Orders of Tate-Shafarevich groups for the cubic twists of
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  and by Watkins . 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
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 .
(Birch and Swinnerton-Dyer) (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 , who showed that if , then the group is finite. Gross and Zagier  showed that if has a first-order zero at , then has a rational point of infinite order. Rubin  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  proved that, if , then and is finite. Very recently, Bhargava, Skinner and Zhang  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  for all primes which do not divide the order of the group of roots of unity of . Coates et al. , and Gonzalez-Avilés  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 (, 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 ). Our calculations extend those given by Zagier and Kramarz  and by Watkins . 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 , 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 )
, , 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  and . 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 .
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   and a family of Neumann-Setzer type elliptic curves .
Remark. Watkins claims (, 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   for the case of quadratic twists of elliptic curves, and  for the case of a family of Neumann-Setzer type elliptic curves).
Conjecture. For any positive integer there are constants anf such that
Remark. Park, Poonen, Voight and Wood  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
5 Cohen-Lenstra heuristics for the order of X
Delaunay  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
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  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 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 , ). We also have good conjecture for a family of rank zero Neumann-Setzer type elliptic curves .
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
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.
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