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

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

2008-04-10

arxiv.org/abs/0804.1658v1

Mathematics

Number Theory

20 pages, to appear in J. Number Theory

Scientific paper

10.1016/j.jnt.2007.11.007

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.

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