Osaka Journal of Mathematics

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

Mayumi Kawachi

Full-text: Open access

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 $v_{L}(c_{p})$.

Article information

Source
Osaka J. Math., Volume 48, Number 3 (2011), 691-708.

Dates
First available in Project Euclid: 26 September 2011

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1317044942

Mathematical Reviews number (MathSciNet)
MR2837676

Zentralblatt MATH identifier
1271.11064

Subjects
Primary: 11G07: Elliptic curves over local fields [See also 14G20, 14H52]
Secondary: 14L05: Formal groups, $p$-divisible groups [See also 55N22]

Citation

Kawachi, Mayumi. Leading coefficients of isogenies of degree $p$ over $\mathbb{Q}_{p}$. Osaka J. Math. 48 (2011), no. 3, 691--708. https://projecteuclid.org/euclid.ojm/1317044942


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