Algebraic points on Shimura curves of Γ₀(p)-type (III)

2012/05/16 by Keisuke Arai, Arai, Keisuke, Fumiyuki Momose
Mathematics · Social Sciences · #11G10 #11G15 (Secondary) #11G18 #14G05 (Primary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.1303.5270

openalex publication_date 2013/02/22 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

In previous articles, we classified the characters associated to algebraic points on Shimura curves of Γ₀(p)-type, and over number fields in a certain large class we showed that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number p. In this article, we prove the non-existence of elliptic points on Shimura curves of Γ₀(p)-type under a mild assumption. We also give an explicit example.

Citations

• On the Rasmussen-Tamagawa conjecture for QM-abelian surfaces
• An effective bound of p for algebraic points on Shimura curves of Γ₀(p)-type
• Non-existence of certain CM abelian varieties with prime power torsion

Related

• Algebraic points on Shimura curves of Γ₀(p)-type (II)
• Algebraic points on Shimura curves of Γ₀(p)-type
• An effective bound of p for algebraic points on Shimura curves of Γ₀(p)-type
• Non-existence of points rational over number fields on Shimura curves
• On Shimura curves generated by families of Galois G-covers of curves