The greatest common valuation of ${\varphi }_n$ and ${\psi }_n^2$ at points on elliptic curves

Abstract

Given a minimal model of an elliptic curve, $E/K$, over a finite extension, $K$, of ${\mathbb{Q}}_p$ for any rational prime, $p$, and any point $P\in E(K)$ of infinite order, we determine precisely $\min \left(v\left({\varphi }_n(P)\right),v\left({\psi }_n^2(P)\right)\right)$, where $v$ is a normalised valuation on $K$ and ${\varphi }_n(P)$ and ${\psi }_n(P)$ are polynomials arising from multiplication by $n$ for this model of the curve.

accepted version (J. Number Theory), main result linked to local heights. Comments very welcome!

Cited by in corpus (2)

• Common valuations of division polynomials
• Explicit valuation of elliptic nets for elliptic curves with complex multiplication