Multiplicity one for wildly ramified representations. (English) Zbl 1471.11280

Summary: Let \(F\) be a totally real field in which \(p\) is unramified. Let \(\bar{r}:G_F\to \operatorname{GL}_2(\overline{\mathbb{F}}_p)\) be a modular Galois representation which satisfies the Taylor-Wiles hypotheses and is generic at a place \(v\) above \(p\). Let \(\mathfrak{m}\) be the corresponding Hecke eigensystem. We show that the \(\mathfrak{m}\)-torsion in the \(\bmod p\) cohomology of Shimura curves with full congruence level at \(v\) coincides with the \(\operatorname{GL}_2(k_v)\)-representation \(D_0(\bar{r}|_{G_{F_v}})\) constructed by C. Breuil and V. Paškūnas [Towards a modulo \(p\) Langlands correspondence for \(\mathrm{GL}_{2}\). Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1245.22010)]. 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, Y. Hu and H. Wang [Math. Res. Lett. 25, No. 3, 843–873 (2018; Zbl 1454.11106)], 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.


11R39 Langlands-Weil conjectures, nonabelian class field theory
11S37 Langlands-Weil conjectures, nonabelian class field theory
11F80 Galois representations
Full Text: DOI arXiv


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