## 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

### Organizers:

### Overview:

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

### Schedule

- Saturday October 20, 2018, 8:30 a.m.-11:20 a.m.

Special Session on From Hyperelliptic to Superelliptic Curves, I

Room 2455, Mason Hall- 8:30 a.m.
*On Prym varieties for the coverings of some singular plane curves.*

Lubjana Beshaj*, US Military Academy, West Point

Takuya Yamauchi, Mathematical Inst. Tohoku University

(1143-14-31) - 9:00 a.m.
*Geometry of (1,2)-polarized Kummer surfaces and theta identities.*

Andreas Malmendier*, Utah State University (USU)

Adrian Clingher, University Of Missouri - St. Louis

(1143-14-566) - 9:30 a.m.
*Triangular Schlesinger Systems, Elliptic, Hyperelliptic, and Superelliptic Curves.*

Vladimir Dragovic*, Department of Mathematical Sciences, The University of Texas at Dallas

(1143-14-389) - 10:00 a.m.
*Division by 1−ζ on superelliptic curves and jacobians.*

Vishal Arul*, Massachusetts Institute of Technology

(1143-14-352) - 10:30 a.m.
*Automorphism Groups of Cyclic n-gonal Surfaces.*

Aaron D Wootton*, University of Portland

Sean A Broughton, Rose-Hulman Institute of Technology

(1143-14-149) - 11:00 a.m.
*p-adic analogue of Hyperelliptic lambda functions.*

Yaacov Kopeliovich*, University of Connecticut

(1143-14-138)

- 8:30 a.m.
- Saturday October 20, 2018, 2:00 p.m.-4:15 p.m.

Special Session on From Hyperelliptic to Superelliptic Curves, II

Room 2455, Mason Hall- 2:00 p.m.
*Explicit descent for Jacobians of prime power cyclic covers of the projective line.*

Edward F Schaefer*, Santa Clara University

(1143-11-26) - 2:30 p.m.
*Zeta Functions of Superelliptic Curves in Large Characteristic.*

Vishal Arul, Massachusetts Institute of Technology

Alex Best, Boston University

Edgar Costa, Massachusetts Institute of Technology

Richard Magner, Boston University

Nicholas Triantafillou*, Massachusetts Institute of Technology

(1143-11-55) - 3:00 p.m.
*Rational points and unipotent fundamental groups.*

Daniel Rayor Hast*, Rice University

(1143-11-117) - 3:30 p.m.
*The homogeneous double ramification cycle.*

Aaron Pixton*, Massachusetts Institute of Technology

(1143-14-537)

- 2:00 p.m.
- Sunday October 21, 2018, 8:00 a.m.-10:15 a.m.

Special Session on From Hyperelliptic to Superelliptic Curves, III

Room 2455, Mason Hall- 8:00 a.m.
*Nodal curves and a class of solutions of the Lax equations for shock clustering and Burgers turbulence.*

Luen-Chau Li*, Pennsylvania State University

(1143-14-456) - 9:00 a.m.
*Double ramification cycles.*

Felix Janda*, University of Michigan

(1143-14-434) - 9:30 a.m.
*Conductor-Discriminant inequalities for hyperelliptic curves.*

Andrew Obus*, Baruch College (CUNY)

Padmavathi Srinivasan, Georgia Institute of Technology

(1143-14-104)

- 8:00 a.m.
- Sunday October 21, 2018, 1:00 p.m.-1:50 p.m.

Special Session on From Hyperelliptic to Superelliptic Curves, IV

Room 2455, Mason Hall- 1:00 p.m.
*Heights on weighted projective spaces.*

Tony Shaska*, Oakland University

Andreas Malmendier, Utah State University

(1143-14-388) - 1:30 p.m.
*On a Berest conjecture for elliptic and hyperelliptic curves.*

Alexander Zheglov*, Moscow State University

(1143-13-445)

- 1:00 p.m.