Uniform behavior of families of Galois representations on Siegel modular forms and the Endoscopy Conjecture

TL;DR

This paper establishes a uniformity principle for Galois representations attached to genus two Siegel cusp forms, leading to a proof of the Endoscopy Conjecture through the reducibility criterion.

Contribution

It proves a uniformity principle linking reducibility of Galois representations to the Endoscopy Conjecture for Siegel modular forms.

Findings

If one Galois representation in the family is reducible, all are reducible.

The uniformity principle applies to forms with weight greater than 3.

The result confirms the Endoscopy Conjecture assuming Serre's conjecture.

Abstract

We prove the following uniformity principle: if one of the Galois representations in the family attached to a genus two Siegel cusp form of weight $k>3$, "semistable" and with multiplicity one, is reducible (for an odd prime $p$),then all the representations in the family are reducible. This, combined with Serre's conjecture (which is now a theorem) gives a proof of the Endoscopy Conjecture.