## Algebra & Number Theory

### Multiplicity one for wildly ramified representations

Daniel Le

#### Abstract

Let $F$ be a totally real field in which $p$ is unramified. Let $r̄:GF→ GL2(F¯p)$ 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 $modp$ cohomology of Shimura curves with full congruence level at $v$ coincides with the $GL2(kv)$-representation $D0(r̄|GFv)$ constructed by Breuil and Paškūnas. In particular, it depends only on the local representation $r̄|GFv$, 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 $r̄|GFv$ 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.

#### Article information

Source
Algebra Number Theory, Volume 13, Number 8 (2019), 1807-1827.

Dates
Received: 19 October 2017
Revised: 13 February 2019
Accepted: 27 May 2019
First available in Project Euclid: 29 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.ant/1572314505

Digital Object Identifier
doi:10.2140/ant.2019.13.1807

Mathematical Reviews number (MathSciNet)
MR4017535

Zentralblatt MATH identifier
07118653

#### Citation

Le, Daniel. Multiplicity one for wildly ramified representations. Algebra Number Theory 13 (2019), no. 8, 1807--1827. doi:10.2140/ant.2019.13.1807. https://projecteuclid.org/euclid.ant/1572314505

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