\(3\)-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field.

*(English)*Zbl 1442.14084The quadratic twist \(A_s\) of an abelian variety over a number field \(F\) is determined by a class \(s\in F^*/F^{*2}\). Geometrically \(A_s\) is
the same as \(A\) (they are isomorphic over \(F(\sqrt{s})\)) but arithmetically they can vary considerably. Because of the absence of
complications arising from moduli, the families \(A_s\) are natural ones to study from the point of view of arithmetic, but even very basic
information such as the statistics of the Mordell-Weil ranks are essentially unknown. It is expected, at least in some cases, that the average rank should be bounded: for elliptic curves \(E\) over \(\mathbb{Q}\) it is expected to be \(\frac{1}{2}\) and it is known to be at most
\(\frac{1}{2}\) if \(E[2](\mathbb{Q})\) is maximal, i.e. is the Klein 4-group.

This paper improves the position dramatically by proving that \(A_s(F)\) has bounded average rank as long as \(A\) has a 3-selfisogeny: slightly more generally, if there is an isogeny \(A\to A\) that factors as a composition of 3-isogenies over \(F\). In particular this is the case if \(E\) is an elliptic curve over \(\mathbb{Q}\) with a 3-torsion point, so this provides examples of families without full 2-torsion for which the average rank of \(A_s\) is bounded, as well as many examples in dimension \(>1\).

The main invariant actually studied in the paper is not in fact the Mordell-Weil rank directly, but the 3-Selmer groups mentioned in the title. If \(\phi\colon A\to A'\) is an isogeny defined over \(F\) there is an associated Selmer group \({\operatorname{Sel}}_\phi(A)\) (which reduces to the usual \(n\)-Selmer group in the case where \(A\) is an elliptic curve and \(\phi\) is mutiplication by \(n\)), and there is also an isogeny \(\phi_s\colon A_s\to A'_s\). The main technical result is an estimate of the average size of \({\operatorname{Sel}}_{\phi_s}(A_s)\) as \(s\) varies in a (mildly restricted) subset \(\Sigma\subset F^*/F^{*2}\), ordered by a height. The point is that the size of \({\operatorname{Sel}}_{\phi_s}(A_s)\) should, in view of the results of [M. Bhargava et al., J. Lond. Math. Soc., II. Ser. 101, No. 1, 299–327 (2020; Zbl 1458.11091)], be determined by an Euler product, but the local factors turn out to be equal to \(1\) away from \(3\), \(\infty\) and the conductor of \(A\), and this allows a direct computation of the average size of \({\operatorname{Sel}}_{\phi_s}(A_s)\), which is even a rational number.

Several corollaries are listed, some of them conditional on conjectures about the 3-rank of the Tate-Shafarevich group. The precise statements need some care, but one, which gives the general flavour, says (unconditionally) that if \(E\) is an elliptic curve over a totally real field \(F\) and admits a 3-isogeny, then the proportion of the \(E_s\) having Mordell-Weil rank zero is positive.

The proofs rest on a correspondence between elements of \({\operatorname{Sel}}_\phi\) and binary cubic forms over \(F\), whose origins can be traced back to [E. S. Selmer, Acta Math. 85, 203–362 (1951; Zbl 0042.26905)] but which is new even for elliptic curves. One may identify \({\operatorname{SL}}_2(F)\) orbits of \(F\)-valued binary quadratic forms (elements of \(\operatorname{Sym}^2(\mathbb{Z})^3(F)\)) of discriminant \(4D\) with the kernel of the norm map \(K^*/K^{*3}\to F^*/F^{*3}\), where \(K=F[t]/(t^2-D)\). But this kernel (replacing \(D\) with \(-3D\)) is also isomorphic to \(H^1({\operatorname{Gal}}(\overline{F}/F), A[\phi])\) for a 3-isogeny \(\phi\), and this classifies \(\phi\)-coverings of \(A\).

This paper improves the position dramatically by proving that \(A_s(F)\) has bounded average rank as long as \(A\) has a 3-selfisogeny: slightly more generally, if there is an isogeny \(A\to A\) that factors as a composition of 3-isogenies over \(F\). In particular this is the case if \(E\) is an elliptic curve over \(\mathbb{Q}\) with a 3-torsion point, so this provides examples of families without full 2-torsion for which the average rank of \(A_s\) is bounded, as well as many examples in dimension \(>1\).

The main invariant actually studied in the paper is not in fact the Mordell-Weil rank directly, but the 3-Selmer groups mentioned in the title. If \(\phi\colon A\to A'\) is an isogeny defined over \(F\) there is an associated Selmer group \({\operatorname{Sel}}_\phi(A)\) (which reduces to the usual \(n\)-Selmer group in the case where \(A\) is an elliptic curve and \(\phi\) is mutiplication by \(n\)), and there is also an isogeny \(\phi_s\colon A_s\to A'_s\). The main technical result is an estimate of the average size of \({\operatorname{Sel}}_{\phi_s}(A_s)\) as \(s\) varies in a (mildly restricted) subset \(\Sigma\subset F^*/F^{*2}\), ordered by a height. The point is that the size of \({\operatorname{Sel}}_{\phi_s}(A_s)\) should, in view of the results of [M. Bhargava et al., J. Lond. Math. Soc., II. Ser. 101, No. 1, 299–327 (2020; Zbl 1458.11091)], be determined by an Euler product, but the local factors turn out to be equal to \(1\) away from \(3\), \(\infty\) and the conductor of \(A\), and this allows a direct computation of the average size of \({\operatorname{Sel}}_{\phi_s}(A_s)\), which is even a rational number.

Several corollaries are listed, some of them conditional on conjectures about the 3-rank of the Tate-Shafarevich group. The precise statements need some care, but one, which gives the general flavour, says (unconditionally) that if \(E\) is an elliptic curve over a totally real field \(F\) and admits a 3-isogeny, then the proportion of the \(E_s\) having Mordell-Weil rank zero is positive.

The proofs rest on a correspondence between elements of \({\operatorname{Sel}}_\phi\) and binary cubic forms over \(F\), whose origins can be traced back to [E. S. Selmer, Acta Math. 85, 203–362 (1951; Zbl 0042.26905)] but which is new even for elliptic curves. One may identify \({\operatorname{SL}}_2(F)\) orbits of \(F\)-valued binary quadratic forms (elements of \(\operatorname{Sym}^2(\mathbb{Z})^3(F)\)) of discriminant \(4D\) with the kernel of the norm map \(K^*/K^{*3}\to F^*/F^{*3}\), where \(K=F[t]/(t^2-D)\). But this kernel (replacing \(D\) with \(-3D\)) is also isomorphic to \(H^1({\operatorname{Gal}}(\overline{F}/F), A[\phi])\) for a 3-isogeny \(\phi\), and this classifies \(\phi\)-coverings of \(A\).

Reviewer: G. K. Sankaran (Bath)

##### 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]. |

[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. · Zbl 1377.11045 |

[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. · Zbl 1307.11071 |

[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. · Zbl 1317.11038 |

[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. · Zbl 1311.11052 |

[11] | N. Bruin and B. Nasserden, Arithmetic aspects of the Burkhardt quartic threefold, J. Lond. Math. Soc. (2) 98 (2018), no. 3, 536-556. · Zbl 1446.11111 |

[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. · Zbl 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. · Zbl 1208.11072 |

[15] | H. Cohen and F. Pazuki, Elementary \(3\)-descent with a \(3\)-isogeny, Acta Arith. 140 (2009), no. 4, 369-404. · Zbl 1253.11063 |

[16] | S. Comalada, Twists and reduction of an elliptic curve, J. Number Theory 49 (1994), no. 1, 45-62. · Zbl 0873.11034 |

[17] | T. Dokchitser and V. Dokchitser, Local invariants of isogenous elliptic curves, Trans. Amer. Math. Soc. 367 (2015), no. 6, 4339-4358. · Zbl 1395.11094 |

[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. · Zbl 1044.11587 |

[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. · Zbl 0815.11032 |

[22] | K. James, \(L\)-series with nonzero central critical value, J. Amer. Math. Soc. 11 (1998), no. 3, 635-641. · Zbl 0904.11015 |

[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. · Zbl 1300.11061 |

[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. · Zbl 1285.11085 |

[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. · Zbl 1300.11063 |

[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. · Zbl 1316.11045 |

[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. · Zbl 1396.11097 |

[28] | D. Kriz and C. Li, Goldfeld’s conjecture and congruences between Heegner points, Forum Math. Sigma 7 (2019), e15. · Zbl 1448.11105 |

[29] | S. Lang, Complex Multiplication, Grundlehren Math. Wiss. 255, Springer, New York, 1983. · Zbl 0536.14029 |

[30] | Z. K. Li, Quadratic twists of elliptic curves with 3-Selmer rank 1, Int. J. Number Theory 10 (2014), no. 5, 1191-1217. · Zbl 1297.14037 |

[31] | A. Morgan, Quadratic twists of abelian varieties and disparity in Selmer ranks, Algebra Number Theory 13 (2019), no. 4, 839-899. · Zbl 1455.11087 |

[32] | V. K. Murty and V. M. Patankar, Splitting of abelian varieties, Int. Math. Res. Not. IMRN 2008, no. 12, art. ID rnn033. · Zbl 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. · Zbl 0991.11028 |

[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. · Zbl 0859.11034 |

[36] | E. F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), no. 3, 447-471. · Zbl 0889.11021 |

[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. · Zbl 1095.11033 |

[38] | E. S. Selmer, The Diophantine equation \(ax^3+by^3+cz^3=0\), Acta Math. 85 (1951), 203-362. · Zbl 0042.26905 |

[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. · Zbl 0172.46101 |

[41] | A. Shankar and X. Wang, Rational points on hyperelliptic curves having a marked non-Weierstrass point, Compos. Math. 154 (2018), 188-222. · Zbl 1387.11035 |

[42] | A. N. Shankar, \(2\)-Selmer groups of hyperelliptic curves with two marked points, Trans. Amer. Math. Soc. 372 (2019), no. 1, 267-304. · Zbl 1444.11138 |

[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. · Zbl 1242.11041 |

[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. · Zbl 1416.11044 |

[47] | V. Vatsal, Rank-one twists of a certain elliptic curve, Math. Ann. 311 (1998), no. 4, 791-794. · Zbl 0933.11034 |

[48] | T. Wang, Selmer groups and ranks of elliptic curves, senior thesis, Princeton Univ., 2012. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.