Finite Weil restriction of curves

TL;DR

This paper introduces a framework for the finite Weil restriction of curves, connecting to elliptic curve techniques, and fully analyzes the case when the curve has genus one, providing insights into rational points over number fields.

Contribution

It develops a general construction for Weil restrictions of curves over number fields, extending elliptic curve Chabauty methods to higher genus cases.

Findings

The set of rational points can be infinite only for genus ≤ 1.

Complete analysis of genus 1 case.

Framework includes higher genus versions of Chabauty techniques.

Abstract

Given number fields $L \supset K$, smooth projective curves $C$ defined over $L$ and $B$ defined over $K$, and a non-constant $L$-morphism $h \colon C \to B_L$,we consider the curve $C_h$ defined over $K$ whose $K$-rational points parametrize the $L$-rational points on $C$ whose images under $h$ are defined over $K$. Our construction provides a framework which includes as a special case that used in Elliptic Curve Chabauty techniques and their higher genus versions. The set $C_h(K)$ can be infinite only when $C$ has genus at most 1; we analyze completely the case when $C$ has genus 1.