Dual abelian varieties over a local field have equal volumes

Dual abelian varieties over a local field have equal volumes

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Vadim Vologodsky, the National Research University Higher School of Economics
Fine Hall 314

A top degree differential form on a smooth algebraic variety X over a local field K gives rise to a (real valued) measure on X(K). The Serre duality yields a natural isomorphism between 1-dimensional spaces of global top degree forms on an abelian variety and the dual abelian variety. I will prove that the corresponding volumes are equal.