Computational SectionThe distribution of multiples of real points on an elliptic curve
Introduction
Let be an elliptic curve with , and suppose that P is an element of infinite order in the group . In this paper we investigate the statistics of the coordinates of nP for . The set of points which satisfy the equation for E form either one or two connected subsets of , depending on whether the polynomial has one or three real roots. In the case where has three real roots, the coordinates of points making up one of the connected subsets are bounded, while in the other the coordinates are unbounded. In this case we will say that has two connected components, and we will refer to them as the “bounded component” and “unbounded component”. If instead has only one real root, then we will say that has only one component, we will refer to it as the “unbounded component”.
Let ω be the holomorphic differential on E. We will say that the periods of E are any two complex numbers and with the property that for any closed loop C in , there exist integers m and n such that . As described in [11], there are contours and which enclose exactly two of the three roots of such that and . Moreover, it is always possible for and to be chosen such that , , and (if E has two connected components) or (if E has only one connected component), as described in algorithm 7.4.7 of [2].
In section 3, we prove theorems which explain how large the coordinates of nP get as a function of n:
Theorem 1.1 Suppose that has periods and , chosen such that and . Then for every point P of infinite order in the unbounded component of , there exist infinitely many n such that If P is instead a point of infinite order on the bounded component of (in the case where has two connected components), then there exist infinitely many n such that The implied constants depend only on E. Theorem 1.2 Let ψ be a non-decreasing function from to . If diverges, then for all points P in except for a set of points of Lebesgue measure zero, there exist infinitely many positive integers n such that while if converges, then the set of points P in for which there exist infinitely many such n has measure zero. Theorem 1.3 For any E and any function , there exists a point P in such that, for infinitely many positive integers n,
Variants of these theorems can be given for general , and not just for . For example, Theorem 1.4 Let P be a point in of infinite order. Then where the implied constants depends only on E.
The proofs of these theorems rely on the work of Hurwitz [6], Khinchin [7] [8], and Dirichlet (see [5], theorem 200) in the field of Diophantine approximation. The correspondence between results in Diophantine approximation and asymptotics for the size of the coordinates of nP can be extended further.
In section 4, we investigate the full distribution of the x and y coordinates of nP. Let and be the periods of , chosen such that and . Let Λ be the lattice in with basis . Then is parameterized by elements z of via , where and is the derivative . We prove the following regarding the distribution of integer multiples of a fixed in section 4, which states essentially that these integer multiples of P are “equidistributed” in a sense which is clarified in section 4.
Theorem 1.5 Let P be a point of infinite order in , and let be the preimage of P under the parameterization . Let and be the periods of , chosen such that and . Let Λ be the lattice in with basis . Define as follows: where denotes the interval of real numbers. Then, for any , we have where μ is the Lebesgue measure.
Corollary 1.7 Fix and . For all of infinite order, the natural density of integers n for which is where if both P and are on the unbounded component of , if P is on the bounded component of , and if is on the bounded component of but P is not. The implied constant depends only on E and .
We then obtain the following spacing law:
Corollary 1.9 Let be an elliptic curve, let be an arbitrary fixed point in , and let d be an arbitrary real number. Define Let be the real solutions to . Then, for any point P in of infinite order, the distribution of the values as n varies over the integers is proportional to the function , defined as where indicates that, if P is on the unbounded component of , then the sum omits the for which is not the x-coordinate of any point on the unbounded component of .
Two examples of the distributions arising from Corollary 1.9 are shown in Fig. 1.10, Fig. 1.11.
We also show in Corollary 5.1, Corollary 5.2 that the raw moments of the function diverge, and give an upper bound for the associated partial sums.
As an application of these growth and distribution results, we explain certain numerical observations of Bremner and Macleod made in [1]. There, for every integer , Bremner and Macleod find the positive integer solutions to the equation Solutions to (1) are given by certain rational points on certain elliptic curves . If has rank 1 and P is a generator for , then Bremner and Macleod make numerical observations regarding the set of for which nP yields a solution to equation (1). In particular, they tabulate the smallest positive integers n which yield solutions as N varies, and ask about the proportion of integers n that yield solutions for fixed N. Using Corollary 1.7 we give the proportion exactly.
Section snippets
Background
Let be an elliptic curve with periods and , chosen such that and , and let Λ be the lattice in with basis . As stated in the introduction, the Weierstrass-℘ function defined by gives a parameterization via . This parameterization is discussed at length in [11], [10], and [4], for example.
The function has a pole of order 2 when and has no other poles. From this it follows that the set of which
Growth rates
Using Lemma 2.1, Lemma 2.2, Lemma 2.3, Lemma 2.4 we can now prove Theorem 1.1, Theorem 1.4, Theorem 1.2, Theorem 1.3.
Proof of Theorem 1.1 First suppose that P is a point of infinite order on the unbounded component of , and let be the preimage of P under the parameterization defined by , where Λ is the lattice in with basis . Then is real modulo Λ. From the observations in section 2, we have where the
Distributions
Next we turn our attention to results about the full distribution of and as n varies. Let X be a topological space with a measure μ. We say that a sequence of elements of X is equidistributed with respect to μ if and only if, for every function , we have Sometimes we will say that a sequence is equidistributed in a space if it's clear what the associated measure is. In particular, we will say that a sequence is equidistributed modulo 1 if and
Spacing
We can also study the statistics of the distances between the points nP and for any fixed Q in . The raw moments of the distribution of distances diverge as more and more multiples of a fixed point P are taken, as described in Corollary 5.1, and an upper bound for their growth in the number of multiples taken is given in Corollary 5.2. We can, however, still find a distribution for these differences, as done in Corollary 1.9. Corollary 5.1 For any points P and Q in and any positive integer r,
An equation of Bremner and Macleod
In [1], Bremner and Macleod give positive integer solutions to the equation where N is an integer. Bremner and Macleod show that solutions to this equation are in bijection with rational points on the elliptic curve with X-coordinate satisfying either or Theorem 1.6 implies that for any of infinite order on the bounded connected component of , the point nP
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