Elsevier

Journal of Number Theory

Volume 211, June 2020, Pages 530-544
Journal of Number Theory

Computational Section
The distribution of multiples of real points on an elliptic curve

https://doi.org/10.1016/j.jnt.2019.12.010Get rights and content

Abstract

Given an elliptic curve E and a point P in E(R), we investigate the distribution of the points nP as n varies over the integers, giving bounds on the x and y coordinates of nP and determining the natural density of integers n for which nP lies in an arbitrary open subset of R2. Our proofs rely on a connection to classical topics in the theory of Diophantine approximation.

Introduction

Let E:y2=4x3g2xg3 be an elliptic curve with g2,g3R, and suppose that P is an element of infinite order in the group E(R). In this paper we investigate the statistics of the coordinates x(nP),y(nP)R of nP for nZ. The set of points (x,y)R2 which satisfy the equation for E form either one or two connected subsets of R2, depending on whether the polynomial 4x3g2xg3 has one or three real roots. In the case where 4x3g2xg3 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 E(R) has two connected components, and we will refer to them as the “bounded component” and “unbounded component”. If instead 4x3g2xg3 has only one real root, then we will say that E(R) has only one component, we will refer to it as the “unbounded component”.

Let ω be the holomorphic differential dxy on E. We will say that the periods of E are any two complex numbers ω1 and ω2 with the property that for any closed loop C in C, there exist integers m and n such that Cω=mω1+nω2. As described in [11], there are contours C1 and C2 which enclose exactly two of the three roots of 4x3g2xg3 such that ω1=C1ω and ω2=C2ω. Moreover, it is always possible for ω1 and ω2 to be chosen such that ω1R>0, Im(ω2)>0, and Re(ω2)=0 (if E has two connected components) or 12ω1 (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 E/C has periods ω1 and ω2, chosen such that ω1R>0 and Im(ω2)>0. Then for every point P of infinite order in the unbounded component of E(R), there exist infinitely many n such thatx(nP)>5ω12n2+O(n2)andy(nP)>2532ω13n3+O(n1). If P is instead a point of infinite order on the bounded component of E(R) (in the case where E(R) has two connected components), then there exist infinitely many n such thatx(nP)>54ω12n2+O(n2)andy(nP)>5324ω13n3+O(n1). The implied constants depend only on E.

 

Theorem 1.2

Let ψ be a non-decreasing function from N to R>0. If n=1ψ(n)1 diverges, then for all points P in E(R) except for a set of points of Lebesgue measure zero, there exist infinitely many positive integers n such thatx(nP)>ψ(n)2andy(nP)>ψ(n)3, while if n=1ψ(n)1 converges, then the set of points P in E(R) for which there exist infinitely many such n has measure zero.

Theorem 1.3

For any E and any function ψ:NR>0, there exists a point P in E(R) such that, for infinitely many positive integers n,x(nP)>ψ(n)2andy(nP)>ψ(n)3.

Variants of these theorems can be given for general PE(C), and not just for PE(R). For example,

Theorem 1.4

Let P be a point in E(C) of infinite order. Then|x(nP)|nand|y(nP)|n32, 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 ω1 and ω2 be the periods of E/C, chosen such that ω1R>0 and Im(ω2)>0. Let Λ be the lattice in C with basis ω1,ω2. Then E/C is parameterized by elements z of C/Λ via z((z),(z)), where(z):=1z2+λΛλ0(1(zλ)21λ2) and is the derivative ddz. We prove the following regarding the distribution of integer multiples of a fixed PE(R) 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 E(R), and let zP be the preimage of P under the parameterization z((z),(z)). Let ω1 and ω2 be the periods of E/C, chosen such that ω1R>0 and Im(ω2)>0. Let Λ be the lattice in C with basis ω1,ω2. Define IPC/Λ as follows:IP:={[0,ω1],Im(zP)=0modΛ,[0,ω1]([0,ω1]+ω22),Im(zP)=Im(ω22)modΛ, where [0,ω1] denotes the interval of real numbers. Then, for any UR2, we havelimn12n#{|k|<n:(x(kP),y(kP))U}=μ({zIP:((z),(z))U})μ(IP), where μ is the Lebesgue measure.

Corollary 1.7

Fix P0=(x0,y0)E(R) and ε>0. For all PE(R) of infinite order, the natural density of integers n for which (x(nP)x0)2+(y(nP)y0)2<ε2 is2η(ε+O(ε2))ω1y02+(6x02g22)2, where η=1 if both P and P0 are on the unbounded component of E(R), η=12 if P is on the bounded component of E(R), and η=0 if P0 is on the bounded component of E(R) but P is not. The implied constant depends only on E and P0.

We then obtain the following spacing law:

Corollary 1.9

Let E:y2=x3+ax+b be an elliptic curve, let Q=(xQ,yQ) be an arbitrary fixed point in E(R), and let d be an arbitrary real number. DefineF±,Q(x):=(±x3+ax+byQxxQ)22xxQandρ(x):=1x3+ax+b. Let x1±,,xk±± be the real solutions to F±,Q(x)=d. Then, for any point P in E(R) of infinite order, the distribution of the values x(nP+Q)x(nP) as n varies over the integers is proportional to the function f(d), defined asf(d):=i=1k+ρ(xi+)F+,Q(xi+)+i=1kρ(xi)F,Q(xi), where indicates that, if P is on the unbounded component of E(R), then the sum omits the xi± for which xi± is not the x-coordinate of any point on the unbounded component of E(R).

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 f(d) 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 N1000, Bremner and Macleod find the positive integer solutions a,b,c to the equationab+c+ba+c+ca+b=N. Solutions to (1) are given by certain rational points on certain elliptic curves EN. If EN has rank 1 and P is a generator for EN, then Bremner and Macleod make numerical observations regarding the set of nZ 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 E/C be an elliptic curve with periods ω1 and ω2, chosen such that ω1R>0 and Im(ω2)>0, and let Λ be the lattice in C with basis ω1,ω2. As stated in the introduction, the Weierstrass-℘ function defined by(z):=1z2+λΛλ0(1(zλ)21λ2) gives a parameterization C/ΛE(C) via z((z),(z)). This parameterization is discussed at length in [11], [10], and [4], for example.

The function (z) has a pole of order 2 when zΛ and has no other poles. From this it follows that the set of zC/Λ 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 E(R), and let zP be the preimage of P under the parameterization C/ΛE(C) defined by z((z),(z)), where Λ is the lattice in C with basis ω1,ω2. Then zP is real modulo Λ. From the observations in section 2, we have(nzP)=ω12{nzPω1}2+O({nzPω1}2)and(nzP)=2ω13{nzPω1}3+O({nzPω1}), where the

Distributions

Next we turn our attention to results about the full distribution of x(nP) and y(nP) as n varies. Let X be a topological space with a measure μ. We say that a sequence (an) of elements of X is equidistributed with respect to μ if and only if, for every function f:XC, we havelimn1nk=1nf(ak)=Xf(x)dμ. 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 nP+Q for any fixed Q in E(R). 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 E(R) and any positive integer r,

An equation of Bremner and Macleod

In [1], Bremner and Macleod give positive integer solutions a,b,c to the equationab+c+ba+c+ca+b=N, where N is an integer. Bremner and Macleod show that solutions to this equation are in bijection with rational points on the elliptic curveEN:Y2=X3+(4N2+12N3)X2+32(N+3)X with X-coordinate satisfying either312N4N2(2N+5)4N2+4N152<X<2(N+3)(N+N24) or2(N+3)(NN24)<X<4N+3N+2. Theorem 1.6 implies that for any PEN(Q) of infinite order on the bounded connected component of EN(R), the point nP

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