Common valuations of division polynomials

Abstract

In this note we prove a formula for the cancellation exponent $k_{v,n}$ between division polynomials ${\psi }_n$ and ${\varphi }_n$ associated with a sequence $\{nP{\}}_{n\in \mathbb{N}}$ of points on an elliptic curve $E$ defined over a discrete valuation field $K$. The formula is identical with the result of Yabuta-Voutier for the case of finite extension of ${\mathbb{Q}}_p$ and generalizes to the case of non-standard Kodaira types for non-perfect residue fields.

Final version of the paper

References in corpus (3)

• Distribution of Mordell--Weil ranks of families of elliptic curves
• Divisibility sequences of polynomials and heights estimates
• The greatest common valuation of ${\varphi }_n$ and ${\psi }_n^2$ at points on elliptic curves

Cited by in corpus (2)

• Remarks on a theorem of Silverman
• Explicit valuation of elliptic nets for elliptic curves with complex multiplication