Multiplicity one for wildly ramified representations

Le, Daniel

doi:10.2140/ant.2019.13.1807

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Multiplicity one for wildly ramified representations

Daniel Le

Vol. 13 (2019), No. 8, 1807–1827

DOI: 10.2140/ant.2019.13.1807

Abstract

Let $F$ be a totally real field in which $p$ is unramified. Let $\bar{r} : G_{F} \to {GL}_{2} ({\bar{F}}_{p})$ be a modular Galois representation which satisfies the Taylor–Wiles hypotheses and is generic at a place $v$ above $p$ . Let $m$ be the corresponding Hecke eigensystem. We show that the $m$ -torsion in the $mod p$ cohomology of Shimura curves with full congruence level at $v$ coincides with the ${GL}_{2} (k_{v})$ -representation $D_{0} (\bar{r} |_{G_{F_{v}}})$ constructed by Breuil and Paškūnas. In particular, it depends only on the local representation $\bar{r} |_{G_{F_{v}}}$ , and its Jordan–Hölder factors appear with multiplicity one. This builds on and extends work of the author with Morra and Schraen and, independently, Hu–Wang, which proved these results when $\bar{r} |_{G_{F_{v}}}$ was additionally assumed to be tamely ramified. The main new tool is a method for computing Taylor–Wiles patched modules of integral projective envelopes using multitype tamely potentially Barsotti–Tate deformation rings and their intersection theory.

Keywords

Galois deformations, mod p Langlands program

Mathematical Subject Classification 2010

Primary: 11S37

Milestones

Received: 19 October 2017

Revised: 13 February 2019

Accepted: 27 May 2019

Published: 9 October 2019

Authors

Daniel Le
Department of Mathematics University of Toronto ON Canada

Vol. 13, No. 8, 2019