Computational Algebra Seminar: Voight -- 17T7 is a Galois group over the rationals

Speaker: John Voight (Magma, Sydney)
Title: 17T7 is a Galois group over the rationals
Time & Place: 15:00 -- 16:00, Thursday 31/10/24, SMRI Seminar Room
Abstract:
Using Magma, we prove that the transitive permutation group 17T7 is a Galois group
over the rationals, completing the list of transitive subgroups ordered by degree
up to 23 (leaving the Mathieu group on 23 letters as the next missing group).
We exhibit such a Galois extension as the field of definition of 2-torsion on an
abelian fourfold with real multiplication defined over a real quadratic field with
Galois alignment.
We find such fourfolds using Hilbert modular forms. Finally, building upon work of
Dembele, we show how to (conjecturally) reconstruct the period matrix for abelian
variety attached to a Hilbert modular form; we then use this to construct an explicit
degree 17 polynomial with Galois group 17T7.
This is joint work with Raymond van Bommel, Edgar Costa, Noam Elkies,
Timo Keller, and Sam Schiavone.