Finite Weil restriction of curves

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 $$h :C \rightarrow B_{L}$$ , we denote by $$C_{h}$$ the curve defined over $$K$$ whose $$K$$ -rational points parametrize the $$L$$ -rational points on $$C$$ whose images under $$h$$ are defined over $$K$$ . We compute the geometric genus of the curve $$C_{h}$$ and give a criterion for the applicability of the Chabauty method to find the points of the curve $$C_{h}$$ . We provide 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.