Thursday 14 Nov 2013On the arithmetic of abelian varieties (joint with Akio Tamagawa).

Mohamed Saidi - University of Exeter

Harrison 101 15:00-16:00

We investigate the arithmetic of abelian varieties over a large class of fields including function fields over number fields and p-adic local fields. We prove finiteness of the N-torsion of the Selmer and the Shafarevich-Tate groups (which in this case are defined by localizing cohomology classes at closed points of a
curve with function field the given base field). For an isotrivial abelian variety we prove that the full Shafarevich-Tate group (relative to a proper base curve) is finite. One of the key ingrediants in proving this results is a new specialisation theorem \`a la N'eron for the first Galois cohomology group of the Tate module (which generalises N\'eron specialisation Theorem for rational points).

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