Orders of Tate-Shafarevich groups for the Neumann-Setzer type elliptic curves

2019

Banach Center Publ.

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This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of a given elliptic curve, and for the family of the Neumann-Setzer type elliptic curves. Here we present the results of our search for the (analytic) orders of Tate-Shafarevich groups for the cubic twists of X 0 (27). Our calculations extend those given by Zagier and Kramarz [20] and by Watkins [19]. Our main observations concern the asymptotic formula for the frequency of orders of Tate-Shafarevich groups. In the last section we propose a similar asymptotic formula for the class numbers of real quadratic fields.

Section: Introductionsupporting

confidence: 79%

Section: Distributions Of |X(e M )|mentioning

confidence: 89%

See 1 more Smart Citation

Orders of Tate–Shafarevich groups for the cubic twists of $X_0(27)$

2019

Banach Center Publ.

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“…Very recently, Bhargava, Skinner and Zhang [1] proved that at least 66.48% of all elliptic curves over Q, when ordered by height, satisfy the weak form of the Birch and Swinnerton-Dyer conjecture, and have finite Tate-Shafarevich group.When E has complex multiplication by the ring of integers of an imaginary quadratic field K and L(E, 1) is non-zero, the p-part of the Birch and Swinnerton-Dyer conjecture has been established by Rubin [27] for all primes p which do not divide the order of the group of roots of unity of K. Coates et al [5] [4], and Gonzalez-Avilés [17] showed that there is a large class of explicit quadratic twists of X 0 (49) whose complex L-series does not vanish at s = 1, and for which the full Birch and Swinnerton-Dyer conjecture is valid (covering the case p = 2 when K = Q( √ −7)). The deep results by Skinner-Urban ([29], Theorem 2) (see also Theorem 7 in section 8.4 below) allow, in specific cases (still assuming L(E, 1) is non-zero), to establish p-part of the Birch and Swinnerton-Dyer conjecture for elliptic curves without complex multiplication for all odd primes p (see examples in section 8.4 below, and section 3 in [10]). The numerical studies and conjectures by Conrey-Keating-Rubinstein-Snaith [7], Delaunay [12][13], Watkins [31], Radziwi l l-Soundararajan [25] (see also the papers [11][10] [9], and references therein) substantially extend the systematic tables given by Cremona.This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of the curve X 0 (49).…”

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confidence: 99%

Behaviour of the Order of Tate–Shafarevich Groups for the Quadratic Twists of $$X_0(49)$$

Jędrzejak

Springer Proceedings in Mathematics &Amp; Statistics

2016

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We present the results of our search for the orders of Tate-Shafarevich groups for the quadratic twists of elliptic curves. We formulate a general conjecture, giving for a fixed elliptic curve E over Q and positive integer k, an asymptotic formula for the number of quadratic twists E d , d positive square-free integers less than X, with finite group E d (Q) and |X(E d (Q))| = k 2 . This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of the curve X 0 (49). In section 8 we exhibit 88 examples of rank zero elliptic curves with |X(E)| > 63408 2 , which was the largest previously known value for any explicit curve. Our record is an elliptic curve E with |X(E)| = 1029212 2 .Birch and Swinnerton-Dyer conjecture relates the arithmetic data of E to the behaviour of L(E, s) at s = 1.Conjecture 1 (Birch and Swinnerton-Dyer) (i) L-function L(E, s) has a zero of order r = rank E(Q) at s = 1,If X(E) is finite, the work of Cassels and Tate shows that its order must be a square.The first general result in the direction of this conjecture was proven for elliptic curves E with complex multiplication by Coates and Wiles in 1976 [6], who showed that if L(E, 1) = 0, then the group E(Q) is finite. Gross and Zagier [18] showed that if L(E, s) has a first-order zero at s = 1, then E has a rational point of infinite order. Rubin [26] proves that if E has complex multiplication and L(E, 1) = 0, then X(E) is finite. Let g E be the rank of E(Q) and let r E the order of the zero of L(E, s) at s = 1. Then Kolyvagin [20] proved that, if r E ≤ 1, then r E = g E and X(E) is finite. Very recently, Bhargava, Skinner and Zhang [1] proved that at least 66.48% of all elliptic curves over Q, when ordered by height, satisfy the weak form of the Birch and Swinnerton-Dyer conjecture, and have finite Tate-Shafarevich group.When E has complex multiplication by the ring of integers of an imaginary quadratic field K and L(E, 1) is non-zero, the p-part of the Birch and Swinnerton-Dyer conjecture has been established by Rubin [27] for all primes p which do not divide the order of the group of roots of unity of K. Coates et al. [5] [4], and Gonzalez-Avilés [17] showed that there is a large class of explicit quadratic twists of X 0 (49) whose complex L-series does not vanish at s = 1, and for which the full Birch and Swinnerton-Dyer conjecture is valid (covering the case p = 2 when K = Q( √ −7)). The deep results by Skinner-Urban ([29], Theorem 2) (see also Theorem 7 in section 8.4 below) allow, in specific cases (still assuming L(E, 1) is non-zero), to establish p-part of the Birch and Swinnerton-Dyer conjecture for elliptic curves without complex multiplication for all odd primes p (see examples in section 8.4 below, and section 3 in [10]). The numerical studies and conjectures by Conrey-Keating-Rubinstein-Snaith [7], Delaunay [12][13], Watkins [31], Radziwi l l-Soundararajan [25] (see also the papers [11][10] [9], and references therein) substantially extend the systematic tables given by...

“…In earlier papers (see [11], [7], [8], [9], [10]), we have investigated some numerical examples of E defined over Q for which L(E, 1) is non-zero and the order of Ш(E) is large.…”

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confidence: 99%

Elliptic curves with exceptionally large analytic order of the Tate–Shafarevich groups

2021

Colloq. Math.

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We exhibit 88 examples of rank zero elliptic curves over the rationals with |Ш(E)| > 63408 2 , which was the largest previously known value for any explicit curve. Our record is an elliptic curve E with |Ш(E)| = 1029212 2 = 2 4 • 79 2 • 3257 2. We use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that, in some cases, these orders are the true orders of Ш. For instance, 410536 2 is the true order of Ш(E) for E = E4(21, −233) from the table in Section 2.3.