A note on elliptic curves with ordinary reduction
type: Item
instance of: scholarly article
title: A note on elliptic curves with ordinary reduction
zbMATH Open document ID: 0779.14006
DOI: 10.1007/BF01202309
author: Tetsuo Nakamura
published in: Archiv der Mathematik
publication date: 1993-11-15T00:00:00Z
review text: Let \(p\) be a prime number and \(K\) be a complete unramified extension of \(\mathbb{Q}_ p\). Let \(E\) be an elliptic curve over \(K\) with ordinary reduction. Denote by \(\tilde E\) the reduction mod \(p\) of \(E\). Let \(j\) be the \(j\)-invariant of \(E\) and we assume \(j\) mod \(p\) is algebraic over \(\mathbb{F}_ q\). Let \(j_ 0\) be the \(j\)-invariant of the canonical lifting \(E_ 0\) of \(\tilde E\), i.e. \(E_ 0\) is an elliptic curve over \(K\) such that \(\text{End} E_ 0\cong \text{End} \tilde E(=R)\) and \(E_ 0\equiv E\) mod \(p\). Put \(E_ p=\text{Ker}[p]_ E\) and \(m={1\over 2}| R^ \times|\). Theorem: \(K(E_ p)/K\) is tamely ramified if and only if \(j\equiv j_ 0\) mod \(p^{m+1}\). The theorem is proved in a letter of \textit{J.-P. Serre} (1988). His proof depends deeply on the theory of Serre-Tate \(q\) (Dwork \(q)\). Another proof of the theorem which uses modular polynomial and the theory of complex multiplication is given.
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G07
zbMATH DE Number: 446372
zbMATH Keywords: ramification of function field
zbMATH Keywords: reduction mod \(p\)
zbMATH Keywords: elliptic curve
zbMATH Keywords: ordinary reduction
zbMATH Keywords: modular polynomial
zbMATH Keywords: complex multiplication
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01202309
OpenAlex ID: W2051301363
description: scientific article; zbMATH DE number 446372