A non-selfdual 4-dimensional Galois representation

TL;DR

This paper constructs specific 4-dimensional Galois representations with particular Hodge types, verifies their L-functions numerically, and discusses their properties under certain cohomological hypotheses.

Contribution

It introduces a method to construct non-selfdual 4-dimensional Galois representations with specified Hodge types under a cohomological hypothesis.

Findings

Computed first 80,000 coefficients of the L-function.

Numerically verified the functional equation of the L-function.

Proposed a candidate for the conductor of the representation.

Abstract

In this paper it is explained how one can construct non-selfdual 4-dimensional $\ell$-adic Galois representations of Hodge type $h^{3,0}=h^{2,1}=h^{1,2}=h^{0,3}=1$, assuming a hypothesis concerning the cohomology of a certain threefold. For one such a representation the first 80000 coefficients of its $L$-function are computed, and it is numerically verified that this $L$-function satisfies a functional equation. Also a candidate for the conductor is obtained.