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On ranks of twists of elliptic curves and power-free values of binary forms. (English) Zbl 0857.11026

Let \(E\) be an elliptic curve over \(\mathbb{Q}\) and \(r_E\) the rank of the Mordell-Weil group \(E(\mathbb{Q})\). Since A. Néron [Proc. Internat. Congr. Math. 1954 Amsterdam 3, 481-488 (1956; Zbl 0074.15901)] the question of how large \(r_E\) can be has been studied. J.-F. Mestre [C. R. Acad. Sci., Paris, Sér. I 314, 919-922 (1992; Zbl 0766.14023)] developed Néron’s methods to produce an example of an infinite family of elliptic curves \(E\) defined over \(\mathbb{Q}\) whose rank is at least 12. On the other hand computational evidence has shown that a typical elliptic curve has in general a small rank, zero or one [A. Brumer and O. McGuinness [Bull. Am. Math. Soc., New Ser. 23, 375-382 (1990; Zbl 0741.14010)]. A. Brumer proved [Invent. Math. 109, 445-472 (1992; Zbl 0783.14019)] that supposing the Taniyama-Shimura and the Birch and Swinnerton-Dyer conjectures and the generalized Riemann hypothesis that the average rank of elliptic curves over \(\mathbb{Q}\) ordered by their Faltings’ height is 2.3.
In this paper the authors study how \(r_E\) varies over quadratic, cubic, quartic and sextic twists of a given elliptic curve over \(\mathbb{Q}\). Let \(y^2 = x^3+ax + b\) be a Weierstrass equation for \(E\). Given a non-zero integer \(d\), the quadratic twist \(E_d\) of \(E\) by \(d\) is given by \(dy^2 = x^3+ax + b\). Let \(x\) be a positive real number. Under the hypothesis of the validity of the Taniyama-Shimura and the Birch and Swinnerton-Dyer conjectures, D. Goldfeld [Lect. Notes Math. 751, 108-118 (1979; Zbl 0417.14031)] conjectured that \[ \sum_{0<|d |\leq x} r_{E_d} \sim {1\over 2} \sum_{0< |d |\leq x} 1. \] Moreover, T. Honda conjectured even more: if \(E\) is defined over a number field \(K\), \(E'\) is any twist of \(E\) and \(r_{E'}\) denotes the rank of \(E'(K)\), then \(r_{E'}\) depends just on \(E\) and \(K\) [Jap. J. Math. 30, 84-101 (1960; Zbl 0109.39602)]. In [J. Indian Math. Soc., New. Ser. 52, 51-69 (1987; Zbl 0688.14016)] D. Zagier and G. Kramarz computed the \(L\)-function and their derivatives at 1 for the cubic twists \(E_d\): \(x^3 + y^3=d\) of the elliptic curve \(E:x^3 + y^3=1\) up to \(d=70,000\). Subject to the Birch and Swinnerton-Dyer conjecture the calculations showed that a positive proportion of cubic twists \(E_d\) have even rank at least 2 and a positive proportion have odd rank at least 3. Later, F. Q. Gouvêa and B. Mazur proved the first theoretical results in this direction for quadratic twists \(E_d:dy^2 = x^3+ax+b\) of an elliptic curve \(E:y^2 = x^3+ax+b\) defined over \(\mathbb{Q}\) [J. Am. Math. Soc. 4, 1-23 (1991; Zbl 0725.11027)]. Given a real number \(T\), denote by \(S(T)\) the number of square-free integers \(d\) with \(|d |\leq T\) such that \(E_d\) has even rank at least 2. They proved under the validity of the parity conjecture that for any \(\varepsilon>0\) there exist positive numbers \(C_0\) and \(C_1\) depending only on \(E\) and \(\varepsilon\) such that if \(T>C_0\) then \(S(T)>C_1T^{1/2-\varepsilon}\). L. Mai [Can. J. Math. 45, 847-862 (1993; Zbl 0811.11041)] extended their result for cubic twists of \(E:x^3 + y^3=1\). He proved that for any \(\varepsilon > 0\) there exist two positive real numbers \(C_2\) and \(C_3\) such that for any real number \(T\) the number \(S(T)\) of cube-free integers \(d\) such that the twist \(E_d:x^3 + y^3=d\) of \(E\) has even rank at least 2, if \(T>C_2\), then \(S(T) > C_3T^{2/3-\varepsilon}\), under the hypothesis of the parity conjecture for cubic twists.
In this paper the authors generalize both results without the hypothesis of the validity of the parity conjecture. Furthermore, they give an affirmative answer to a question of Gouvêa and Mazur, who asked whether the exponent of \(T\) measuring the average of elliptic curves with even rank at least 2 is positive. For example, they prove that there exists a positive real number \(C_4\) such that if \(T>657\) then the number of cube-free integers \(d\) such that \(x^3+y^3=d\) has rank at least 3 is greater than \(C_4 T^{1/6}\). They use a construction due to J.-F. Mestre [loc. cit.] to show that if \(E\) is an elliptic curve over \(\mathbb{Q}\) with \(j\)-invariant not 0 nor 1728 then there exist positive real numbers \(C_5\) and \(C_6\) depending on \(E:y^2 = x^3+ax+b\) such that if \(T>C_5\) then the number \(S(T)\) of square-free integers \(d\) such that \(E_d:dy^2 = x^3+ax+b\) with rank at least 2 is greater than \(C_6T^{1/7}/(\log T)^2\).
Their strategy is to provide curves which are cyclic coverings of degrees 2, 3, 4 and 6 of the projective line with the property that their Jacobian varieties, contain over \(\mathbb{Q}\) (up to isogenies) many copies of a given elliptic curve. This strategy goes back to J. Tate and I. R. Shafarevich for elliptic curves over finite fields [Sov. Math. Dokl. 8, 917-920 (1967); translation from Dokl. Akad. Nauk SSSR 175, 770-773 (1967; Zbl 0168.42201)]. The authors count the number of twists with large rank by determining the number of \(k\)-free integers below a certain bound assumed by a binary quadratic form with integral coefficients and integral arguments.

MSC:

11G05 Elliptic curves over global fields
14H52 Elliptic curves
11N36 Applications of sieve methods
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[1] Jan Stienstra and Frits Beukers, On the Picard-Fuchs equation and the formal Brauer group of certain elliptic \?3-surfaces, Math. Ann. 271 (1985), no. 2, 269 – 304. · Zbl 0539.14006
[2] B. J. Birch and N. M. Stephens, The parity of the rank of the Mordell-Weil group, Topology 5 (1966), 295 – 299. · Zbl 0146.42401
[3] B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7 – 25. · Zbl 0118.27601
[4] B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79 – 108. · Zbl 0147.02506
[5] H. P. F. Swinnerton-Dyer and B. J. Birch, Elliptic curves and modular functions, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1975, pp. 2 – 32. Lecture Notes in Math., Vol. 476. · Zbl 1214.11081
[6] Armand Brumer, The average rank of elliptic curves. I, Invent. Math. 109 (1992), no. 3, 445 – 472. · Zbl 0783.14019
[7] Armand Brumer and Oisín McGuinness, The behavior of the Mordell-Weil group of elliptic curves, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 375 – 382. · Zbl 0741.14010
[8] Maurice Craig, A type of class group for imaginary quadratic fields, Acta Arith. 22 (1973), 449 – 459. (errata insert). · Zbl 0227.12003
[9] Maurice Craig, A construction for irregular discriminants, Osaka J. Math. 14 (1977), no. 2, 365 – 402. · Zbl 0366.12007
[10] D. S. Dummit, Néron models, Tate curves and Mestre’s technique of using elliptic curves to construct quadratic fields with non-trivial \( 5\)-rank, Number Theory Seminar, Montréal, 1989.
[11] P. Erdős and K. Mahler, On the number of integers which can be represented by a binary form, J. London Math. Soc. 13 (1938), 134-139. · Zbl 0018.34401
[12] G. Frey, On the Selmer group of twists of elliptic curves with \?-rational torsion points, Canad. J. Math. 40 (1988), no. 3, 649 – 665. · Zbl 0646.14024
[13] M. Fried, Constructions arising from Néron’s high rank curves, Trans. Amer. Math. Soc. 281 (1984), no. 2, 615 – 631. · Zbl 0609.14016
[14] Dorian Goldfeld, Conjectures on elliptic curves over quadratic fields, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 108 – 118. · Zbl 0417.14031
[15] F. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4 (1991), no. 1, 1 – 23. · Zbl 0725.11027
[16] George Greaves, Power-free values of binary forms, Quart. J. Math. Oxford Ser. (2) 43 (1992), no. 169, 45 – 65. · Zbl 0768.11034
[17] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. · Zbl 0423.10001
[18] Taira Honda, Isogenies, rational points and section points of group varieties, Japan. J. Math. 30 (1960), 84 – 101. · Zbl 0109.39602
[19] C. Hooley, On the power free values of polynomials, Mathematika 14 (1967), 21 – 26. · Zbl 0166.05203
[20] C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge University Press, Cambridge-New York-Melbourne, 1976. Cambridge Tracts in Mathematics, No. 70. · Zbl 0327.10044
[21] Jun-ichi Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. (2) 72 (1960), 612 – 649. · Zbl 0122.39002
[22] Kenneth Kramer, Arithmetic of elliptic curves upon quadratic extension, Trans. Amer. Math. Soc. 264 (1981), no. 1, 121 – 135. · Zbl 0471.14020
[23] K. Kramer and J. Tunnell, Elliptic curves and local \?-factors, Compositio Math. 46 (1982), no. 3, 307 – 352. · Zbl 0496.14030
[24] J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: \?-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 409 – 464. · Zbl 0362.12011
[25] E. Liverance, Heights of Heegner points in a family of elliptic curves, Ph.D. Thesis, University of Maryland, 1992.
[26] Kurt Mahler, Zur Approximation algebraischer Zahlen. I, Math. Ann. 107 (1933), no. 1, 691 – 730 (German). · Zbl 0006.10502
[27] Kurt Mahler, Zur Approximation algebraischer Zahlen. III, Acta Math. 62 (1933), no. 1, 91 – 166 (German). Über die mittlere Anzahl der Darstellungen grosser Zahlen durch binäre Formen. · Zbl 0008.19801
[28] Liem Mai, The analytic rank of a family of elliptic curves, Canad. J. Math. 45 (1993), no. 4, 847 – 862. · Zbl 0811.11041
[29] Jean-François Mestre, Courbes elliptiques et groupes de classes d’idéaux de certains corps quadratiques, J. Reine Angew. Math. 343 (1983), 23 – 35 (French). · Zbl 0502.12004
[30] Jean-François Mestre, Rang de courbes elliptiques d’invariant donné, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 12, 919 – 922 (French, with English summary). · Zbl 0920.11036
[31] Jean-François Mestre, Courbes elliptiques de rang \ge 12 sur \?(\?), C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 4, 171 – 174 (French, with English summary). · Zbl 0749.14026
[32] L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. · Zbl 0188.34503
[33] Shin Nakano, Construction of pure cubic fields with large 2-class groups, Osaka J. Math. 25 (1988), no. 1, 161 – 170. · Zbl 0649.12001
[34] A. Néron, Propriétés arithmétiques de certaines familles de courbes algébriques, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Erven P. Noordhoff N.V., Groningen; North-Holland Publishing Co., Amsterdam, 1956, pp. 481 – 488 (French).
[35] U. Schneiders and H. G. Zimmer, The rank of elliptic curves upon quadratic extension, Computational number theory (Debrecen, 1989) de Gruyter, Berlin, 1991, pp. 239 – 260. · Zbl 0743.14023
[36] Chad Schoen, Bounds for rational points on twists of constant hyperelliptic curves, J. Reine Angew. Math. 411 (1990), 196 – 204. · Zbl 0702.14015
[37] R. J. Schoof, Class groups of complex quadratic fields, Math. Comp. 41 (1983), no. 163, 295 – 302. · Zbl 0516.12002
[38] Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15, Friedr. Vieweg & Sohn, Braunschweig, 1989. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. · Zbl 0676.14005
[39] Tetsuji Shioda, An infinite family of elliptic curves over \? with large rank via Néron’s method, Invent. Math. 106 (1991), no. 1, 109 – 119. · Zbl 0766.14024
[40] Joseph H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197 – 211. · Zbl 0505.14035
[41] Joseph H. Silverman, Divisibility of the specialization map for families of elliptic curves, Amer. J. Math. 107 (1985), no. 3, 555 – 565. · Zbl 0613.14029
[42] -, The arithmetic of elliptic curves, Graduate Texts in Math., vol. 106, Springer-Verlag, Berlin, New York, and Heidelberg, 1985.
[43] C. L. Stewart, On the number of solutions of polynomial congruences and Thue equations, J. Amer. Math. Soc. 4 (1991), no. 4, 793 – 835. · Zbl 0744.11016
[44] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1975, pp. 33 – 52. Lecture Notes in Math., Vol. 476. · Zbl 1214.14020
[45] J. T. Tate and I. R. Shafarevich, The rank of elliptic curves, Soviet Math. Dokl. 8 (1967), 917-920. · Zbl 0168.42201
[46] J. Top, Néron’s proof of the existence of elliptic curves over \( \mathbb{Q}\) with rank at least 11, R. U. Utrecht Dept. of Math. preprint series, no. 476, 1987.
[47] D. Zagier and G. Kramarz, Numerical investigations related to the \?-series of certain elliptic curves, J. Indian Math. Soc. (N.S.) 52 (1987), 51 – 69 (1988). · Zbl 0688.14016
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