Visualizing elements of Sha[3] in Genus 2 jacobians

Nils Bruin; Sander R. Dahmen

doi:10.1007/978-3-642-14518-6_12

Visualizing elements of Sha[3] in Genus 2 jacobians

Nils Bruin^*, Sander R. Dahmen

^*Corresponding author for this work

Mathematics

Research output: Chapter in Book / Report / Conference proceeding › Conference contribution › Academic › peer-review

Abstract

Mazur proved that any element ξ of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that ξ lies in the kernel of the natural homomorphism between the cohomology groups H¹(Gal(k̄/k), E) → H¹(Gal(k̄/k), A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.

Original language	English
Title of host publication	Algorithmic Number Theory - 9th International Symposium, ANTS-IX, Proceedings
Pages	110-125
Number of pages	16
DOIs	https://doi.org/10.1007/978-3-642-14518-6_12
Publication status	Published - 2010
Event	9th International Symposium on Algorithmic Number Theory, ANTS-IX - Nancy, France Duration: 19 Jul 2010 → 23 Jul 2010

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	6197 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	9th International Symposium on Algorithmic Number Theory, ANTS-IX
Country/Territory	France
City	Nancy
Period	19/07/10 → 23/07/10

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10.1007/978-3-642-14518-6_12

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Bruin, N., & Dahmen, S. R. (2010). Visualizing elements of Sha[3] in Genus 2 jacobians. In Algorithmic Number Theory - 9th International Symposium, ANTS-IX, Proceedings (pp. 110-125). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6197 LNCS). https://doi.org/10.1007/978-3-642-14518-6_12

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title = "Visualizing elements of Sha[3] in Genus 2 jacobians",

abstract = "Mazur proved that any element ξ of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that ξ lies in the kernel of the natural homomorphism between the cohomology groups H1(Gal({\=k}/k), E) → H1(Gal({\=k}/k), A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.",

author = "Nils Bruin and Dahmen, {Sander R.}",

year = "2010",

doi = "10.1007/978-3-642-14518-6_12",

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series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

pages = "110--125",

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Bruin, N & Dahmen, SR 2010, Visualizing elements of Sha[3] in Genus 2 jacobians. in Algorithmic Number Theory - 9th International Symposium, ANTS-IX, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6197 LNCS, pp. 110-125, 9th International Symposium on Algorithmic Number Theory, ANTS-IX, Nancy, France, 19/07/10. https://doi.org/10.1007/978-3-642-14518-6_12

Visualizing elements of Sha[3] in Genus 2 jacobians. / Bruin, Nils; Dahmen, Sander R.
Algorithmic Number Theory - 9th International Symposium, ANTS-IX, Proceedings. 2010. p. 110-125 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6197 LNCS).

Research output: Chapter in Book / Report / Conference proceeding › Conference contribution › Academic › peer-review

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AU - Bruin, Nils

AU - Dahmen, Sander R.

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N2 - Mazur proved that any element ξ of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that ξ lies in the kernel of the natural homomorphism between the cohomology groups H1(Gal(k̄/k), E) → H1(Gal(k̄/k), A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.

AB - Mazur proved that any element ξ of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that ξ lies in the kernel of the natural homomorphism between the cohomology groups H1(Gal(k̄/k), E) → H1(Gal(k̄/k), A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.

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Visualizing elements of Sha[3] in Genus 2 jacobians

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