Projectivity of Rigid Group Actions on Complex Tori
Friday, 19.10.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk, we shall discuss a result already obtained by Torsten Ekedahl around 1999, stating that every complex torus \(T\) admitting a rigid group action of a finite group \(G\) is in fact projective, i.e., an Abelian variety. Firstly, we shall explain the notion of “deformations of the pair \((T,G)\)“; afterwards the proof of Ekedahl’s Theorem will be outlined and the projectivity of \(T\) will be shown explicitly. If time allows, applications of Ekedahl’s result will be explained towards the end of the seminar talk. This is (partly) joint work with Fabrizio Catanese.\n\n
Finiteness of perfect torsion points of an abelian variety
Friday, 26.10.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
I will report on a joint work with Emiliano Ambrosi. Let k be a field\nthat\nis finitely generated over the algebraic closure of a finite field. As\na\nconsequence of the theorem of Lang-Néron, for every abelian variety\nover k\nwhich does not contain any isotrivial abelian variety, the group of\nk-rational torsion points is finite. We show that if k^perf is a\nperfect\nclosure of k, the group of k^perf-rational torsion points is finite as\nwell. This gives a positive answer to a question asked by Hélène\nEsnault in\n2011. To prove the theorem we translate the problem to a certain\nquestion\non morphisms of F-isocrystals. Subsequently, we handle the question\nstudying the monodromy groups of the F-isocrystals involved. We can\nprove\nthat a certain monodromy group is "big" via an argument with Frobenius\ntori. Then class field theory and some considerations on the slopes\nconclude the proof. As an additional outcome of our work we prove a\nweak\n(weak) semi-simplicity statement for p-adic representations coming\nfrom\npure overconvergent F-isocrystals.\n