×

zbMATH — the first resource for mathematics

Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. (English) Zbl 1307.11071
In this paper it is proved that the average rank of elliptic curves over \(\mathbb{Q}\) is bounded. To be more precise, consider curves \[ E_{A,B}:\,y^2=x^3+Ax+B \] with integer coefficients \(A,B\) normalized such that \(A^3\) and \(B^2\) have no nontrivial common 12-th power divisor. Set \(H(E_{A,B}):=\max(4|A|^3\,,\,27B^2)\). Then \[ \limsup_{X\rightarrow\infty}\frac{\sum_{H(E_{A,B})\leq X} \mathrm{Rank}(E_{A,B})}{\#\{E_{A,B}:H(E_{A,B})\leq X\}}\leq \frac{3}{2}. \] This result follows immediately from an assertion about the average size of the 2-Selmer group: \[ \sum_{H(E_{A,B})\leq X}\# S_2(E_{A,B})\sim 3\#\{E_{A,B}:H(E_{A,B})\}\leq X \] as \(X\rightarrow\infty\). Indeed the same is true when one restricts the coefficients \(A\) and \(B\) by any finite set of congruence conditions, or even a suitable natural set of infinitely many congruence conditions.
The proof handles the 2-Selmer group by counting binary quartic forms \(f(x,y)=ax^4+bx^3y+cx^2y^2+dxy^3+ey^4\) such that the curve \(f(x,y)=z^2\) has points everywhere locally. In order to relate such forms \(f\) to elliptic curves with \(H(E_{A,B})\leq X\), one counts forms for which the invariants \(I(f),J(f)\) satisfy \(H(I,J):=\max(|I|^3,J^2/4)\leq Y\), where again one may impose additional congruence conditions on \(f\). Such congruence conditions allow one to account for the local solvability constraints. Write \(h^{(j)}(I,J)\) for the number of equivalence classes under \(\mathrm{GL}_2(\mathbb{Z})\) of forms \(f\) having invariants \(I\) and \(J\), and having exactly \(4-2j\) real roots. Then it is shown for example that \[ \sum_{H(I,J)\leq X}h^{(0)}(I,J)=\frac{4}{135}\zeta(2)X^{5/6} +O(X^{3/4+\varepsilon}), \] for any fixed \(\varepsilon>0\). There are similar results for the other counting functions \(h^{(j)}(I,J)\), and for the situation when there are congruence restrictions on the forms \(f\).
The equivalence classes of binary quartic forms are counted by producing a suitable fundamental domain for the action of \(\mathrm{GL}_2(\mathbb{Z})\). However this fundamental domain is unbounded, so that it is not straightforward to count integer points in its dilates. This difficulty is addressed using ideas developed from the first author’s earlier work [Ann. Math. (2) 162, No. 2, 1031–1063 (2005; Zbl 1159.11045); ibid. 172, No. 3, 1559–1591 (2010; Zbl 1220.11139)].

MSC:
11G05 Elliptic curves over global fields
11E76 Forms of degree higher than two
11P21 Lattice points in specified regions
Software:
ecdata
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] B. Bektemirov, B. Mazur, W. Stein, and M. Watkins, ”Average ranks of elliptic curves: tension between data and conjecture,” Bull. Amer. Math. Soc., vol. 44, iss. 2, pp. 233-254, 2007. · Zbl 1190.11032
[2] M. Bhargava, ”Higher composition laws. III. The parametrization of quartic rings,” Ann. of Math., vol. 159, iss. 3, pp. 1329-1360, 2004. · Zbl 1169.11045
[3] M. Bhargava, ”The density of discriminants of quartic rings and fields,” Ann. of Math., vol. 162, iss. 2, pp. 1031-1063, 2005. · Zbl 1159.11045
[4] M. Bhargava, ”The density of discriminants of quintic rings and fields,” Ann. of Math., vol. 172, iss. 3, pp. 1559-1591, 2010. · Zbl 1220.11139
[5] M. Bhargava, The Ekedahl sieve and the density of squarefree values of invariant polynomials. · Zbl 1220.11139
[6] M. Bhargava and W. Ho, Coregular spaces and genus one curves. · Zbl 1342.14074
[7] B. J. Birch and H. P. F. Swinnerton-Dyer, ”Notes on elliptic curves. I,” J. Reine Angew. Math., vol. 212, pp. 7-25, 1963. · Zbl 0118.27601
[8] A. Borel and Harish-Chandra, ”Arithmetic subgroups of algebraic groups,” Ann. of Math., vol. 75, pp. 485-535, 1962. · Zbl 0107.14804
[9] A. Brumer, ”The average rank of elliptic curves. I,” Invent. Math., vol. 109, iss. 3, pp. 445-472, 1992. · Zbl 0783.14019
[10] A. Brumer and K. Kramer, ”The rank of elliptic curves,” Duke Math. J., vol. 44, iss. 4, pp. 715-743, 1977. · Zbl 0376.14011
[11] J. E. Cremona, Algorithms for Modular Elliptic Curves, Second ed., Cambridge: Cambridge Univ. Press, 1997. · Zbl 0872.14041
[12] J. E. Cremona, ”Reduction of binary cubic and quartic forms,” LMS J. Comput. Math., vol. 2, pp. 64-94, 1999. · Zbl 0927.11020
[13] J. E. Cremona and T. A. Fisher, ”On the equivalence of binary quartics,” J. Symbolic Comput., vol. 44, iss. 6, pp. 673-682, 2009. · Zbl 1159.14301
[14] M. Stoll and J. E. Cremona, ”Minimal models for 2-coverings of elliptic curves,” LMS J. Comput. Math., vol. 5, pp. 220-243, 2002. · Zbl 1067.11031
[15] H. Davenport, ”On a principle of Lipschitz,” J. London Math. Soc., vol. 26, pp. 179-183, 1951. · Zbl 0042.27504
[16] H. Davenport, ”On the class-number of binary cubic forms. I,” J. London Math. Soc., vol. 26, pp. 183-192, 1951. · Zbl 0044.27002
[17] H. Davenport and H. Heilbronn, ”On the density of discriminants of cubic fields. II,” Proc. Roy. Soc. London Ser. A, vol. 322, iss. 1551, pp. 405-420, 1971. · Zbl 0212.08101
[18] A. J. de Jong, ”Counting elliptic surfaces over finite fields,” Mosc. Math. J., vol. 2, iss. 2, pp. 281-311, 2002. · Zbl 1031.11033
[19] C. Delaunay, ”Heuristics on class groups and on Tate-Shafarevich groups: the magic of the Cohen-Lenstra heuristics,” in Ranks of Elliptic Curves and Random Matrix Theory, Cambridge: Cambridge Univ. Press, 2007, vol. 341, pp. 323-340. · Zbl 1231.11129
[20] B. Delaunay, ”Über die Darstellung der Zahlen durch die binären kubischen Formen von negativer Diskriminante,” Math. Z., vol. 31, iss. 1, pp. 1-26, 1930. · JFM 55.0722.02
[21] B. N. Delone and D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Amer. Math. Soc., 1964, vol. 10. · Zbl 0133.30202
[22] T. Ekedahl, ”An infinite version of the Chinese remainder theorem,” Comment. Math. Univ. St. Paul., vol. 40, iss. 1, pp. 53-59, 1991. · Zbl 0749.11004
[23] J. -H. Evertse, ”On the representation of integers by binary cubic forms of positive discriminant,” Invent. Math., vol. 73, iss. 1, pp. 117-138, 1983. · Zbl 0494.10009
[24] É. Fouvry, ”Sur le comportement en moyenne du rang des courbes \(y^2=x^3+k\),” in Séminaire de Théorie des Nombres, Paris, 1990-91, Birkhäuser, Boston, 1993, vol. 108, pp. 61-84. · Zbl 0814.11034
[25] D. Goldfeld, ”Conjectures on elliptic curves over quadratic fields,” in Number Theory, Carbondale 1979, New York: Springer-Verlag, 1979, vol. 751, pp. 108-118. · Zbl 0417.14031
[26] D. R. Heath-Brown, ”The size of Selmer groups for the congruent number problem,” Invent. Math., vol. 111, iss. 1, pp. 171-195, 1993. · Zbl 0808.11041
[27] D. R. Heath-Brown, ”The average analytic rank of elliptic curves,” Duke Math. J., vol. 122, iss. 3, pp. 591-623, 2004. · Zbl 1063.11013
[28] D. Kane, ”On the ranks of the 2-Selmer groups of twists of a given elliptic curve,” Algebra Number Theory, vol. 7, iss. 5, pp. 1253-1279, 2013. · Zbl 1300.11061
[29] N. M. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc., 1999, vol. 45. · Zbl 0958.11004
[30] R. P. Langlands, ”The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups,” in Algebraic Groups and Discontinuous Subgroups, Providence, R.I.: Amer. Math. Soc., 1966, pp. 143-148. · Zbl 0218.20041
[31] F. Mertens, ”Ueber einige asymptotische Gesetze der Zahlentheorie,” J. reine angew Math., vol. 77, pp. 289-338, 1874. · JFM 06.0114.02
[32] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, Boston, 1994, vol. 139. · Zbl 0841.20046
[33] B. Poonen, ”Squarefree values of multivariable polynomials,” Duke Math. J., vol. 118, iss. 2, pp. 353-373, 2003. · Zbl 1047.11021
[34] B. Poonen and E. Rains, ”Random maximal isotropic subspaces and Selmer groups,” J. Amer. Math. Soc., vol. 25, iss. 1, pp. 245-269, 2012. · Zbl 1294.11097
[35] P. Schneider, \(p\)-adic Lie Groups, New York: Springer-Verlag, 2011, vol. 344. · Zbl 1223.22008
[36] . J-P. Serre, A Course in Arithmetic, New York: Springer-Verlag, 1973. · Zbl 0256.12001
[37] C. L. Siegel, ”The average measure of quadratic forms with given determinant and signature,” Ann. of Math., vol. 45, pp. 667-685, 1944. · Zbl 0063.07007
[38] P. Swinnerton-Dyer, ”The effect of twisting on the 2-Selmer group,” Math. Proc. Cambridge Philos. Soc., vol. 145, iss. 3, pp. 513-526, 2008. · Zbl 1242.11041
[39] M. E. M. Wood, Moduli spaces for rings and ideals, Ann Arbor, MI: ProQuest LLC, 2009.
[40] M. M. Wood, ”Quartic rings associated to binary quartic forms,” Int. Math. Res. Not., vol. 2012, p. no. 6 (2012), 1300-1320. · Zbl 1254.11094
[41] M. P. Young, ”Low-lying zeros of families of elliptic curves,” J. Amer. Math. Soc., vol. 19, iss. 1, pp. 205-250, 2006. · Zbl 1086.11032
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.