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

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 χ-eigenspace (with χ a complex ring class character) provided that the projection onto this eigenspace of a suitable Drinfeld–Heegner point is non-zero. This represents the analogue in the function field setting of a theorem for elliptic curves over Q 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.