An Application of Maeda's Conjecture to the Inverse Galois Problem

TL;DR

This paper demonstrates that Maeda's conjecture implies the existence of certain Galois groups as Galois groups over number fields ramifying only at specific primes, linking modular forms conjectures to inverse Galois problems.

Contribution

It establishes a novel connection between Maeda's conjecture and the inverse Galois problem, showing how conjectural properties of eigenforms influence Galois realizations.

Findings

For every positive even d, PSL_2(F_{p^d}) occurs as a Galois group for a density-one set of primes p.

Maeda's conjecture implies these Galois groups can be realized with ramification only at p.

The result applies to a broad class of finite simple groups arising from projective special linear groups.

Abstract

It is shown that Maeda's conjecture on eigenforms of level 1 implies that for every positive even d and every p in a density-one set of primes, the simple group PSL_2(F_{p^d}) occurs as the Galois group of a number field ramifying only at p.