# Arithmetic properties of curves of small genus

**Lecturers**

Jordi Guàrdia (UPC)

Email: jordi.guardia-rubies@upc.edu

Joan-C. Lario (UPC)

Email: joan.carles.lario@upc.edu

Christophe Ritzenthaler (Rennes 1)

Email: christophe.ritzenthaler@univ-rennes1.fr

**Dates:**

**14/02 10h-13h** Ritzenthaler – Invariant theory (I) +

Introduccion to Magma

**15/02 10h-13h** C. Ritzenthaler, J. Guàrdia – Invariant theory (II)+

Theta function (I)

**16/02 10h-13h** C. Ritzenthaler, J. Guàrdia – Invariant theory (II) +

Theta function (II)

**17/02 10h-13h** C. Ritzenthaler, J.C Lario – Genus 3 curves +

L-series (I)

**23/02 10h-14h** J.Guàrdia, JC. Lario – Theta function (III) +

L-series (II)

**24/02 10h-14h** C. Ritzenthaler, J. Guàrdia – Construction of CM curves of genus 3

**Location**

room 102, FME (Facultat de Matemàtiques i Estadística, UPC), Barcelona.

**Website**

###### NA

##### Summary

The course is an introduction to theoretical and computational aspects of curves of small genus as well as their Jacobians. One of the main goals is to discuss the algorithmic characterization for the decision problem on when two small genus curves are isomorphic, or when two Abelian varieties of small dimension are isomorphic or isogenous.

The course also will serve as a preparation for attending the workshop we are organizing in the frame of the Foundations of Computational Mathematics meeting, to be held in BCN on July 2017.

##### Contents

**Theta functions (6h)**

Introduction to abelian varieties and Jacobians of curves. Siegel moduli space and the symplectic modular group.Thetanullwerte.Theta functions will help us to establish a link between curves and their Jacobians. Moreover their values at zero can be considered as coordinates on the moduli space of principally polarized abelian varieties and are therefore a nice tool to investigate this space.

**Invariant theory (6h)**

Introduction. Invariants of curves of genus 1 and 2. Fields of moduli. Fields of definition. Invariant theory will be used to give an explicit description of the points of the moduli space of curves in terms of invariants. We will also study related arithmetic questions such as the smallest field over which a representative of the isomorphism class can be defined.

**Plane quartic curves (6h)**

Classification. Invariants. Automorphism group. Torelli problem for genus 3 curves. The first examples of non-hyperelliptic curves appear in genus 3, and they are the focus of many present computational problems. We will review the classical results for non-singular plane quartic curves in order to study their arithmetical properties.

**Serre’s obstruction in genus 3 (2h)**.

Serre’s obstruction. Solution for genus 3 curves. Maximal curves in genus 3. Serre’s obstruction determines whether an abelian variety over a field is the Jacobian of a non hyperelliptic curve over the algebraic closure of the field. It is trivial for dimensions 1 and 2. Theta functions allow its explicit description for dimension 3, We will apply this to try to compute what the maximal number of points of a curve of genus 3 over a finite field is. This question is still largely open.

##### Recommended prerequisites

Target audience: young researchers, advanced undergraduate students or master students acquainted with some background of algebraic geometry, commutative algebra, and number theory.