17T7 is a Galois group over the rationals

TL;DR

This paper proves that the transitive permutation group 17T7 is realizable as a Galois group over the rationals, using abelian fourfolds with real multiplication and Hilbert modular forms, and constructs an explicit polynomial with this Galois group.

Contribution

It demonstrates that 17T7 is a Galois group over the rationals and provides a method to explicitly construct such polynomials using advanced algebraic geometry and modular forms.

Findings

17T7 is a Galois group over the rationals.

Explicit degree 17 polynomial with Galois group 17T7 constructed.

Connection established between abelian fourfolds, Hilbert modular forms, and Galois groups.

Abstract

We prove that the transitive permutation group 17T7, isomorphic to a split extension of $C_2$ by $\mathrm{PSL}_2(\mathbb{F}_{16})$, is a Galois group over the rationals. The group arises from the field of definition of the $2$-torsion on an abelian fourfold with real multiplication defined over a real quadratic field. We find such fourfolds using Hilbert modular forms. Finally, building upon work of Demb\'el\'e, we show how to conjecturally reconstruct a period matrix for an abelian variety attached to a Hilbert modular form; we then use this to exhibit an explicit degree $17$ polynomial with Galois group 17T7.