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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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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

abelian varieties Selmer groups geometry of numbers rational points quadratic twists


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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  • [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].
  • [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].
  • [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.
  • [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.
  • [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].
  • [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.
  • [11] N. Bruin and B. Nasserden, Arithmetic aspects of the Burkhardt quartic threefold, J. Lond. Math. Soc. (2) 98 (2018), no. 3, 536–556.
  • [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.
  • [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.
  • [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.
  • [17] T. Dokchitser and V. Dokchitser, Local invariants of isogenous elliptic curves, Trans. Amer. Math. Soc. 367 (2015), no. 6, 4339–4358.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [31] A. Morgan, Quadratic twists of abelian varieties and disparity in Selmer ranks, Algebra Number Theory 13 (2019), no. 4, 839–899.
  • [32] V. K. Murty and V. M. Patankar, Splitting of abelian varieties, Int. Math. Res. Not. IMRN 2008, no. 12, art. ID rnn033.
  • [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.
  • [36] E. F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), no. 3, 447–471.
  • [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.
  • [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.
  • [41] A. Shankar and X. Wang, Rational points on hyperelliptic curves having a marked non-Weierstrass point, Compos. Math. 154 (2018), 188–222.
  • [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].
  • [44] A. Smith, $2^{\infty }$-Selmer groups, $2^{\infty }$-class groups, and Goldfeld’s conjecture, preprint, arXiv:1702.02325 [math.NT].
  • [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.
  • [47] V. Vatsal, Rank-one twists of a certain elliptic curve, Math. Ann. 311 (1998), no. 4, 791–794.
  • [48] T. Wang, Selmer groups and ranks of elliptic curves, senior thesis, Princeton Univ., 2012.