Elliptic divisibility sequence


In mathematics, an elliptic divisibility sequence 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
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 elliptic divisibility sequence is a sequence of integers
defined recursively by four initial values
,,,,
with ≠ 0 and with subsequent values determined by the formulas
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
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
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

Nonsingular EDS

The discriminant of a normalized EDS is the quantity
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

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
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 such that
Here ψ is the
division polynomial
of ; the roots of ψ are the
nonzero points of order on. There is
a complicated formula
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. One writes the -coordinates of the multiples of as
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 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
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
However, all but finitely many terms in a nonsingular EDS admit a primitive prime
divisor.
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
More precisely, there are elements and in F* such that
The values of and are related to the
Tate pairing of the point on the associated elliptic curve.

Applications of EDS

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 Stange
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.

Further material