p-adic properties of division polynomials and elliptic divisibility sequences
Number Theory
2007-07-09 v1 Algebraic Geometry
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 (F_1(P),F_2(P),F_3(P),...) of the division polynomials of E at P. If K is a finite field, we prove that the sequence is periodic. If K/Q_p is a local field, we prove (under certain hypotheses) that there is a power q=p^e so that for all m > 0, the limit as k goes to infinity of F_{mq^{k}}(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.
Cite
@article{arxiv.math/0404412,
title = {p-adic properties of division polynomials and elliptic divisibility sequences},
author = {Joseph H. Silverman},
journal= {arXiv preprint arXiv:math/0404412},
year = {2007}
}
Comments
30 pages