p-adic properties of division polynomials and elliptic divisibility sequences

Abstract

Abstract. 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 K/F p , we prove that the sequence is periodic. For a local field K/Q p , we prove (under certain hypotheses) that there is a power q = p e so that for all m ≥ 1, the limit of F mq k (P ) as k → ∞ exists in K and is algebraic over Q(E). We apply this result to prove an analogous p-adic limit and algebraicity result for elliptic divisibility sequences.

Primitive Divisors on Twists of Fermat's Cubic

“…One can verify that this binary form divides φ left with the possibilities (W 1 , W 2 ) = (9,27) or (9,9). These yield (s, t) = (7, −1) or (3, −3), the latter of which may be discarded as gcd(s, t) = 1 by construction.…”

Analytic solutions and integrability for bilinear recurrences of order six

“…There has been a considerable amount of further interest in them recently [19][20][21][22][23][24][25], especially because they have been used to resolve Hilbert's tenth problem for larger subrings of Q than the integers [26]. As observed by Robinson [27], more general Somos sequences, such as the Somos-4 sequence (1.9),…”

Sigma function solution of the initial value problem for Somos 5 sequences

“…Whether this result can lead to an effective solution to the initial value problem for these higher order recurrences is the subject of further investigation. It would also be interesting to see how some of our results on Somos 4 and 5 sequences could be modified in the p-adic setting [27].…”