Elliptic Divisibility Sequences Research Articles

RECURRENCE RELATIONS FOR ELLIPTIC SEQUENCES: EVERY SOMOS 4 IS A SOMOS LOWERCASE $k$

In his celebrated memoir, Morgan Ward's definition of elliptic divisibility sequences has the remarkable feature that it does not become at all clear until deep into the paper that there exist nontrivial examples of such sequences. Even then, Ward's proof of the coherence of his definition relies on displaying a sequence of values of quotients of Weierstraß σ-functions. We give a direct proof of coherence and show, rather more generally, that a sequence defined by a so-called Somos relation of width 4 is always also given by three-term Somos relations of all larger widths 5, 6, 7, …. 2000 Mathematics Subject Classification 11B83, 11G05 (primary), 11A55, 14H05, 14H52 (secondary).

Primitive divisors of elliptic divisibility sequences

Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. As an application, we show that every term beyond the fourth of the Somos-4 sequence has a primitive divisor.

Addendum to “p-adic properties of division polynomials and elliptic divisibility sequences”

After the (online) publication of [4], the author became aware of a significant overlap of results, although not of methodology, with a beautiful paper of Mohamed Ayad [2]. (See also [1].) In this addendum we briefly compare the results of [2] and [4], using the notation from the latter. Thus Ayad’s elliptic divisibility sequence
m = dm−1 m(M) is denoted instead by Fm(P ). We fix an odd prime p such that P ∈ E(Q) is nonsingular modulo p, and we write r for the rank of apparition of the sequence modulo p, or equivalently, for the order of P modulo p. Assuming that r ≥ 3, Ayad proves [2, Theoreme D] that the elliptic divisibility sequence is periodic modulo p for all μ ≥ 1. This may be compared with the author’s similar result [4, Theorem 8] over number fields and for r ≥ 2, but under the assumption that the elliptic curve E has good ordinary reduction. Ayad also gives a fairly explicit formula for the period πμ, which allows him to prove that πμ = min{1, pμ−ν}π1 for the constant ν = ordp(Fr(P )). This relation between the periods modulo various powers of p is Ayad’s main objective in [2] (“notre objectif principal dans ce travail”), and as an application he obtains nontrivial results for S-integer points on rank one subgroups of E(Q). The author’s purpose in [4] was rather different. The main goal was to prove the existence and algebraicity of the p-adic limit of certain subsequences of Fm(P ). More precisely, Theorem 7 of [4] (over Q) says that if E has good ordinary reduction, then there is a power q = p so that for every m ≥ 1, the limit

Prime powers in elliptic divisibility sequences

Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In some cases the methods prove that only finitely many terms are divisible by a bounded number of distinct primes.

P-adic properties of division polynomials and elliptic divisibility sequences

For a fixed rational point P ∈ E (K) on an elliptic curve, we consider the sequence of values (F n (P)) n ≥1 of the division polynomials of E at P. For a finite field we prove that the sequence is periodic. For a local field we prove (under certain hypotheses) that there is a power q=p e so that for all m≥1, the limit of exists in K and is algebraic over We apply this result to prove an analogous p-adic limit and algebraicity result for elliptic divisibility sequences.

Common divisors of elliptic divisibility sequences over function fields

Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R ∈ E(k(T)), write xR=AR/DR2 with relatively prime polynomials AR(T),DR(T) ∈ k[T]. The sequence {DnR}n≥ 1 is called the elliptic divisibility sequence of R. Let P,Q ∈ E(k(T)) be independent points. We conjecture that deg (gcd(DnP, DmQ)) is bounded for m, n ≥ 1, and that gcd(DnP, DnQ) = gcdDP, DQ) for infinitely many n ≥ 1. We prove these conjectures in the case that j(E) ∈ k. More generally, we prove analogous statements with k(T) replaced by the function field of any curve and with P and Q allowed to lie on different elliptic curves. If instead k is a finite field of characteristic p and again assuming that j(E) ∈ k, we show that deg (gcd(DnP, DnQ)) is as large as Open image in new window for infinitely many n≢0 (mod p).

Primes in Elliptic Divisibility Sequences

AbstractMorgan Ward pursued the study of elliptic divisibility sequences, originally initiated by Lucas, and Chudnovsky and Chudnovsky subsequently suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here, from both a theoretical and a practical viewpoint. We show calculations, together with a heuristic argument, to suggest that these sequences contain only finitely many primes.

The Apparition Problem for Equianharmonic Divisibility Sequences

The apparition problem consists in finding an arithmetical relation between a rational prime p and its rank of apparition in an elliptic divisibility sequence.' In this report the elliptic divisibility sequences parameterized by equianharmonic functions are investigated, and a partial answer to the apparition problem for such sequences is given (Theorem 2). If (x2/Wl = p = 1/2( -1 -+ /-3), the Weierstrass functions for the parallelogram 0, 2w1 2co2, 2co + 2w2 are called equianharmonic functions and have the following expansions: ^?J/ u U 1 U2 a(u) = u II 1 1exp +

The law of repetition of primes in an elliptic divisibility sequence

The law of repetition of primes in an elliptic divisibility sequence

Memoir on Elliptic Divisibility Sequences

Memoir on Elliptic Divisibility Sequences