A non-solvable extension of $\Q$ unramified outside 7

TL;DR

This paper constructs a specific non-solvable Galois extension of the rational numbers ramified only at 7 by analyzing a mod 7 Galois representation attached to a genus 2 Siegel cuspform, demonstrating the realization of PGSp(4,7) as a Galois group.

Contribution

It provides a new example of a non-solvable Galois extension of Q ramified only at 7 by explicitly analyzing a genus 2 Siegel cuspform and its associated Galois representation.

Findings

Realization of PGSp(4,7) as a Galois group over Q

Construction of a non-solvable extension ramified only at 7

Explicit analysis of Fourier coefficients and eigenvalues

Abstract

We consider a mod 7 Galois representation attached to a genus 2 Siegel cuspforms of level 1 and weight 28 and using some of its Fourier coefficients and eigenvalues computed by N. Skoruppa and the classification of maximal subgroups of PGSp(4,p) we show that its image is as large as possible. This gives a realization of PGSp(4,7) as a Galois group over $\Q$ and the corresponding number field provides a non-solvable extension of $\Q$ which ramifies only at 7.