On Galois representations and Hilbert-Siegel modular forms

TL;DR

This paper constructs Galois representations for certain automorphic forms on GSp(4) over totally real fields, linking local and global Langlands correspondences and relating monodromy rank to parahoric fixed spaces.

Contribution

It introduces a novel method to associate Galois representations to GSp(4) automorphic forms using a combination of base change, Harris-Taylor techniques, and descent, extending previous work on motives.

Findings

Galois representations are associated to GSp(4) automorphic forms.

The rank of the monodromy operator relates to parahoric fixed spaces.

Compatibility with the local Langlands correspondence is established.

Abstract

In this paper, we associate Galois representations to globally generic cuspidal automorphic representations on GSp(4), over a totally real field F, which are Steinberg at some finite place. This association is compatible with the local Langlands correspondence for GSp(4) studied recently in a preprint of Gan and Takeda. As a corollary, we relate the rank of the monodromy operator at p to the dimensions of the parahoric fixed spaces at p. The Galois representations are constructed by first passing to GL(4) over a CM extension, then applying the book of Harris-Taylor plus a refinement due to Taylor-Yoshida, and finally descending to F by a delicate patching argument. This is a variation of the techniques used by Blasius and Rogawski in order to attach motives to Hilbert modular forms.