Pip Goodman

Superelliptic curves with large Galois images

Let K be a number field. The inverse Galois problem for K asks if for every finite group G there exists a Galois extension L/K whose Galois group is isomorphic to G. Many people have used torsion points on abelian varieties to realise symplectic similitude groups (GSp_n(F_p)) over Q.
In this talk, we will see how superelliptic curves may be used to realise general linear and unitary similitude groups over cyclotomic fields. A variety of techniques are involved, including those from group theory, CM theory, Odlyzko bounds, and models of curves.