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Genus two curves with full \(\sqrt{3}\)-level structure and Tate–Shafarevich groups

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Abstract

We give an explicit rational parameterization of the surface \(\mathcal {H}_3\) over \({\mathbb Q}\) whose points parameterize genus 2 curves \(C\) with full \(\sqrt{3}\)-level structure on their Jacobian J. We use this model to construct abelian surfaces A with the property that for a positive proportion of quadratic twists \(A_d\). In fact, for \(100\%\) of \(x \in \mathcal {H}_3({\mathbb Q})\), this holds for the surface \(A = {\textrm{Jac}}(C_x)/\langle P \rangle \), where P is the marked point of order 3. Our methods also give an explicit bound on the average rank of \(J_d({\mathbb Q})\), as well as statistical results on the size of \(\#C_d({\mathbb Q})\), as d varies through squarefree integers.

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Notes

  1. Even so, the large discriminant of C prevents its appearance in the current version of the LMFDB.

References

  1. Alpöge, L., Bhargava, M., Shnidman, A.: A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so. Available at arXiv:2011.01186 (2020)

  2. Wieb, B., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. Symb. Comput. 24(3–4), 235–265 (1997)

    MathSciNet  MATH  Google Scholar 

  3. Balakrishnan, J.S., Dogra, N.: An effective Chabauty–Kim theorem. Compos. Math. 155(6), 1057–1075 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bhargava, M., Klagsbrun, Z., Lemke Oliver, R.J.: 3-Iogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field. Duke Math. J. 168(15), 2951–2989 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bhargava, M., Klagsbrun, Z., Lemke Oliver, R.J., Shnidman, A.: Elements of given order in Tate–Shafarevich groups of abelian varieties in quadratic twist families. Algebra Number Theory 15(3), 627–655 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21. Springer, Berlin (1990)

    Google Scholar 

  7. Bruin, N., Doerksen, K.: The arithmetic of genus two curves with \((4,4)\)-split Jacobians. Can. J. Math. 63(5), 992–1024 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bruin, N., Flynn, V., Shnidman, A.: Sage code related to this paper. http://math.huji.ac.il/~shnidman/BFScode.sage (2021)

  9. Bruin, N., Flynn, E.V., Testa, D.: Descent via \((3,3)\)-isogeny on Jacobians of genus 2 curves. Acta Arithmetica 165(3), 201–223 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bruin, N., Nasserden, B.: Arithmetic aspects of the Burkhardt quartic threefold. J. Lond. Math. Soc. 98(3), 536–556 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Castella, F., Grossi, G., Lee, J., Skinner, C.: On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes. Invent. Math. 227(2), 517–580 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dokchitser, T., Dokchitser, V., Maistret, C., Morgan, A.: Arithmetic of hyperelliptic curves over local fields. Available at arXiv:1808.02936 (2018)

  13. Delaunay, C.: Heuristics on class groups and on Tate–Shafarevich groups: the magic of the Cohen–Lenstra heuristics, Ranks of elliptic curves and random matrix theory. London Math. Soc. Lecture Note Ser., 341. Cambridge Univ. Press, Cambridge, pp. 323–340 (2007)

  14. Dokchitser, T., Dokchitser, V.: Local invariants of isogenous elliptic curves. Trans. Am. Math. Soc. 367(6), 4339–4358 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. El-Baz, D., Loughran, D., Sofos, E.: Multivariate normal distribution for integral points on varieties. Trans. Am. Math. Soc. 375(5), 3089–3128 (2022)

    MathSciNet  MATH  Google Scholar 

  16. Elkies, N., Kumar, A.: K3 surfaces and equations for Hilbert modular surfaces. Algebra Number Theory 8(10), 2297–2411 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. González, J., Guàrdia, J., Rotger, V.: Abelian surfaces of \({\rm GL}_2\)-type as Jacobians of curves. Acta Arith. 116(3), 263–287 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Groupes de monodromie en géométrie algébrique.: I, French, Lecture Notes in Mathematics, Vol. 288, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I); Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim, Springer, Berlin (1972)

  19. Hirzebruch, F., Zagier, D.: Classification of Hilbert Modular Surfaces. Complex Analysis and Algebraic Geometry, pp. 43–77. Iwanami Shoten, Tokyo (1977)

    Book  MATH  Google Scholar 

  20. Klosin, K., Papikian, M.: On Ribet’s isogeny for \(J_0(65)\). Proc. Am. Math. Soc. 146(8), 3307–3320 (2018)

    Article  MATH  Google Scholar 

  21. Oliver, R.L., Loughran, D., Shnidman, A.: Normal distribution for bad reduction types (2022). (in preparation)

  22. Müller, J.S., Stoll, M.: Canonical heights on genus-2 Jacobians. Algebra Number Theory 10(10), 2153–2234 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Namikawa, Y., Ueno, K.: The complete classification of fibres in pencils of curves of genus two. Manuscr. Math. 9, 143–186 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  24. Raynaud, M.: Courbes sur une variété abélienne et points de torsion. Invent. Math. 71(1), 207–233 (1983). (French)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shnidman, A.: Quadratic twists of abelian varieties with real multiplication. Int. Math. Res. Not. IMRN 5, 3267–3298 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Smith, A.: \(2^\infty \)-Selmer groups, \(2^\infty \)-class groups, and Goldfeld’s conjecture. Available at arXiv:1702.02325 (2017)

  27. The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.4), 2011. https://www.sagemath.org

  28. van Bommel, R.: Efficient computation of BSD invariants in genus 2. Available at arXiv:2002.04667 (2020)

  29. van der Geer, G.: Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 16. Springer, Berlin (1988)

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Acknowledgements

Thanks go to Michael Stoll for organizing Rational Points 2017, where the authors first discussed this project. We also thank Levent Alpöge, Netan Dogra, Robert Lemke Oliver, Dino Lorenzini, and Daniel Loughran for helpful conversations. The authors used sage [27] and Magma [2] for many computations in this paper.

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

    Nils Bruin

  2. Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford, OX1 3LB, UK

    E. Victor Flynn

  3. Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

    Ari Shnidman

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  1. Nils Bruin
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  2. E. Victor Flynn
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  3. Ari Shnidman
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Correspondence to Ari Shnidman.

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Nils Bruin acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2018-04191. A. Shnidman was supported by the Israel Science Foundation (Grant No. 2301/20).

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Bruin, N., Flynn, E.V. & Shnidman, A. Genus two curves with full \(\sqrt{3}\)-level structure and Tate–Shafarevich groups. Sel. Math. New Ser. 29, 42 (2023). https://doi.org/10.1007/s00029-023-00839-w

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