Genus two curves with full √3 -level structure and Tate–Shafarevich groups

Nils Bruin; E. Victor Flynn; Ari Shnidman

doi:10.1007/s00029-023-00839-w

Genus two curves with full √3 -level structure and Tate–Shafarevich groups

Nils Bruin, E. Victor Flynn, Ari Shnidman

Einstein Institute of Mathematics

The Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

Abstract

We give an explicit rational parameterization of the surface H₃ over Q whose points parameterize genus 2 curves C with full 3 -level structure on their Jacobian J. We use this model to construct abelian surfaces A with the property that [InlineEquation not available: see fulltext.] for a positive proportion of quadratic twists A_d . In fact, for 100 % of x∈ H₃(Q) , this holds for the surface A= Jac (C_x) / ⟨ P⟩ , where P is the marked point of order 3. Our methods also give an explicit bound on the average rank of J_d(Q) , as well as statistical results on the size of # C_d(Q) , as d varies through squarefree integers.

Original language	English
Article number	42
Journal	Selecta Mathematica, New Series
Volume	29
Issue number	3
DOIs	https://doi.org/10.1007/s00029-023-00839-w
State	Published - Jul 2023

Keywords

Higher genus curves
Jacobians
Tate–Shafarevich group

All Science Journal Classification (ASJC) codes

General Mathematics
General Physics and Astronomy

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10.1007/s00029-023-00839-w

Cite this

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title = "Genus two curves with full √3 -level structure and Tate–Shafarevich groups",

abstract = "We give an explicit rational parameterization of the surface H3 over Q whose points parameterize genus 2 curves C with full 3 -level structure on their Jacobian J. We use this model to construct abelian surfaces A with the property that [InlineEquation not available: see fulltext.] for a positive proportion of quadratic twists Ad . In fact, for 100 \% of x∈ H3(Q) , this holds for the surface A= Jac (Cx) / ⟨ P⟩ , where P is the marked point of order 3. Our methods also give an explicit bound on the average rank of Jd(Q) , as well as statistical results on the size of \# Cd(Q) , as d varies through squarefree integers.",

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author = "Nils Bruin and Flynn, \{E. Victor\} and Ari Shnidman",

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doi = "10.1007/s00029-023-00839-w",

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TY - JOUR

T1 - Genus two curves with full √3 -level structure and Tate–Shafarevich groups

AU - Bruin, Nils

AU - Flynn, E. Victor

AU - Shnidman, Ari

PY - 2023/7

Y1 - 2023/7

N2 - We give an explicit rational parameterization of the surface H3 over Q whose points parameterize genus 2 curves C with full 3 -level structure on their Jacobian J. We use this model to construct abelian surfaces A with the property that [InlineEquation not available: see fulltext.] for a positive proportion of quadratic twists Ad . In fact, for 100 % of x∈ H3(Q) , this holds for the surface A= Jac (Cx) / ⟨ P⟩ , where P is the marked point of order 3. Our methods also give an explicit bound on the average rank of Jd(Q) , as well as statistical results on the size of # Cd(Q) , as d varies through squarefree integers.

AB - We give an explicit rational parameterization of the surface H3 over Q whose points parameterize genus 2 curves C with full 3 -level structure on their Jacobian J. We use this model to construct abelian surfaces A with the property that [InlineEquation not available: see fulltext.] for a positive proportion of quadratic twists Ad . In fact, for 100 % of x∈ H3(Q) , this holds for the surface A= Jac (Cx) / ⟨ P⟩ , where P is the marked point of order 3. Our methods also give an explicit bound on the average rank of Jd(Q) , as well as statistical results on the size of # Cd(Q) , as d varies through squarefree integers.

KW - Higher genus curves

KW - Jacobians

KW - Tate–Shafarevich group

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DO - 10.1007/s00029-023-00839-w

M3 - مقالة

SN - 1022-1824

VL - 29

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

IS - 3

M1 - 42

ER -

Genus two curves with full √3 -level structure and Tate–Shafarevich groups

Abstract

Keywords

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