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

• Nils Bruin ^[1] ; E. Victor Flynn ^[2] ; Ari Shnidman ^[3]
  1. [1] Simon Fraser University

     Simon Fraser University

     Canadá

     
  2. [2] University of Oxford

     University of Oxford

     Oxford District, Reino Unido

     
  3. [3] Hebrew University of Jerusalem

     Hebrew University of Jerusalem

     Israel

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 29, Nº. 3, 2023
• Idioma: inglés
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• Resumen

  • 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 X(Ad )[3] = 0 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.