I will explain how to put certain natural geometric structures on Tate-Shafarevich groups and other related groups attached to abelian varieties over function fields. We can refine arithmetic duality theorems by taking these geometric structures into account. This has applications to Weil-etale cohomology, the Birch-Swinnerton-Dyer conjecture and Iwasawa theory. Partially based on joint work with Geisser and with Lai, Longhi, Tan and Trihan.
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