Optimal quotients and surjections of Mordell-Weil groups

TL;DR

This paper investigates the conditions under which certain endomorphisms of Jacobian varieties induce surjective maps on Mordell-Weil groups, providing a counterexample to a previously posed question.

Contribution

It demonstrates that endomorphisms with connected kernels do not always induce surjective maps on Mordell-Weil groups, answering a question posed by Ed Schaefer.

Findings

Counterexample to surjectivity of certain endomorphisms

Connected kernel does not imply surjective map on Mordell-Weil groups

Clarifies the relationship between automorphisms and Mordell-Weil group maps

Abstract

Answering a question of Ed Schaefer, we show that if J is the Jacobian of a curve C over a number field, if s is an automorphism of J coming from an automorphism of C, and if u lies in the subring Z[s] of End J and has connected kernel, then it is not necessarily the case that u gives a surjective map from the Mordell-Weil group of J to the Mordell-Weil group of its image.