Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties

TL;DR

This paper proves that for any even n and positive d, certain symplectic groups occur as Galois groups over the rationals, using automorphic representations to construct compatible Galois systems with specific local properties.

Contribution

It establishes the existence of compatible Galois representations with prescribed local properties, leading to new realizations of symplectic groups as Galois groups over Q.

Findings

PSp_n(F_{l^d}) or PGSp_n(F_{l^d}) occur as Galois groups over Q for a positive density of primes l.

Construction of automorphic representations with specific local types.

Application of automorphic methods to the inverse Galois problem for symplectic groups.

Abstract

This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse Galois problem for symplectic groups. For any even positive integer n and any positive integer d, PSp_n(F_{l^d}) or PGSp_n(F_{l^d}) occurs as a Galois group over the rational numbers for a positive density set of primes l. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of GL_n(A_Q) with local types chosen so as to obtain a compatible system of Galois representations to which the results from Part II of this series apply.