Leading coefficients of isogenies of degree $p$ over $\mathbb{Q}_{p}$

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}_{p}$ which has potentially supersingular good reduction. Let $L/\mathbb{Q}_{p}$ be a totally ramified extension such that $E$ has good reduction over $L$ and $\tilde{E}$ be the reduction of $E$ mod $\pi$, where $\pi$ is a prime element of the ring of integers $\mathcal{O}_{L}$ of $L$. Let $\hat{E}$ be the formal group over $\mathcal{O}_{L}$ associated to $E/\mathcal{O}_{L}$. The multiplication by $p$ map $[p]\colon \hat{E} \to \hat{E}$ is written by power series $[p](x) = px +c_{2}x^{2} + \cdots + c_{p}x^{p}+ \cdots + c_{p^{2}} x^{p^{2}} + \cdots{} \in \mathcal{O}_{L}[[x]]$. By using the liftings over $\mathcal{O}_{L}$ of the Dieudonné module of $p$-divisible group $\tilde{E}(p)$ over $\mathbb{F}_{p}$, we determine the values of $v_{L}(c_{p})$.