Orders of Tate-Shafarevich groups for the cubic twists of X_{0}(27)

# Orders of Tate-Shafarevich groups for the cubic twists of X0(27)

Andrzej Dąbrowski and Lucjan Szymaszkiewicz

Abstract. 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 . 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.

Key words: elliptic curves, cubic twists, Tate-Shafarevich group, Cohen-Lenstra heuristics, distribution of central -values, class numbers of real quadratic fields

2010 Mathematics Subject Classification: 11G05, 11G40, 11Y50

## 1 Introduction

Let be an elliptic curve defined over of conductor , and let denote its -series. Let be the Tate-Shafarevich group of , the group of rational points, and the regulator, with respect to the Néron-Tate height pairing. Finally, let be the least positive real period of the Néron differential of a global minimal Weierstrass equation for , and define or according as is connected or not, and let denote the product of the Tamagawa factors of at the bad primes. The Euler product defining converges for . The modularity conjecture, proven by Wiles-Taylor-Diamond-Breuil-Conrad, implies that has an analytic continuation to an entire function. The Birch and Swinnerton-Dyer conjecture relates the arithmetic data of to the behaviour of at .

###### Conjecture 1

(Birch and Swinnerton-Dyer) (i) -function has a zero of order at ,

(ii) is finite, and

 lims→1L(E,s)(s−1)r=C∞(E)Cfin(E)R(E)|\cyrfont X(E)||E(Q)tors|2.

If 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 with complex multiplication by Coates and Wiles in 1976 [3], who showed that if , then the group is finite. Gross and Zagier [11] showed that if has a first-order zero at , then has a rational point of infinite order. Rubin [16] proves that if has complex multiplication and , then is finite. Let be the rank of and let the order of the zero of at . Then Kolyvagin [13] proved that, if , then and is finite. Very recently, Bhargava, Skinner and Zhang [1] proved that at least of all elliptic curves over , when ordered by height, satisfy the weak form of the Birch and Swinnerton-Dyer conjecture, and have finite Tate-Shafarevich group.

When has complex multiplication by the ring of integers of an imaginary quadratic field and is non-zero, the -part of the Birch and Swinnerton-Dyer conjecture has been established by Rubin [17] for all primes which do not divide the order of the group of roots of unity of . Coates et al. [2], and Gonzalez-Avilés [10] showed that there is a large class of explicit quadratic twists of whose complex -series does not vanish at , and for which the full Birch and Swinnerton-Dyer conjecture is valid (covering the case when ). The deep results by Skinner-Urban ([18], Theorem 2) allow, in specific cases (still assuming is non-zero), to establish -part of the Birch and Swinnerton-Dyer conjecture for elliptic curves without complex multiplication for all odd primes .

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 . These analytic orders are the true ones if are coprime to (by [17]). Our calculations extend those given by Zagier and Kramarz [20] and by Watkins [19]. Our main observations concern the asymptotic formulae in sections 3 (frequency of orders of X) and 4 (asymptotics for the sums in the rank zero case), and the distributions of and () in sections 6 and 7. In section 8 we propose a variant of the asymptotic formula from section 3 for the class numbers of real quadratic fields.

This research was supported in part by PL-Grid Infrastructure. Our computations were carried out in 2016 and 2017 on the Prometheus supercomputer via PL-Grid infrastructure.

## 2 Formula for the order of \cyrfont X(Em), when L(Em,1)≠0

Let be any cubefree positive integer. Let denote the cubic twist of . Then it is plain to see that has the Weierstrass equation , and . Let () denote its -series. If , then the analytic order of may be expressed as follows (see [20])

 |\cyrfont X(Em)|=L(Em,1)⋅TmCfin(Em)⋅C∞(Em),

where

1. , , and for any ;

2. if , and if ;

3. , where if , if , if , if , and if or .

If , then the central -value is given by sum of the approximating series

 L(Em,1)=2∞∑n=1am(n)ne−2πn/√NEm,

where is the conductor of . The coefficients can be computed as in [20] and [19]. In order to compute with appropriate accuracy, we need to calculate terms of the approximating series (and, hence such a number of coefficients ) for some constant .

Definition. We say, that a positive cube-free integer satisfies condition , if .

## 3 Frequency of orders of X

Our data contains values of for all positive cubic-free integers satisfying . Our calculations strongly suggest that for any positive integer there are infinitely many positive cube-free integers satisfying , such that has rank zero and . Below we will state a more precise conjecture.

Let denote the number of cube-free integers , satisfying () and such that . Let denote the number of cube-free integers , satisfying () and such that . We obtain the following graph of the function .

We expect that tends to a constant (). Using ([19], Question 1.4.1, and [4]), we believe the following asymptotic formula holds

 g(X)∼c⋅X5/6(logX)d,X→∞,

with some positive and real . We therefore expect a similar asymptotic formula for . Compare a similar phenomena for the cases of quadratic twists of elliptic curves [5] [7] and a family of Neumann-Setzer type elliptic curves [6].

Remark. Watkins claims ([19], Question 1.4.1 and comments after it), that if we restrict to the cubic twists by primes congruent to modulo , then we can take and . Our calculations suggest (see the figures below) that the constant is . Let denote the number of primes , satisfying () and such that .

Let us also include the graph of the function .

Now let denote the number of cube-free integers , satisfying () and such that . Let . We obtain the following graphs of the functions for .

The above calculations suggest the following general conjecture (compare [5] [7] for the case of quadratic twists of elliptic curves, and [6] for the case of a family of Neumann-Setzer type elliptic curves).

Conjecture. For any positive integer there are constants anf such that

 f(k,X)∼ckX5/6(logX)dk,X→∞.

Remark. Park, Poonen, Voight and Wood [15] have formulated an analogous (but less precise) conjecture for the family of all elliptic curves over the rationals, ordered by height.

## 4 Variant of Delaunay’s asymptotic formula

Let , where the sum is over primes , satisfying (*) and , and denotes the number of terms in the sum. Similarly, let , where the sum is over positive cube-free integers , satisfying (*) and , and denotes the number of terms in the sum. Let , and . We obtain the following picture

Note similarity with the predictions by Delaunay [8] for the case of quadratic twists of a given elliptic curve (and numerical evidences in [5] [7]), and with a variant of this phenomenon in the case of the family of Neumann-Setzer type elliptic curves [6].

## 5 Cohen-Lenstra heuristics for the order of X

Delaunay [9] have considered Cohen-Lenstra heuristics for the order of Tate-Shafarevich group. He predicts, among others, that in the rank zero case, the probability that of a given elliptic curve over is divisible by a prime should be Hence, , , , , and so on.

Let denote the number of cube-free satisfying and , and let denote the number of such ’s satisfying . Let , We obtain the following table (in the last row we restrict to prime twists)

The numerical values of exceed the expected value , but for the values seem to tend to ; additionally restricting to prime twists tends to speed convergence to the expected values.

## 6 Distributions of L(Em,1)

It is a classical result (due to Selberg) that the values of follow a normal distribution.

Let be any elliptic curve defined over . Let denote the set of all fundamental discriminants with and , where is the root number of and . Keating and Snaith [12] have conjectured that, for , the quantity has a normal distribution with mean and variance .

Below we consider the case of cubic twists of . Our data suggest that the values also follow an approximate normal distribution. Let and for . We create histograms with bins from the data . Below we picture this histogram.

## 7 Distributions of |\cyrfont X(Em)|

It is an interesting question to find results (or at least a conjecture) on distribution of the order of the Tate-Shafarevich group in family of elliptic curves. It turns out that in a case of rank zero quadratic twists of a fixed elliptic curve the values of are the natural ones to consider (compare the numerical experiments in [5], [7]). We also have good conjecture for a family of rank zero Neumann-Setzer type elliptic curves [6].

Let us consider the family of cubic twists of . In this case we will create histograms for the values , separately. Let and for . We create histograms with bins from the data . Below we picture these histograms.

## 8 Observations concerning the class numbers of real quadratic fields

Consider a real quadratic field ( a positive square-free integer); let denote its class number. We calculated the values for all positive square-free integers . Our observations suggest that ’s behave in a similar way to the orders of Tate-Shafarevich groups in some families of rank zero elliptic curves (i.e. quadratic or cubic twists of a given one).

Let denote the number of positive square-free integers such that . Let . We obtain the following graphs of the functions for .

Now let us consider graphs of the functions , .

The above calculations suggest the following (optimistic) conjecture.

Conjecture. For any positive integer there are positive constants , such that

 h(k,X)∼rkX5/6(logX)sk,X→∞.

Remark. The Gauss’ class-number one problem for real quadratic fields states that there are infinitely real quadratic fields with trivial ideal class group. It is still an open problem; note that it is not even known if there are infinitely many number fields with a given class number. Therefore the above conjecture is a highly optimistic version of these open questions.

## References

• [1] M. Bhargava, Ch. Skinner, W. Zhang, A majority of elliptic curves over satisfy the Birch and Swinnerton-Dyer conjecture, arxiv.org/abs/1407.1826
• [2] J. Coates, Y. Li, Y. Tian, S. Zhai, Quadratic twists of elliptic curves, Proc. London Math. Soc. 110 (2015), 357-394
• [3] J. Coates, A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), 223-251
• [4] J. B. Conrey, J. P. Keating, M. O. Rubinstein, N. C. Snaith, On the frequency of vanishing of quadratic twists of modular -functions. In: Number Theory for the millennium, I, (Urbana, IL, 2000) (ed. by M. A. Bennett et al.), 301-315
• [5] A. Dąbrowski, T. Jędrzejak, L. Szymaszkiewicz, Behaviour of the order of Tate-Shafarevich groups for the quadratic twists of , In: Elliptic Curves, Modular Forms and Iwasawa Theory (in honour of John Coates’ 70th birthday), Springer Proceedings in Mathematics and Statistics 188 (2016), 125-158
• [6] A. Dąbrowski, L. Szymaszkiewicz, Orders of Tate-Shafarevich groups for the Neumann-Setzer type elliptic curves, Math. Comput. 87 (2018)
• [7] A. Dąbrowski, L. Szymaszkiewicz, Behaviour of the order of Tate-Shafarevich groups for the quadratic twists of elliptic curves, arXiv:1611.07840
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• [10] C.D. Gonzalez-Avilés, On the conjecture of Birch and Swinnerton-Dyer, Trans. Amer. Math. Soc. 349 (1997), 4181-4200
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• [14] The PARI Group, PARI/GP version 2.7.2, Bordeaux, 2014, http://pari.math.u-bordeaux.fr/
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• [20] D. Zagier, G. Kramarz, Numerical investigations related to the -series of certain elliptic curves, J. Indian Math. Soc. (N.S.) 52 (1987), 51-69

Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland; E-mail addresses: andrzej.dabrowski@usz.edu.pl and dabrowskiandrzej7@gmail.com; lucjansz@gmail.com

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