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October, 2014 The $\mathbb{Q}$-rational cuspidal group of $J_{1}(2p)$
Toshikazu TAKAGI
J. Math. Soc. Japan 66(4): 1249-1301 (October, 2014). DOI: 10.2969/jmsj/06641249


Let $p$ be a prime not equal to 2 or 3. In this paper we study the $\mathbb{Q}$-rational cuspidal group $\mathcal{C}_{\mathbb{Q}}$ of the jacobian $J_{1}(2p)$ of the modular curve $X_{1}(2p)$. We prove that the group $\mathcal{C}_{\mathbb{Q}}$ is generated by the $\mathbb{Q}$-rational cusps. We determine the order of $\mathcal{C}_{\mathbb{Q}}$, and give numerical tables for all $p\leq127$. These tables give also other cuspidal class numbers for the modular curves $X_{1}(2p)$ and $X_{1}(p)$. We give a basis of the group of the principal divisors supported on the $\mathbb{Q}$-rational cusps, and using this we determine the explicit structure of $\mathcal{C}_{\mathbb{Q}}$ for all $p\leq127$. We determine the structure of the Sylow $p$-subgroup of $\mathcal{C}_{\mathbb{Q}}$, and the explicit structure for all $p\leq4001$.


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Toshikazu TAKAGI. "The $\mathbb{Q}$-rational cuspidal group of $J_{1}(2p)$." J. Math. Soc. Japan 66 (4) 1249 - 1301, October, 2014.


Published: October, 2014
First available in Project Euclid: 23 October 2014

zbMATH: 1320.11055
MathSciNet: MR3272599
Digital Object Identifier: 10.2969/jmsj/06641249

Primary: 11G18
Secondary: 11F03 , 14G05 , 14G35 , 14H40

Keywords: cuspidal class number , Jacobian variety , modular curve , modular unit , rational point , torsion subgroup

Rights: Copyright © 2014 Mathematical Society of Japan


Vol.66 • No. 4 • October, 2014
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