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Thursday, March 5, 2020

**Abstract:** Given a connected reductive group $G$ defined over a number field $F$, the Langlands program predicts a connection between suitable automorphic representations of $G(\mathbb A_F)$ and geometric $p$-adic Galois representations $\mathrm{Gal}(\overline{F}/F) \to {}^LG$ into the L-group of $G$. Striking work of Arno Kret and Sug Woo Shin constructs the automorphic-to-Galois direction when $G$ is the group $\mathrm{GSp}_{2n}$ over a totally real field $F$, and $\pi$ is a cuspidal automorphic representation of $\mathrm{GSp}_{2n}(\mathbb A_F)$ that is discrete series at all infinite places and is a twist of the Steinberg representation at some finite place: To such a $\pi$, they attach geometric $p$-adic Galois representations $\rho_{\pi}: \mathrm{Gal}(\overline{F}/F) \to \mathrm{GSpin}_{2n+1}$. In this work we establish a partial converse, proving a potential automorphy theorem, and some applications, for suitable $\mathrm{GSpin}_{2n+1}$-valued Galois representations. In this talk, I will explain the background materials and the known results in this direction before touching upon the main theorems of this work.