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The Cassels-Tate Pairing on Polarized Abelian Varieties

Bjorn Poonen and Michael Stoll
Annals of Mathematics
Second Series, Vol. 150, No. 3 (Nov., 1999), pp. 1109-1149
Published by: Annals of Mathematics
DOI: 10.2307/121064
Stable URL: http://www.jstor.org/stable/121064
Page Count: 41
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The Cassels-Tate Pairing on Polarized Abelian Varieties
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Abstract

Let (A, λ ) be a principally polarized abelian variety defined over a global field k, and let III(A) be its Shafarevich-Tate group. Let III(A)nd denote the quotient of III(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing III(A)nd× III(A)nd→ Q/Z. If A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on III(A)nd. These criteria are expressed in terms of an element c ∈ III(A)nd that is canonically associated to the polarization λ . In the case that A is the Jacobian of some curve, a down-to-earth version of the result allows us to determine effectively whether #III(A) (if finite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperelliptic curves of even genus g ≥ 2 over Q have a Jacobian with nonsquare #III (if finite). For example, it appears that this density is about 13% for curves of genus 2. The proof makes use of a general result relating global and local densities; this result can be applied in other situations.

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