The theory of elliptic and hyperelliptic curves in the development of algebraic geometry has been fundamental. Almost all important ideas in the area took as examples elliptic or hyperelliptic curves, whether it was elliptic or hyperelliptic integrals, theta functions, Thomae’s formula, the concept of Jacobians, etc.

The goal of this session is to focus on the natural generalization of the theory of hyperelliptic curves to superelliptic curves (i.e., smooth projective models of plane affine curves y^n=f(x) ) and all the open problems that come with this generalization.

We will also explore applications and recent developments in the theory of moduli spaces of curves and Abelian varieties. We will focus both on the algebraic and arithmetic sides of the theory.

Topics of the session include, but are not limited to:

Hyperelliptic and superelliptic curves

Endomorphisms and isogenies of Abelian varieties, Galois properties of torsion points and Tate modules

Hyperelliptic and superelliptic Jacobians, and their endomorphism rings

Abelian varieties over fields of positive characteristic

Brauer group of abelian varieties, K3 surfaces and generalized Kummer varieties

Néron models of superelliptic jacobians

Minimal models of superelliptic curves

Curves and Jacobians over function fields

Brill-Noether theory

Subloci and stratifications of the moduli spaces of curves and abelian varieties

Theta-nulls for superelliptic curves

Equations of curves over their minimal field of definition