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Ann Arbor 2018

From hyperelliptic to superelliptic curves

Fall Central Sectional Meeting 2018
University of Michigan, Ann Arbor, Ann Arbor, MI
October 20-21, 2018 (Saturday - Sunday)
Meeting #1143


 T. Shaska
T. Shaska, Oakland University
 N. Tarasca
N. Tarasca, Rutgers University
 Y. Zarhin
Y. Zarhin, Pennsylvania State University


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
  • Field of moduli versus the field of definition
  • Models of curves with minimal height
  • Moduli height of curves
  • Counting of curves in the moduli space
  • Jacobians of curves and their decompositions
  • Algebraic curves and mathematical physics