Xiaoyu (Coco) Huang: The Mathieu group M23 is a Galois group over Q

Abstract: Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over $\mathbb{Q}$ during 1984--1989. We complete this program by proving that the last remaining sporadic group, the Mathieu group $M_{23}$, occurs as a Galois group over $\mathbb{Q}$. In fact, we produce an explicit degree-$23$ polynomial with rational coefficients whose splitting field has Galois group $M_{23}$ over $\mathbb{Q}$. To accomplish this, we discover an unanticipated splitting of the Nielsen class associated to a non-rigid triple of conjugacy classes of $M_{23}$.  We compute the Belyi maps associated with this Nielsen class to construct an explicit regular Galois extension of $\mathbb{Q}(t)$ with Galois group $M_{23}$. Essential for our computation is the numerical Belyi map algorithm developed and implemented by Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark. As a bonus, one specialization of the regular extension yields an $M_{22}$-extension of $\mathbb{Q}$. This is joint work with Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang.