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arXiv · 2002.08750

The greatest common valuation of $ϕ_{n}$ and $ψ_{n}^{2}$ at points on elliptic curves

Paul Voutier·Minoru Yabuta
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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 $\min \left( v \left( ϕ_{n}(P) \right), v \left( ψ_{n}^{2}(P) \right) \right)$, where $v$ is a normalised valuation on $K$ and $ϕ_{n}(P)$ and $ψ_{n}(P)$ are polynomials arising from multiplication by $n$ for this model of the curve.

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Paul Voutier, Minoru Yabuta. 2021-06-14. The greatest common valuation of $ϕ_{n}$ and $ψ_{n}^{2}$ at points on elliptic curves. https://doi.org/10.1016/j.jnt.2021.04.015

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