Results 1 to 10 of about 540 (74)
On binary quartics and the Cassels–Tate pairing [PDF]
Research in Number Theory, 2022 AbstractWe use the invariant theory of binary quartics to give a new formula for the Cassels–Tate pairing on the 2-Selmer group of an elliptic curve. Unlike earlier methods, our formula does not require us to solve any conics. An important role in our construction is played by a certain K3 surface defined by a (2, 2, 2)-form.
Tom Fisher, Fisher Tom
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The Cassels-Tate Pairing on Polarized Abelian Varieties [PDF]
Annals of Mathematics, 1999 Let (A,λ) be a principally polarized abelian variety defined over a global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing \Sha(A)_\nd \times \Sha(A)_\nd \rightarrow \Q/\Z.
Björn Poonen, Michael Stoll
exaly +5 more sources
Computing the Cassels–Tate pairing on 3-isogeny Selmer groups via cubic norm equations [PDF]
Acta Arithmetica, 2018 We explain a method for computing the Cassels-Tate pairing on the 3-isogeny Selmer groups of an elliptic curve. This improves the upper bound on the rank of the elliptic curve coming from a descent by 3-isogeny, to that coming from a full 3-descent. One ingredient of our work is a new algorithm for solving cubic norm equations, that avoids the need for
Tom Fisher
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The yoga of the Cassels–Tate pairing [PDF]
LMS Journal of Computation and Mathematics, 2010 AbstractCassels has described a pairing on the 2-Selmer group of an elliptic curve which shares some properties with the Cassels–Tate pairing. In this article, we prove that the two pairings are the same.
Tom A. Fisher +2 more
openaire +3 more sources
Computing the Cassels–Tate pairing on the 3-Selmer group of an elliptic curve [PDF]
International Journal of Number Theory, 2014 We extend the method of Cassels for computing the Cassels–Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank ...
Fisher, Tom, Newton, Rachel
openaire +4 more sources
Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian
, 2023 We describe a method for computing the Cassels-Tate pairing on the 2-Selmer group of the Jacobian of a genus 2 curve. This can be used to improve the upper bound coming from 2-descent for the rank of the group of rational points on the Jacobian. Our method remains practical regardless of the Galois action on the Weierstrass points of the genus 2 curve.
Fisher, Tom, Yan, Jiali
core +4 more sources
The Cassels-Tate pairing for finite Galois modules
, 2021 Given a global field $F$ with absolute Galois group $G_F$, we define a category $SMod_F$ whose objects are finite $G_F$-modules decorated with local conditions. We define this category so that `taking the Selmer group' defines a functor $Sel$ from $SMod_F$ to $Ab$.
Morgan, Adam, Smith, Alexander
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The Cassels-Tate pairing on 2-Selmer groups of elliptic curves
, 2023 We explicitly compute the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve using the Albanese-Albanese definition of the pairing given by Poonen and Stoll. This leads to a new proof that a pairing defined by Cassels on the 2-Selmer groups of elliptic curves agrees with the Cassels-Tate pairing.
Shukla, Himanshu, Stoll, Michael
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Field change for the Cassels–Tate pairing and applications to class groups
Research in Number TheoryAdam Morgan, Alexander Smith
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Computing the Cassels-Tate Pairing in the Case of a Richelot Isogeny
, 2021 In this paper, we study the Cassels-Tate pairing on Jacobians of genus two curves admitting a special type of isogenies called Richelot isogenies. Let $ϕ: J \rightarrow \widehat{J}$ be a Richelot isogeny between two Jacobians of genus two curves. We give an explicit formula as well as a practical algorithm to compute the Cassels-Tate pairing on $\text ...
openaire +2 more sources