Vol. 13, No. 8, 2019

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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

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 mod p 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.

Galois deformations, mod p Langlands program
Mathematical Subject Classification 2010
Primary: 11S37
Received: 19 October 2017
Revised: 13 February 2019
Accepted: 27 May 2019
Published: 9 October 2019
Daniel Le
Department of Mathematics
University of Toronto