Potential level-lowering for GSp(4)

TL;DR

This paper extends level-lowering techniques for GSp(4) over totally real fields by establishing congruences between automorphic forms with different local properties, using trace formulas and types theory.

Contribution

It introduces a new level-lowering method for GSp(4) automorphic forms via congruences, leveraging an analogue of Jacquet-Langlands and Roche's types theory.

Findings

Established congruences between automorphic forms with different local behaviors.

Proved an analogue of the Jacquet-Langlands correspondence for GSp(4).

Applied trace formula and types theory to achieve level-lowering results.

Abstract

In this preprint, we explore a beautiful idea of Skinner and Wiles in the context of GSp(4) over a totally real field. The main result provides congruences between automorphic forms which are Iwahori-spherical at a certain place w, and forms with a tamely ramified principal series at w. Thus, after base change to a totally real finite solvable extension, one can often lower the level at w. For the proof, we first establish an analogue of the Jacquet-Langlands correspondence, using the stable trace formula. The congruences are then obtained on inner forms, which are compact at infinity mod center, and split at all finite places. The crucial ingredient allowing us to do so, is an important result of Roche on types for principal series representations of split reductive groups.