Wojciech J. Gajda

Abstract:

(Joint work with Arno Fehm) We will explain the proof of the following Theorem. Let K be an ample field which is not algebraic over a finite field. Then rank(A(K)) = ∞ for every non-zero abelian variety A/K. This is a common generalization of a variety of infinite rank results for abelian varieties over certain types of “large” fields. The theorem is known to be true in the case char(K) = 0 for a couple of years. The case char(K) > 0 is more involved, however, and could be established only recently. On the way we point out that a (slight generalization of) a recent theorem of Roessler together with work of Ghioca-Moosa imply the dimension one case of the full Mordell-Lang conjecture. Notes from the talk.