##
**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.

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.

Reviewer: Amílcar Pacheco (Rio de Janeiro)

### MSC:

11G05 | Elliptic curves over global fields |

14H52 | Elliptic curves |

11N36 | Applications of sieve methods |

### Keywords:

twists of elliptic curves; cyclic coverings of the projective line; \(k\)-free integers assumed by a binary quadratic form; even rank; twists with large rank### Citations:

Zbl 0074.15901; Zbl 0766.14023; Zbl 0741.14010; Zbl 0783.14019; Zbl 0417.14031; Zbl 0109.39602; Zbl 0688.14016; Zbl 0725.11027; Zbl 0811.11041; Zbl 0168.42201
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\textit{C. L. Stewart} and \textit{J. Top}, J. Am. Math. Soc. 8, No. 4, 943--973 (1995; Zbl 0857.11026)

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### References:

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