November 13, 2024: Inverse Galois theory for 17T7 over the rationals

Why this matters. The inverse Galois problem asks which finite groups occur as the Galois group of a polynomial with rational coefficients. It is wide open, and even when a group is known to occur, an explicit polynomial realizing it is often out of reach. We realize the group 17T7 over $\mathbf{Q}$ and, going beyond a pure existence statement, exhibit an explicit degree- $17$ polynomial with this Galois group.

With Raymond van Bommel, Edgar Costa, Noam D. Elkies, Sam Schiavone, and John Voight, we constructed an explicit polynomial with Galois group 17T7 (a split extension of $\operatorname{Gal}(\mathbf{F}_{16}/\mathbf{F}_4) \cong C_2$ by $\operatorname{PSL}_2(\mathbf{F}_{16})$ ) over the rationals: $\begin{aligned} x^{17} &- 2 x^{16} + 12 x^{15} - 28 x^{14} + 60 x^{13} - 160 x^{12} + 200 x^{11} - 500 x^{10} + 705 x^{9} - 886 x^{8}\\ &+ 2024 x^{7} - 604 x^{6} + 2146 x^{5} + 80 x^{4} - 1376 x^{3} - 496 x^{2} - 1013 x - 490 \end{aligned}$

The existence follows from the existence of a suitable Hilbert modular form $f$ . For the explicit polynomial, we construct the period lattice of the conjectural abelian variety of dimension $4$ over a real quadratic base field $F$ of $f$ . The $2$ -torsion of that abelian variety realizes $\operatorname{PSL}_2(\mathbf{F}_{16})$ over $F$ , and, by Hilbert descent, 17T7 over $\mathbf{Q}$ .

We assume the Eichler–Shimura conjecture (part of the Langlands philosophy) that one can attach an abelian variety $A_f/F$ to $f$ having isomorphic mod- $2$ representations, and use some form of the BSD conjecture to construct the period lattice of $A_f$ . In the end, we quickly verify that our polynomial has Galois group 17T7 by a Magma computation.

The next smallest open case for the inverse Galois problem is the Mathieu group $M_{23}$ .