Finite Weil restriction of curves

Abstract

Given number fields L ⊃ K , smooth projective curves C defined over L and B defined over K , and a non-constant L-morphism h : C → 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.

Elliptic curves over real quadratic fields are modular

“…We would like to describe the points P ∈ X(F) such that P k = π k (P ) for k = 1, 2. Here we follow the description of Flynn and Testa [33,Section 2]. Write γ = µ 1 (P 1 ) = µ 2 (P 2 ) ∈ P 1 (F).…”