Period and index for higher genus curves

TL;DR

This paper investigates the relationship between period and index of higher genus curves over fields, establishing existence results for curves with prescribed invariants under certain divisibility constraints.

Contribution

It proves the existence of curves with specified period, index, and genus, extending Lichtenbaum's divisibility conditions to higher genus cases with a new existence theorem.

Findings

Existence of curves with given period, index, and genus under certain conditions.

The index cannot be divisible by 4 for the existence result to hold.

Provides a constructive approach to realizing prescribed invariants.

Abstract

Given a curve $C$ over a field $K$, the period of $C/K$ is the gcd of degrees of $K$-rational divisor classes, while the index is the gcd of degrees of $K$-rational divisors. S. Lichtenbaum showed that the period and index must satisfy certain divisibility conditions. For given admissible period, index, and genus, we show that there exists a curve $C$ and a number field $K$ with these desired invariants, as long as the index is not divisible by $4$.