P-adic properties of division polynomials and elliptic divisibility sequences

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 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.