17T7 is a Galois group over the rationals

We prove that the transitive permutation group 17T7, isomorphic to a split extension of C2 by PSL2(F16), 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 Dembélé, 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.

In the talk, we will give a general overview of the topic and the methods involved, and then more details of our construction.

This is joint work with Raymond van Bommel, Edgar Costa, Noam D. Elkies, Sam Schiavone, and John Voight.