On mod $p$ local-global compatibility for $\mathrm{GL}_n(\mathbf{Q}_p)$ in the ordinary case

TL;DR

This paper proves that the local Galois representation at p for certain automorphic forms over CM fields is determined by the action of Hecke operators and Jacobi sum operators on mod p automorphic forms, under genericity and weight elimination assumptions.

Contribution

It establishes a mod p local-global compatibility result for GL_n over CM fields in the ordinary case, linking Galois representations to automorphic forms via Hecke and Jacobi sum operators.

Findings

Galois representation is determined by Hecke algebra action.

Wildly ramified part is captured by Jacobi sum operators.

Results depend on a weight elimination theorem by Bao V. Le Hung.

Abstract

Let $p$ be a prime number, $n>2$ an integer, and $F$ a CM field in which $p$ splits completely. Assume that a continuous automorphic Galois representation $\overline{r}:\mathrm{Gal}(\overline{\mathbf{Q}}/F)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$ is upper-triangular and satisfies certain genericity conditions at a place $w$ above $p$, and that every subquotient of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ of dimension $>2$ is Fontaine--Laffaille generic. In this paper, we show that the isomorphism class of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ is determined by $\mathrm{GL}_n(F_w)$-action on a space of mod $p$ algebraic automorphic forms cut out by the maximal ideal of a Hecke algebra associated to $\overline{r}$, assuming a weight elimination result which is a theorem of Bao V. Le Hung in his forthcoming paper~\cite{LeH}. In particular, we show that the wildly ramified part of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ is determined by the action of Jacobi sum operators (seen as elements of $\mathbf{F}_p[\mathrm{GL}_n(\mathbf{F}_p)]$) on this space.