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December 2019 On Congruence Relations and Equations of Shimura Curves
Akira KURIHARA
Tokyo J. Math. 42(2): 525-550 (December 2019). DOI: 10.3836/tjm/1502179308

Abstract

On a Shimura curve, the reduction modulo a prime $p$ of the Hecke correspondence $T(p)$ yields the congruence relation $\Pi\cup\Pi'$ with $\Pi$ being the graph of the Frobenius mapping from the Shimura curve modulo $p$ to itself, and $\Pi'$ its transpose. Starting with a curve $C$ of genus $g \geq 2$ over $\mathbb{F}_p$ together with a subset $\mathfrak{S}\subset C(\mathbb{F}_{p^2})$, Ihara studied the liftability to characteristic $0$ of $\Pi\cup\Pi'$ so that $\Pi$ and $\Pi'$ are separated outside $\mathfrak{S}$ in the lifting. In some case, Ihara obtained the uniqueness of the liftability to characteristic $0$ and gave some necessary and sufficient condition, described by some differential form on $C$, for $(C,\mathfrak{S})$ to be liftable to modulo $p^2$. In this paper, in case when $C$ is defined over $\mathbb{F}_{p^2}$, we compute complete tables of such $(C,{\mathfrak S})$ liftable to modulo $p^2$ for $g=2$ and $3\leq p \leq 13$ using computer, and as an application of this uniqueness, we identify some particular Shimura curve by its equation.

Citation

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Akira KURIHARA. "On Congruence Relations and Equations of Shimura Curves." Tokyo J. Math. 42 (2) 525 - 550, December 2019. https://doi.org/10.3836/tjm/1502179308

Information

Published: December 2019
First available in Project Euclid: 2 June 2020

zbMATH: 07209632
MathSciNet: MR4106591
Digital Object Identifier: 10.3836/tjm/1502179308

Subjects:
Primary: 11G18
Secondary: 14-04, 14G35

Rights: Copyright © 2019 Publication Committee for the Tokyo Journal of Mathematics

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Vol.42 • No. 2 • December 2019
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