Elliptic divisibility sequence

In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan Ward^[1] in the 1940s. They attracted only sporadic attention until around 2000, when EDS were taken up as a class of nonlinear recurrences that are more amenable to analysis than most such sequences. This tractability is due primarily to the close connection between EDS and elliptic curves. In addition to the intrinsic interest that EDS have within number theory, EDS have applications to other areas of mathematics including logic and cryptography.

A (nondegenerate) elliptic divisibility sequence (EDS) is a sequence of integers (W_n)_{n ≥ 1} defined recursively by four initial values W₁, W₂, W₃, W₄, with W₁W₂W₃ ≠ 0 and with subsequent values determined by the formulas

${\displaystyle {\begin{aligned}W_{2n+1}W_{1}^{3}&=W_{n+2}W_{n}^{3}-W_{n+1}^{3}W_{n-1},\qquad n\geq 2,\\W_{2n}W_{2}W_{1}^{2}&=W_{n+2}W_{n}W_{n-1}^{2}-W_{n}W_{n-2}W_{n+1}^{2},\qquad n\geq 3,\\\end{aligned}}}$

It can be shown that if W₁ divides each of W₂, W₃, W₄ and if further W₂ divides W₄, then every term W_n in the sequence is an integer.

An EDS is a divisibility sequence in the sense that

${\displaystyle m\mid n\Longrightarrow W_{m}\mid W_{n}.}$

In particular, every term in an EDS is divisible by W₁, so EDS are frequently normalized to have W₁ = 1 by dividing every term by the initial term.

Any three integers b, c, d with d divisible by b lead to a normalized EDS on setting

${\displaystyle W_{1}=1,\quad W_{2}=b,\quad W_{3}=c,\quad W_{4}=d.}$

It is not obvious, but can be proven, that the condition b | d suffices to ensure that every term in the sequence is an integer.

A fundamental property of elliptic divisibility sequences is that they satisfy the general recursion relation

${\displaystyle W_{n+m}W_{n-m}W_{r}^{2}=W_{n+r}W_{n-r}W_{m}^{2}-W_{m+r}W_{m-r}W_{n}^{2}\quad {\text{for all}}\quad n>m>r.}$

(This formula is often applied with r = 1 and W₁ = 1.)

The discriminant of a normalized EDS is the quantity

${\displaystyle \Delta =W_{4}W_{2}^{15}-W_{3}^{3}W_{2}^{12}+3W_{4}^{2}W_{2}^{10}-20W_{4}W_{3}^{3}W_{2}^{7}+3W_{4}^{3}W_{2}^{5}+16W_{3}^{6}W_{2}^{4}+8W_{4}^{2}W_{3}^{3}W_{2}^{2}+W_{4}^{4}.}$

An EDS is nonsingular if its discriminant is nonzero.

A simple example of an EDS is the sequence of natural numbers 1, 2, 3,... . Another interesting example is (sequence A006709 in the OEIS) 1, 3, 8, 21, 55, 144, 377, 987,... consisting of every other term in the Fibonacci sequence, starting with the second term. However, both of these sequences satisfy a linear recurrence and both are singular EDS. An example of a nonsingular EDS is (sequence A006769 in the OEIS)

${\displaystyle {\begin{aligned}&1,\,1,\,-1,\,1,\,2,\,-1,\,-3,\,-5,\,7,\,-4,\,-23,\,29,\,59,\,129,\\&-314,\,-65,\,1529,\,-3689,\,-8209,\,-16264,\dots .\\\end{aligned}}}$

A sequence (A_n)_{n ≥ 1} is said to be periodic if there is a number N ≥ 1 so that A_n+N = A_n for every n ≥ 1. If a nondegenerate EDS (W_n)_{n ≥ 1} is periodic, then one of its terms vanishes. The smallest r ≥ 1 with W_r = 0 is called the rank of apparition of the EDS. A deep theorem of Mazur^[2] implies that if the rank of apparition of an EDS is finite, then it satisfies r ≤ 10 or r = 12.

Ward proves that associated to any nonsingular EDS (W_n) is an elliptic curve E/Q and a point P ε E(Q) such that

${\displaystyle W_{n}=\psi _{n}(P)\qquad {\text{for all}}~n\geq 1.}$

Here ψ_n is the n division polynomial of E; the roots of ψ_n are the nonzero points of order n on E. There is a complicated formula^[3] for E and P in terms of W₁, W₂, W₃, and W₄.

There is an alternative definition of EDS that directly uses elliptic curves and yields a sequence which, up to sign, almost satisfies the EDS recursion. This definition starts with an elliptic curve E/Q given by a Weierstrass equation and a nontorsion point P ε E(Q). One writes the x-coordinates of the multiples of P as

${\displaystyle x(nP)={\frac {A_{n}}{D_{n}^{2}}}\quad {\text{with}}~\gcd(A_{n},D_{n})=1~{\text{and}}~D_{n}\geq 1.}$

Then the sequence (D_n) is also called an elliptic divisibility sequence. It is a divisibility sequence, and there exists an integer k so that the subsequence ( ±D_nk )_{n ≥ 1} (with an appropriate choice of signs) is an EDS in the earlier sense.

Let (W_n)_{n ≥ 1} be a nonsingular EDS that is not periodic. Then the sequence grows quadratic exponentially in the sense that there is a positive constant h such that

${\displaystyle \lim _{n\to \infty }{\frac {\log |W_{n}|}{n^{2}}}=h>0.}$

The number h is the canonical height of the point on the elliptic curve associated to the EDS.

It is conjectured that a nonsingular EDS contains only finitely many primes^[4] However, all but finitely many terms in a nonsingular EDS admit a primitive prime divisor.^[5] Thus for all but finitely many n, there is a prime p such that p divides W_n, but p does not divide W_m for all m < n. This statement is an analogue of Zsigmondy's theorem.

An EDS over a finite field F_q, or more generally over any field, is a sequence of elements of that field satisfying the EDS recursion. An EDS over a finite field is always periodic, and thus has a rank of apparition r. The period of an EDS over F_q then has the form rt, where r and t satisfy

${\displaystyle r\leq \left({\sqrt {q}}+1\right)^{2}\quad {\text{and}}\quad t\mid q-1.}$

More precisely, there are elements A and B in F_q^* such that

${\displaystyle W_{ri+j}=W_{j}\cdot A^{ij}\cdot B^{j^{2}}\quad {\text{for all}}~i\geq 0~{\text{and all}}~j\geq 1.}$

The values of A and B are related to the Tate pairing of the point on the associated elliptic curve.

Bjorn Poonen^[6] has applied EDS to logic. He uses the existence of primitive divisors in EDS on elliptic curves of rank one to prove the undecidability of Hilbert's tenth problem over certain rings of integers.

Katherine E. Stange^[7] has applied EDS and their higher rank generalizations called elliptic nets to cryptography. She shows how EDS can be used to compute the value of the Weil and Tate pairings on elliptic curves over finite fields. These pairings have numerous applications in pairing-based cryptography.

• G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward. Recurrence sequences, volume 104 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003. ISBN 0-8218-3387-1. (Chapter 10 is on EDS.)
• R. Shipsey. Elliptic divisibility sequences Archived 2011-06-09 at the Wayback Machine. PhD thesis, Goldsmiths College (University of London), 2000.
• K. Stange. Elliptic nets. PhD thesis, Brown University, 2008.
• C. Swart. Sequences related to elliptic curves. PhD thesis, Royal Holloway (University of London), 2003.

• Graham Everest's EDS web page. Archived 2009-04-11 at the Wayback Machine
• Prime Values of Elliptic Divisibility Sequences.
• Lecture on p-adic Properites of Elliptic Divisibility Sequences.