$p$-adic properties of division polynomials and algebraic sigma functions

Abstract

Let $p \geq 5$ be a prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E/K$ be an elliptic curve with good reduction. Let $F_n$ denote the $n$-division polynomial of $E$. Silverman proved that if the reduction is ordinary, then for every $P \in E(K) \setminus \hat{E}(K)$ and a suitable power $q$ of $p$, the sequences $(F_{mq^k}(P))_{k \geq 0}$ converge $p$-adically to limits that are algebraic over the field of definition of $E$. In both the ordinary and supersingular cases, we show that these sequences converge, and determine these limits explicitly in terms of the values of Mumford's algebraic sigma function attached to the Teichm\"uller lift of the prime-to-$p$ torsion component of the reduction of $P$. In particular, the limits are algebraic in the supersingular case as well. As an application, we obtain explicit $p$-adic limit formulas for nonsingular elliptic divisibility sequences.