Common valuations of division polynomials

TL;DR

This paper derives a general formula for the cancellation exponent between division polynomials on elliptic curves over discrete valuation fields, extending previous results to broader cases.

Contribution

It provides a unified formula for the cancellation exponent, generalizing Yabuta-Voutier's result to non-standard Kodaira types and non-perfect residue fields.

Findings

Formula matches Yabuta-Voutier for finite extensions of Q_p

Generalizes to non-standard Kodaira types

Applicable to non-perfect residue fields

Abstract

In this note we prove a formula for the cancellation exponent $k_{v,n}$ between division polynomials $\psi_n$ and $\phi_n$ 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.