Conjecture de type de Serre et formes compagnons pour GSp_4

TL;DR

This paper formulates a Serre-type conjecture for four-dimensional symplectic mod p Galois representations, relating their modularity to cohomological conditions on Siegel modular varieties, especially in the ordinary and tamely ramified cases.

Contribution

It introduces a new conjecture linking Galois representations to specific Serre weights via cohomology, and proposes a construction of cohomology classes using the dual BGG complex.

Findings

Conjecture relates Galois representations to Serre weights via cohomology.

Identification of modular weights in the ordinary and tamely ramified cases.

Proposal of a cohomological construction using the dual BGG complex.

Abstract

We present a Serre-type conjecture on the modularity of four-dimensional symplectic mod p Galois representations. We assume that the Galois representation is irreducible and odd (in the symplectic sense). The modularity condition is formulated using the etale and the algebraic de Rham cohomology of Siegel modular varieties of level prime to p. We concentrate on the case when the Galois representation is ordinary at p and we give a corresponding list of Serre weights. When the representation is moreover tamely ramified at p, we conjecture that all weights of this list are modular, otherwise we describe a subset of weights on the list that should be modular. We propose a construction of de Rham cohomology classes using the dual BGG complex, which should realise some of these weights.