p-adic properties of division polynomials and elliptic divisibility sequences
Joseph H. Silverman
Abstract
For a given point P in the group of K-rational points E(K) of an elliptic curve, we consider the sequence of values (F1(P),F2(P),F3(P),...) of the division polynomials of E at P. If K is a finite field, we prove that the sequence is periodic. If K/Qp is a local field, we prove (under certain hypotheses) that there is a power q=pe so that for all m > 0, the limit as k goes to infinity of Fmqk(P) 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.
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