Tamagawa Torsors of An Abelian Variety

Abstract

For an abelian variety A/K over a number field K, we define the set of Tamagawa torsors of A at a prime υ of K to be the set of principal homogeneous spaces of A over the completion Kυ of K at υ that are split by an unramified extension of Kυ. In this paper, we study some of the arithmetic properties of the Tamagawa torsors.We also give, following Mazurs theory of visibility, conditions under which non-trivial elements of the Tamagawa torsors of A may be interpreted as rational points on another abelian variety B.