On ring class eigenspaces of Mordell-Weil groups of elliptic curves over global function fields

TL;DR

This paper proves that certain Mordell-Weil groups of non-isotrivial elliptic curves over global function fields have 1-dimensional eigenspaces related to ring class characters, extending number field results to function fields.

Contribution

It establishes a function field analogue of a theorem by Bertolini and Darmon, generalizing Brown's results on Heegner modules using Kolyvagin-type methods.

Findings

Mordell-Weil groups have 1-dimensional eigenspaces under specific conditions

Non-vanishing of projected Drinfeld-Heegner points implies eigenspace dimension

Uses Galois cohomology and Igusa's results in positive characteristic

Abstract

If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional eigenspace relative to a fixed complex ring class character provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is nonzero. This represents the analogue in the function field setting of a theorem for rational elliptic curves due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof employs Kolyvagin-type arguments, and the cohomological machinery is started up by the control on the Galois structure of the torsion of E provided by classical results of Igusa in positive characteristic.