Period and index of genus one curves over number fields

TL;DR

This paper constructs examples of genus one curves over number fields with specific prescribed period and index, advancing understanding of their arithmetic properties.

Contribution

It provides explicit constructions of genus one curves with given period and index over certain number fields, a novel achievement in arithmetic geometry.

Findings

Examples of genus one curves with prescribed period and index

Explicit constructions over specific number fields

Enhanced understanding of the relationship between period and index

Abstract

The period of a curve is the smallest positive degree of Galois-invariant divisor classes. The index is the smallest positive degree of rational divisors. We construct examples of genus one curves with prescribed period and index over certain number fields.