Artin representations for GSp_4 attached to real analytic Siegel cusp forms of weight (2,1)

TL;DR

This paper constructs a unique Artin representation associated with certain real analytic Siegel cusp forms of weight (2,1), under specific automorphic and Galois representation assumptions, expanding understanding of GSp_4 representations.

Contribution

It introduces a method to attach a unique GSp_4-type Artin representation to specific Siegel cusp forms under new automorphic and Galois assumptions.

Findings

Construction of a unique GSp_4 Artin representation for the given forms.

Examples provided using transfers and automorphic descent.

Validation of assumptions through explicit cases.

Abstract

Let $F$ be a vector-valued real analytic Siegel cusp eigenform of weight $(2,1)$ with the eigenvalues $-\frac 5{12}$ and 0 for the two generators of the center of the algebra consisting of all $Sp_4(\R)$-invariant differential operators on the Siegel upper half plane of degree 2. Under the assumptions (1) the validity of the transfer of automorphic representations of $GSp_4$ to $GL_4$; (2) the existence of mod $\ell$ Galois representation attached to $F$ and its lift to characteristic zero; (3) rationality of the space consisting of any such $F$; and (4) the integrality of Hecke polynomials of $F$, we construct a unique Artin representation of type $GSp_4$ associated to $F$. Several examples which satisfy these assumptions are given by using various transfers and automorphic descent.