## 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.

### MSC:

 11R39 Langlands-Weil conjectures, nonabelian class field theory 11S37 Langlands-Weil conjectures, nonabelian class field theory 11F80 Galois representations

### Keywords:

Galois deformations; $$\mod p$$ Langlands program

### Citations:

Zbl 1245.22010; Zbl 1454.11106
Full Text:

### References:

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