2-Selmer groups of even hyperelliptic curves over function fields

Thinh, Dao

doi:10.1090/tran/8878

2-Selmer groups of even hyperelliptic curves over function fields
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by Dao Van Thinh;

Trans. Amer. Math. Soc. 376 (2023), 4679-4712

DOI: https://doi.org/10.1090/tran/8878

Published electronically: April 19, 2023

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Abstract:

In this paper, we are going to compute the average size of 2-Selmer groups of families of even hyperelliptic curves over function fields. The result will be obtained by a geometric method which is based on a Vinberg’s representation of the group $G=\text {PSO}(2n+2)$ and a Hitchin fibration. Consistent with the result over $\mathbb {Q}$ of Arul Shankar and Xiaoheng Wang [Compos. Math. 154 (2018), pp. 188–222], we provide an upper bound and a lower bound of the average. However, if we restrict to the family of transversal hyperelliptic curves, we obtain precisely average number 6.

References

Avner Ash, Jos Brakenhoff, and Theodore Zarrabi, Equality of polynomial and field discriminants, Experiment. Math. 16 (2007), no. 3, 367–374. MR 2367325, DOI 10.1080/10586458.2007.10129001
Manjul Bhargava and Benedict H. Gross, The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, Automorphic representations and $L$-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22, Tata Inst. Fund. Res., Mumbai, 2013, pp. 23–91. MR 3156850
Manjul Bhargava and Benedict H. Gross, Arithmetic invariant theory, Symmetry: representation theory and its applications, Progr. Math., vol. 257, Birkhäuser/Springer, New York, 2014, pp. 33–54. MR 3363006, DOI 10.1007/978-1-4939-1590-3_{3}
Manjul Bhargava, Benedict H. Gross, and Xiaoheng Wang, Arithmetic invariant theory II: Pure inner forms and obstructions to the existence of orbits, Representations of reductive groups, Progr. Math., vol. 312, Birkhäuser/Springer, Cham, 2015, pp. 139–171. MR 3495795, DOI 10.1007/978-3-319-23443-4_{5}
M. Bhargava and A. Shankar, The average number of elements in the 4-selmer groups of elliptic curves is 7, 2013, arXiv:1312.7333.
M. Bhargava and A. Shankar, The average size of the 5-selmer group of elliptic curves is 6, and the average rank is less than 1, 2013, arXiv:1312.7859.
M. Bhargava and A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Ann. of Math. (2), 181(1):191–242, 2015.
Manjul Bhargava and Arul Shankar, Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0, Ann. of Math. (2) 181 (2015), no. 2, 587–621. MR 3275847, DOI 10.4007/annals.2015.181.2.4
Manjul Bhargava, Arul Shankar, and Xiaoheng Wang, Squarefree values of polynomial discriminants I, Invent. Math. 228 (2022), no. 3, 1037–1073. MR 4419629, DOI 10.1007/s00222-022-01098-w
Indranil Biswas and Yogish I. Holla, Harder-Narasimhan reduction of a principal bundle, Nagoya Math. J. 174 (2004), 201–223. MR 2066109, DOI 10.1017/S0027763000008850
Bosch, S. 1990. Néron model, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 21, Springer-Verlag.
A. J. de Jong, Counting elliptic surfaces over finite fields, Mosc. Math. J. 2 (2002), no. 2, 281–311. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1944508, DOI 10.17323/1609-4514-2002-2-2-281-311
Feng, T. 2022. The geometric distribution of Selmer groups of elliptic curves over function fields, Math. Ann., 2022.
Q. P. H\grcf{o}, V. B. Lê Hùng, and B. C. Ngô, Average size of 2-Selmer groups of elliptic curves over function fields, Math. Res. Lett. 21 (2014), no. 6, 1305–1339. MR 3335849, DOI 10.4310/MRL.2014.v21.n6.a6
J. Laga, Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics, arXiv e-prints, arXiv:2204.02048, 2022.
Aaron Landesman, The geometric average size of Selmer groups over function fields, Algebra Number Theory 15 (2021), no. 3, 673–709. MR 4261097, DOI 10.2140/ant.2021.15.673
Paul Levy, Involutions of reductive Lie algebras in positive characteristic, Adv. Math. 210 (2007), no. 2, 505–559. MR 2303231, DOI 10.1016/j.aim.2006.07.002
Paul Levy, Vinberg’s $\theta$-groups in positive characteristic and Kostant-Weierstrass slices, Transform. Groups 14 (2009), no. 2, 417–461. MR 2504929, DOI 10.1007/s00031-009-9056-y
Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
Bjorn Poonen, Squarefree values of multivariable polynomials, Duke Math. J. 118 (2003), no. 2, 353–373. MR 1980998, DOI 10.1215/S0012-7094-03-11826-8
Beth Romano and Jack A. Thorne, On the arithmetic of simple singularities of type $E$, Res. Number Theory 4 (2018), no. 2, Paper No. 21, 34. MR 3787911, DOI 10.1007/s40993-018-0110-5
Arul Shankar and Xiaoheng Wang, Rational points on hyperelliptic curves having a marked non-Weierstrass point, Compos. Math. 154 (2018), no. 1, 188–222. MR 3719247, DOI 10.1112/S0010437X17007515
Ananth N. Shankar, 2-Selmer groups of hyperelliptic curves with marked points, Trans. Amer. Math. Soc. 372 (2019), no. 1, 267–304. MR 3968769, DOI 10.1090/tran/7546
T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Math., Vol. 131, Springer, Berlin-New York, 1970, pp. 167–266. MR 268192
Dao Van Thinh, Average size of 2- selmer groups of hyperelliptic curves over function fields, arXiv:1711.06147v1.
Dao Van Thinh, Average size of 2-Selmer groups of Jacobians of odd hyperelliptic curves over function fields, Pacific J. Math. 319 (2022), no. 2, 259–305. MR 4482717, DOI 10.2140/pjm.2022.319.259
Jack A. Thorne, $E_6$ and the arithmetic of a family of non-hyperelliptic curves of genus 3, Forum Math. Pi 3 (2015), e1, 41. MR 3298319, DOI 10.1017/fmp.2014.2
Jack A. Thorne, On the average number of 2-Selmer elements of elliptic curves over $\Bbb F_q(X)$ with two marked points, Doc. Math. 24 (2019), 1179–1223. MR 4012556, DOI 10.4171/dm/702
È. B. Vinberg, The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 3, 488–526, 709 (Russian). MR 430168
Xiaoheng Wang, Pencils of quadrics and Jacobians of hyperelliptic curves, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–Harvard University. MR 3167287