomology of Shimura curves

par Prof. Yongquan HU (Chinese Academy of Sciences, Morningside Center of Mathematics)

Europe/Paris
Centre de conférences Marilyn et James Simons (IHES)

Centre de conférences Marilyn et James Simons

IHES

Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
Description

At present, the mod p (and p-adic) local Langlands correspondence is only well understood for the group GL2(Qp). One of the main difficulties is that little is known about supersingular representations besides this case, and we do know that there is no simple one-to-one correspondence between representations of GL2(K) with two-dimensional representations of Gal(K/K), at least when K/Qp is (non-trivial) finite unramified.

 

However, the Buzzard-Diamond-Jarvis conjecture and the mod p local-global compatibility for GL2/Q suggest that this hypothetical correspondence may be realized in the cohomology of Shimura curves with characteristic p coefficients (cut out by some modular residual global representation r¯). Moreover, the work of Gee, Breuil and Emerton-Gee-Savitt show that, to get information about the GL2(K)-action on the cohomology, one could instead study the geometry of certain Galois deformation rings of the p-component of r¯.

In a work in progress with Haoran Wang, we push forward their analysis of the structure of potentially Barsotti-Tate deformation rings and, as an application, we prove a multiplicity one result of the cohomology at full congruence level when r¯ is reducible generic \emph{non-split} at p. (The semi-simple case was previously proved by Le-Morra-Schraen and by ourselves.)

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