Duke Mathematical Journal

3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

Manjul Bhargava, Zev Klagsbrun, Robert J. Lemke Oliver, and Ari Shnidman

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Abstract

For an abelian variety A over a number field F, we prove that the average rank of the quadratic twists of A is bounded, under the assumption that the multiplication-by-3-isogeny on A factors as a composition of 3-isogenies over F. This is the first such boundedness result for an absolutely simple abelian variety A of dimension greater than 1. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension 1, we deduce that if E/F is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded. If F is totally real, we moreover show that a positive proportion of twists have rank 0 and a positive proportion have 3-Selmer rank 1. These results on bounded average ranks in families of quadratic twists represent new progress toward Goldfeld’s conjecture—which states that the average rank in the quadratic twist family of an elliptic curve over Q should be 1/2—and the first progress toward the analogous conjecture over number fields other than Q. Our results follow from a computation of the average size of the ϕ-Selmer group in the family of quadratic twists of an abelian variety admitting a 3-isogeny ϕ.

Article information

Source
Duke Math. J., Volume 168, Number 15 (2019), 2951-2989.

Dates
Received: 25 April 2018
Revised: 16 April 2019
First available in Project Euclid: 24 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1569290597

Digital Object Identifier
doi:10.1215/00127094-2019-0031

Mathematical Reviews number (MathSciNet)
MR4017518

Subjects
Primary: 14G05: Rational points
Secondary: 11G10: Abelian varieties of dimension > 1 [See also 14Kxx]

Keywords
abelian varieties Selmer groups geometry of numbers rational points quadratic twists

Citation

Bhargava, Manjul; Klagsbrun, Zev; Lemke Oliver, Robert J.; Shnidman, Ari. $3$ -Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field. Duke Math. J. 168 (2019), no. 15, 2951--2989. doi:10.1215/00127094-2019-0031. https://projecteuclid.org/euclid.dmj/1569290597


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References

  • [1] M. Bhargava, N. Elkies, and A. Shnidman, The average size of the $3$-isogeny Selmer groups of elliptic curves $y^{2}=x^{3}+k$, to appear in J. Lond. Math. Soc. (2), preprint, arXiv:1610.05759 [math.NT].
    arXiv: 1610.05759
  • [2] M. Bhargava and B. H. Gross, “Arithmetic invariant theory” in Symmetry: Representation Theory and Its Applications, Progr. Math. 257, Birkhäuser/Springer, New York, 2014, 33–54.
  • [3] M. Bhargava and W. Ho, On the average sizes of Selmer groups in families of elliptic curves, preprint.
  • [4] M. Bhargava, Z. Klagsbrun, R. Lemke Oliver, and A. Shnidman, Sage code related to this paper, http://math.tufts.edu/faculty/rlemkeoliver/code/threeselmer.html (accessed 9 September 2019).
  • [5] M. Bhargava, Z. Klagsbrun, R. Lemke Oliver, and A. Shnidman, Elements of given order in Tate-Shafarevich groups of abelian varieties in quadratic twist families, preprint, arXiv:1904.00116 [math.NT].
    arXiv: 1904.00116
  • [6] 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 (2015), no. 1, 191–242.
    Zentralblatt MATH: 1307.11071
    Digital Object Identifier: doi:10.4007/annals.2015.181.1.3
  • [7] M. Bhargava and A. 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.
    Mathematical Reviews (MathSciNet): MR3275847
    Digital Object Identifier: doi:10.4007/annals.2015.181.2.4
  • [8] M. Bhargava, A. Shankar, and X. Wang, Geometry-of-numbers methods over global fields, I: Prehomogeneous vector spaces, preprint, arXiv:1512.03035 [math.NT].
    arXiv: 1512.03035
  • [9] M. Bhargava, A. Shankar, and X. Wang, Geometry-of-numbers methods over global fields, II: Coregular representations, in preparation.
  • [10] N. Bruin, E. V. Flynn, and D. Testa, Descent via $(3,3)$-isogeny on Jacobians of genus 2 curves, Acta Arith. 165 (2014), no. 3, 201–223.
    Zentralblatt MATH: 1311.11052
    Digital Object Identifier: doi:10.4064/aa165-3-1
  • [11] N. Bruin and B. Nasserden, Arithmetic aspects of the Burkhardt quartic threefold, J. Lond. Math. Soc. (2) 98 (2018), no. 3, 536–556.
    Zentralblatt MATH: 07000627
    Digital Object Identifier: doi:10.1112/jlms.12153
  • [12] J. W. S. Cassels, Arithmetic on curves of genus 1, VIII: On conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 217 (1965), 180–199.
    Zentralblatt MATH: 0241.14017
  • [13] K. Česnavičius, Selmer groups as flat cohomology groups, J. Ramanujan Math. Soc. 31 (2016), no. 1, 31–61.
  • [14] S. Chang, Note on the rank of quadratic twists of Mordell equations, J. Number Theory 118 (2006), no. 1, 53–61.
    Zentralblatt MATH: 1208.11072
    Digital Object Identifier: doi:10.1016/j.jnt.2005.08.004
  • [15] H. Cohen and F. Pazuki, Elementary $3$-descent with a $3$-isogeny, Acta Arith. 140 (2009), no. 4, 369–404.
  • [16] S. Comalada, Twists and reduction of an elliptic curve, J. Number Theory 49 (1994), no. 1, 45–62.
    Zentralblatt MATH: 0873.11034
    Digital Object Identifier: doi:10.1006/jnth.1994.1079
  • [17] T. Dokchitser and V. Dokchitser, Local invariants of isogenous elliptic curves, Trans. Amer. Math. Soc. 367 (2015), no. 6, 4339–4358.
    Zentralblatt MATH: 1395.11094
    Digital Object Identifier: doi:10.1090/S0002-9947-2014-06271-5
  • [18] M. Gealy and Z. Klagsbrun, Minimal differentials of elliptic curves with a $p$-isogeny, to appear in Proc. Amer. Math. Soc.
  • [19] D. Goldfeld, “Conjectures on elliptic curves over quadratic fields” in Number Theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, IL, 1979), Lecture Notes in Math. 75, Springer, Berlin, 1979, 108–118.
  • [20] F. Hazama, Hodge cycles on the Jacobian variety of the Catalan curve, Compositio Math. 107 (1997), no. 3, 339–353.
    Zentralblatt MATH: 1044.11587
    Digital Object Identifier: doi:10.1023/A:1000106427229
  • [21] D. R. Heath-Brown, The size of Selmer groups for the congruent number problem, II, with an appendix by P. Monsky, Invent. Math. 118 (1994), no. 2, 331–370.
    Zentralblatt MATH: 0815.11032
    Digital Object Identifier: doi:10.1007/BF01231536
  • [22] K. James, $L$-series with nonzero central critical value, J. Amer. Math. Soc. 11 (1998), no. 3, 635–641.
  • [23] D. Kane, On the ranks of the 2-Selmer groups of twists of a given elliptic curve, Algebra Number Theory 7 (2013), no. 5, 1253–1279.
    Zentralblatt MATH: 1300.11061
    Digital Object Identifier: doi:10.2140/ant.2013.7.1253
    Project Euclid: euclid.ant/1513730012
  • [24] Z. Klagsbrun, Elliptic curves with a lower bound on 2-Selmer ranks of quadratic twists, Math. Res. Lett. 19 (2012), no. 5, 1137–1143.
    Zentralblatt MATH: 1285.11085
    Digital Object Identifier: doi:10.4310/MRL.2012.v19.n5.a14
  • [25] Z. Klagsbrun, B. Mazur, and K. Rubin, Disparity in Selmer ranks of quadratic twists of elliptic curves, Ann. of Math. (2) 178 (2013), no. 1, 287–320.
    Zentralblatt MATH: 1300.11063
    Digital Object Identifier: doi:10.4007/annals.2013.178.1.5
  • [26] Z. Klagsbrun, B. Mazur, and K. Rubin, A Markov model for Selmer ranks in families of twists, Compos. Math. 150 (2014), no. 7, 1077–1106.
    Zentralblatt MATH: 1316.11045
    Digital Object Identifier: doi:10.1112/S0010437X13007896
  • [27] D. Kriz, Generalized Heegner cycles at Eisenstein primes and the Katz $p$-adic $L$-function, Algebra Number Theory 10 (2016), no. 2, 309–374.
  • [28] D. Kriz and C. Li, Goldfeld’s conjecture and congruences between Heegner points, Forum Math. Sigma 7 (2019), e15.
    Zentralblatt MATH: 07059718
    Digital Object Identifier: doi:10.1017/fms.2019.9
  • [29] S. Lang, Complex Multiplication, Grundlehren Math. Wiss. 255, Springer, New York, 1983.
  • [30] Z. K. Li, Quadratic twists of elliptic curves with 3-Selmer rank 1, Int. J. Number Theory 10 (2014), no. 5, 1191–1217.
    Zentralblatt MATH: 1297.14037
    Digital Object Identifier: doi:10.1142/S1793042114500213
  • [31] A. Morgan, Quadratic twists of abelian varieties and disparity in Selmer ranks, Algebra Number Theory 13 (2019), no. 4, 839–899.
    Zentralblatt MATH: 07059758
    Digital Object Identifier: doi:10.2140/ant.2019.13.839
    Project Euclid: euclid.ant/1558144823
  • [32] V. K. Murty and V. M. Patankar, Splitting of abelian varieties, Int. Math. Res. Not. IMRN 2008, no. 12, art. ID rnn033.
    Zentralblatt MATH: 1152.14043
  • [33] K. Rubin, “Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer” in Arithmetic Theory of Elliptic Curves (Cetraro, 1997), Lecture Notes in Math. 1716, Springer, Berlin, 1999, 167–234.
  • [34] P. Satgé, Groupes de Selmer et corps cubiques, J. Number Theory 23 (1986), no. 3, 294–317.
  • [35] E. F. Schaefer, Class groups and Selmer groups, J. Number Theory 56 (1996), no. 1, 79–114.
    Zentralblatt MATH: 0859.11034
    Digital Object Identifier: doi:10.1006/jnth.1996.0006
  • [36] E. F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), no. 3, 447–471.
    Zentralblatt MATH: 0889.11021
    Digital Object Identifier: doi:10.1007/s002080050156
  • [37] E. F. Schaefer and J. L. Wetherell, Computing the selmer group of an isogeny between abelian varieties using a further isogeny to a Jacobian, J. Number Theory 115 (2005), no. 1, 158–175.
    Zentralblatt MATH: 1095.11033
    Digital Object Identifier: doi:10.1016/j.jnt.2004.11.014
  • [38] E. S. Selmer, The Diophantine equation $ax^{3}+by^{3}+cz^{3}=0$, Acta Math. 85 (1951), 203–362.
  • [39] J.-P. Serre, Galois Cohomology, Springer, Berlin, 1997.
  • [40] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2), 88 (1968), 492–517.
    Zentralblatt MATH: 0172.46101
    Digital Object Identifier: doi:10.2307/1970722
  • [41] A. Shankar and X. Wang, Rational points on hyperelliptic curves having a marked non-Weierstrass point, Compos. Math. 154 (2018), 188–222.
    Zentralblatt MATH: 1387.11035
    Digital Object Identifier: doi:10.1112/S0010437X17007515
  • [42] A. N. Shankar, $2$-Selmer groups of hyperelliptic curves with two marked points, Trans. Amer. Math. Soc. 372 (2019), no. 1, 267–304.
  • [43] A. Shnidman, Quadratic twists of abelian varieties with real multiplication, to appear in Int. Math. Res. Not. IMRN, preprint, arXiv:1710.04086 [math.NT].
    arXiv: 1710.04086
  • [44] A. Smith, $2^{\infty }$-Selmer groups, $2^{\infty }$-class groups, and Goldfeld’s conjecture, preprint, arXiv:1702.02325 [math.NT].
    arXiv: 1702.02325
  • [45] P. Swinnerton-Dyer, The effect of twisting on the $2$-Selmer group, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 3, 513–526.
  • [46] J. A. Thorne, Arithmetic invariant theory and $2$-descent for plane quartic curves, with an appendix by Tasho Kaletha, Algebra Number Theory 10 (2016), no. 7, 1373–1413.
    Zentralblatt MATH: 1416.11044
    Digital Object Identifier: doi:10.2140/ant.2016.10.1373
    Project Euclid: euclid.ant/1510842564
  • [47] V. Vatsal, Rank-one twists of a certain elliptic curve, Math. Ann. 311 (1998), no. 4, 791–794.
    Zentralblatt MATH: 0933.11034
    Digital Object Identifier: doi:10.1007/s002080050209
  • [48] T. Wang, Selmer groups and ranks of elliptic curves, senior thesis, Princeton Univ., 2012.

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