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The Cassels-Tate pairing on polarized abelian varieties

Title data

Poonen, Bjorn ; Stoll, Michael:
The Cassels-Tate pairing on polarized abelian varieties.
In: Annals of Mathematics. Vol. 150 (November 1999) Issue 3 . - pp. 1109-1149.
ISSN 0003-486X
DOI: https://doi.org/10.2307/121064

Official URL: Volltext

Abstract in another language

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.

Further data

Item Type: Article in a journal
Refereed: Yes
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II > Chair Mathematics II - Univ.-Prof. Dr. Michael Stoll
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Result of work at the UBT: No
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 19 Feb 2015 12:49
Last Modified: 19 Feb 2015 12:49
URI: https://eref.uni-bayreuth.de/id/eprint/7145