Lifting of supersingular points on {$X\sb 0(p\sp r)$} and lower bound of ramification index

Abstract

Let $K$ be a finite extension of $\mathbf{Q}_{p}^{\mathit{ur}}$ (= the maximal unramified extension of $\mathbf{Q}_{p}$) of degree $e_{K}$, $\mathcal{O}$ its integer ring, $p$ a rational prime and $r$ a positive integer. If there exists a one parameter formal group defined over $\mathcal{O}$ whose reduction is of height $2$ with a cyclic subgroup $V$ of order $p^{r}$ defined over $\mathcal{O}$, then $e_{K} \geq 2p^{l}$ (resp.~$p^{l}+p^{l-1}$) if $r = 2l+1$ (resp.~$r = 2l$).

We apply this result to a criterion for non-existence of $\mathbf{Q}$-rational point of $X_{0}^{+}(p^{r})$. (This criterion is Momose's theorem in [14] except for the cases $p = 5$ and $p = 13$, but our new proof does not require defining equations of modular curves except for the case $p = 2$.)