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arXiv · 2112.12256

Mod $p$ local-global compatibility for $\mathrm{GSp}_4(\mathbb{Q}_p)$ in the ordinary case

John Enns·Heejong Lee
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Abstract

Let $F$ be a totally real field of even degree in which $p$ splits completely. Let $\overline{r}:G_F \rightarrow \mathrm{GSp}_4(\overline{\mathbb{F}}_p)$ be a modular Galois representation unramified at all finite places away from $p$ and upper-triangular, maximally nonsplit, and of parallel weight at places dividing $p$. Fix a place $w$ dividing $p$. Assuming certain genericity conditions and Taylor--Wiles assumptions, we prove that the $\mathrm{GSp}_4(F_w)$-action on the corresponding Hecke-isotypic part of the space of mod $p$ automorphic forms on a compact mod center form of $\mathrm{GSp}_4$ with infinite level at $w$ determines $\overline{r}|_{G_{F_w}}$.

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John Enns, Heejong Lee. 2021-12-22. Mod $p$ local-global compatibility for $\mathrm{GSp}_4(\mathbb{Q}_p)$ in the ordinary case. https://arxiv.org/abs/2112.12256

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