CRM lectures on curves and Jacobians over function fields

TL;DR

This paper provides lecture notes on the theory of curves and Jacobians over function fields, focusing on rational points and constructions of Jacobians with large Mordell-Weil rank, extending previous work to higher genus curves and broader base fields.

Contribution

It offers a comprehensive overview of results on Jacobians over function fields, including methods to construct Jacobians with high Mordell-Weil rank and generalizations beyond elliptic curves.

Findings

Discussion of rational points on Jacobians over function fields

Construction techniques for Jacobians with large Mordell-Weil rank

Extension of results from elliptic curves to higher genus curves

Abstract

These are notes related to a 12-hour course of lectures given at the Centre de Recerca Mathem\`atica near Barcelona in February, 2010. The aim of the course was to explain results on curves and their Jacobians over function fields, with emphasis on the group of rational points of the Jacobian, and to explain various constructions of Jacobians with large Mordell-Weil rank. They may be viewed as a continuation of my Park City notes (arXiv:1101.1939). In those notes, the focus was on elliptic curves and finite constant fields, whereas here we discuss curves of higher genera and results over more general base fields.