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,听the National Research University Higher School of Economics
A top degree differential form听on a smooth algebraic variety X听over a local field K听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.听