Nagoya Mathematical Journal

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

Fumiyuki Momose and Mahoro Shimura

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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 $p = 2$.)

Article information

Source
Nagoya Math. J., Volume 165 (2002), 159-178.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631703

Mathematical Reviews number (MathSciNet)
MR1892103

Zentralblatt MATH identifier
1032.11024

Subjects
Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 11G07: Elliptic curves over local fields [See also 14G20, 14H52]

Citation

Momose, Fumiyuki; Shimura, Mahoro. Lifting of supersingular points on {$X\sb 0(p\sp r)$} and lower bound of ramification index. Nagoya Math. J. 165 (2002), 159--178. https://projecteuclid.org/euclid.nmj/1114631703


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