Orders of TateShafarevich groups for the cubic twists of
Abstract. This paper continues the authors previous investigations concerning orders of TateShafarevich groups in quadratic twists of a given elliptic curve, and for the family of the NeumannSetzer type elliptic curves. Here we present the results of our search for the (analytic) orders of TateShafarevich 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 TateShafarevich 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, TateShafarevich group, CohenLenstra 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 TateShafarevich group of , the group of rational points, and the regulator, with respect to the NéronTate 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 WilesTaylorDiamondBreuilConrad, implies that has an analytic continuation to an entire function. The Birch and SwinnertonDyer conjecture relates the arithmetic data of to the behaviour of at .
Conjecture 1
(Birch and SwinnertonDyer) (i) function has a zero of order at ,
(ii) is finite, and
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 firstorder 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 SwinnertonDyer conjecture, and have finite TateShafarevich group.
When has complex multiplication by the ring of integers of an imaginary quadratic field and is nonzero, the part of the Birch and SwinnertonDyer 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 GonzalezAvilés [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 SwinnertonDyer conjecture is valid (covering the case when ). The deep results by SkinnerUrban ([18], Theorem 2) allow, in specific cases (still assuming is nonzero), to establish part of the Birch and SwinnertonDyer conjecture for elliptic curves without complex multiplication for all odd primes .
This paper continues the authors previous investigations concerning orders of TateShafarevich groups in quadratic twists of a given elliptic curve, and for the family of the NeumannSetzer type elliptic curves. Here we present the results of our search for the (analytic) orders of TateShafarevich 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 PLGrid Infrastructure. Our computations were carried out in 2016 and 2017 on the Prometheus supercomputer via PLGrid infrastructure.
2 Formula for the order of , when
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])
where

, , and for any ;

if , and if ;

, where if , if , if , if , and if or .
If , then the central value is given by sum of the approximating series
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 cubefree integer satisfies condition , if .
3 Frequency of orders of X
Our data contains values of for all positive cubicfree integers satisfying . Our calculations strongly suggest that for any positive integer there are infinitely many positive cubefree integers satisfying , such that has rank zero and . Below we will state a more precise conjecture.
Let denote the number of cubefree integers , satisfying () and such that . Let denote the number of cubefree 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
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 NeumannSetzer 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 cubefree 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 NeumannSetzer type elliptic curves).
Conjecture. For any positive integer there are constants anf such that
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 cubefree integers , satisfying (*) and , and denotes the number of terms in the sum. Let , and . We obtain the following picture
5 CohenLenstra heuristics for the order of X
Delaunay [9] have considered CohenLenstra heuristics for the order of TateShafarevich 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 cubefree 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)
10000000 
0.4574860107  0.4528351278  0.0797229512  0.0365187357  0.0107055908 
20000000  0.4667861427  0.4665902606  0.0856954224  0.0406883829  0.0126964802 
30000000  0.4720389372  0.4743395107  0.0891666909  0.0430854869  0.0138608186 
40000000  0.4755325884  0.4797263355  0.0916462006  0.0448302849  0.0147494390 
50000000  0.4782835292  0.4838047688  0.0935546233  0.0461842060  0.0154253689 
60000000  0.4804365024  0.4870412651  0.0950607348  0.0472714454  0.0160042804 
70000000  0.4821166758  0.4897452073  0.0963909035  0.0482264317  0.0164998297 
80000000  0.4836581573  0.4920588749  0.0974999561  0.0490436597  0.0169344117 
90000000  0.4849849695  0.4940653891  0.0984979769  0.0497487127  0.0173190511 
100000000  0.4861728066  0.4958441463  0.0993871375  0.0503845401  0.0176658729 
100000000  0.5474977246  0.0713684943  0.1628461726  0.0993604813  0.0467913704 



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
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
It is an interesting question to find results (or at least a conjecture) on distribution of the order of the TateShafarevich 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 NeumannSetzer 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 squarefree integer); let denote its class number. We calculated the values for all positive squarefree integers . Our observations suggest that ’s behave in a similar way to the orders of TateShafarevich 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 squarefree 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
Remark. The Gauss’ classnumber 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.
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Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70451 Szczecin, Poland; Email addresses: andrzej.dabrowski@usz.edu.pl and dabrowskiandrzej7@gmail.com; lucjansz@gmail.com