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

Nils Bruin, E. Victor Flynn, Ari Shnidman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageAmerican English
Article number42
JournalSelecta Mathematica, New Series
Issue number3
StatePublished - Jul 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.


  • Higher genus curves
  • Jacobians
  • Tate–Shafarevich group


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