The constructive inverse Galois problem via Hilbert modular forms: realizing the transitive group 17T7

We show how Hilbert modular forms can be used in the constructive inverse Galois problem over the rationals. In particular, we prove that the transitive permutation group 17T7, isomorphic to a split extension of C_2 by PSL_2(FF_16), is a Galois group over the rationals and exhibit an explicit degree 17 polynomial with this Galois group. 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 a fourfold attached to a Hilbert modular form. Building upon work of Dembele, we describe a method for reconstructing a period matrix attached to a Hilbert modular form, and we use it to construct the 2-isogeny polynomial. We also rigorously identify the relevant fourfold as the Jacobian of a genus 4 Shimura curve and compute explicit equations for this curve.