TY - UNPB

T1 - 17T7 is a Galois group over the rationals

AU - Bommel, Raymond van

AU - Costa, Edgar

AU - Elkies, Noam D.

AU - Keller, Timo

AU - Schiavone, Sam

AU - Voight, John

N1 - 23 pages, comments welcome

PY - 2024/11/12

Y1 - 2024/11/12

N2 - 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.

AB - 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.

KW - math.NT

KW - 12F12 (Primary) 11F80, 11F41, 14G10 (Secondary)

M3 - Preprint

BT - 17T7 is a Galois group over the rationals

ER -