p-adic properties of division polynomials and elliptic divisibility sequences

Details p-adic properties of division polynomials and elliptic divisibility sequences p-adic properties of division polynomials and elliptic divisibility sequences

2004-04-22

arxiv.org/abs/math/0404412v1

Math. Ann. 332 (2005), no. 2, 443--471, addendum 473-474. (MR2178070)

Mathematics

Number Theory

30 pages

Scientific paper

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.

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