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 WardMorgan Ward, Memoir on elliptic divisibility sequences, ''Amer. J. Math.'' 70 (1948), 31–74.
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.

Definition

A (nondegenerate) ''elliptic divisibility sequence'' (EDS) is a sequence of integers
defined recursively by four initial values
, , , ,
with ≠ 0 and with subsequent values determined by the formulas

:$\begin W_W_1^3 &= W_W_n^3 - W_^3W_,\qquad n \ge 2, \\ W_W_2W_1^2 &= W_W_n W_^2 - W_n W_W_^2,\qquad n\ge 3,\\ \end$

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

Divisibility property

An EDS is a divisibility sequence in the sense that
:$m \mid n \Longrightarrow W_m \mid W_n.$
In particular, every term in an EDS is divisible by , so
EDS are frequently ''normalized'' to have = 1 by dividing every term by the initial term.

Any three integers , ,
with divisible by lead to a normalized EDS on setting
:$W_1 = 1,\quad W_2 = b,\quad W_3 = c,\quad W_4 = d.$
It is not obvious, but can be proven, that the condition , suffices to ensure that every term
in the sequence is an integer.

General recursion

A fundamental property of elliptic divisibility sequences
is that they satisfy the general recursion relation
:$W_W_W_r^2 = W_W_W_m^2 - W_W_W_n^2 \quad\text\quad n > m > r.$
(This formula is often applied with = 1 and = 1.)

Nonsingular EDS

The ''discriminant'' of a normalized EDS is the quantity
:$\Delta = W_4W_2^ - W_3^3W_2^ + 3W_4^2W_2^ - 20W_4W_3^3W_2^7 + 3W_4^3W_2^5 + 16W_3^6W_2^4 + 8W_4^2W_3^3W_2^2 + W_4^4.$
An EDS is ''nonsingular'' if its discriminant is nonzero.

Examples

A simple example of an EDS is the sequence of natural numbers 1, 2, 3,... . Another interesting example is 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
:$\begin &1,\, 1,\, -1,\, 1,\, 2,\, -1,\, -3,\, -5,\, 7,\, -4,\, -23,\, 29,\, 59,\, 129,\\ &-314,\, -65,\, 1529,\, -3689,\, -8209,\, -16264,\dots.\\ \end$

Periodicity of EDS

A sequence is said to be ''periodic''
if there is a number so
that = for every ≥ 1.
If a nondegenerate EDS
is periodic, then one of its terms vanishes. The smallest ≥ 1 with = 0 is called the ''rank of apparition'' of the EDS. A deep theorem of Mazur
B. Mazur.
Modular curves and the Eisenstein ideal,
''Inst. Hautes Études Sci. Publ. Math.'' 47:33–186, 1977.

implies that if the rank of apparition of an EDS is finite, then it satisfies ≤ 10 or = 12.

Elliptic curves and points associated to EDS

Ward proves that associated to any nonsingular EDS ()
is an elliptic curve /Q and a point
ε (Q) such that
:$W_n = \psi_n(P)\qquad\text~n \ge 1.$
Here ψ is the
division polynomial
of ; the roots of ψ are the
nonzero points of order on . There is
a complicated formula
This formula is due to Ward. See the appendix to J. H. Silverman and N. Stephens.
The sign of an elliptic divisibility sequence. ''J. Ramanujan Math. Soc.'', 21(1):1–17, 2006.

for and in terms of , , , and .

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 /Q given by a Weierstrass equation and a nontorsion point ε (Q). One writes the -coordinates of the multiples of as
:$x(nP) = \frac \quad \text~\gcd(A_n,D_n)=1~\text~D_n \ge 1.$
Then the sequence () is also called an elliptic divisibility sequence. It is a divisibility sequence, and there exists an integer so that the subsequence ( ± ) _{≥ 1} (with an appropriate choice of signs) is an EDS in the earlier sense.

Growth of EDS

Let be a nonsingular EDS
that is not periodic. Then the sequence grows quadratic exponentially in the sense that there is
a positive constant such that
:$\lim_ \frac = h > 0.$
The number is the canonical height of the point on
the elliptic curve associated to the EDS.

Primes and primitive divisors in EDS

It is conjectured that a nonsingular EDS contains only finitely many
primes
M. Einsiedler, G. Everest, and T. Ward. Primes in elliptic divisibility sequences.
''LMS J. Comput. Math.'', 4:1–13 (electronic), 2001.

However, all but finitely many terms in a nonsingular EDS admit a primitive prime
divisor.
J. H. Silverman. Wieferich's criterion and the ''abc''-conjecture.
''J. Number Theory'', 30(2):226–237, 1988.

Thus for all but finitely many ,
there is a prime such that divides , but does not divide for all < . This statement is an analogue of Zsigmondy's theorem.

EDS over finite fields

An EDS over a finite field F, 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 . The period of an EDS over F then has the form , where and satisfy
:$r \le \left(\sqrt q+1\right)^2 \quad\text\quad t \mid q-1.$
More precisely, there are elements and in F^* such that
:$W_ = W_j\cdot A^ \cdot B^ \quad\text~i \ge 0~\text~j \ge 1.$
The values of and are related to the
Tate pairing of the point on the associated elliptic curve.

Applications of EDS

Bjorn Poonen
B. Poonen. Using elliptic curves of rank one towards the undecidability of
Hilbert's tenth problem over rings of algebraic integers.
In ''Algorithmic number theory (Sydney, 2002)'', volume 2369 of
''Lecture Notes in Comput. Sci.'', pages 33–42. Springer, Berlin, 2002.

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
K. Stange. The Tate pairing via elliptic nets.
In ''Pairing-Based Cryptography (Tokyo, 2007)'', volume 4575 of
''Lecture Notes in Comput. Sci.'' Springer, Berlin, 2007.

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.

References

Further material

* 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''
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.

External links

Lecture on ''p''-adic Properites of Elliptic Divisibility Sequences.

Number theory
Integer sequences