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Computing the Cassels–Tate pairing for genus two jacobians with rational two-torsion points

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  06 December 2024

JIALI YAN
JIALI YAN*
Affiliation:
University of Cambridge, 28 Howards Lane, SW15 6NQ, London. e-mail:[email protected]
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Abstract

In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels–Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where J is the Jacobian variety of a genus two curve under the assumption that all points in J[2] are K-rational. We also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the same assumption. Finally, we include a worked example demonstrating that we can improve the rank bound given by a 2-descent via computing the Cassels–Tate pairing.

MSC classification

Primary: 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields
Secondary: 11G10: Abelian varieties of dimension $> 1$
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 177 , Issue 3 , November 2024 , pp. 415 - 437
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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