A note on elliptic curves with ordinary reduction
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.
scientific article; zbMATH DE number 446372