Algebraic points on Shimura curves of $\Gamma_0(p)$-type

TL;DR

This paper classifies algebraic points on Shimura curves of b0b0-type, showing finiteness of such points over quadratic fields for large primes, and applies results to a conjecture on abelian varieties with restricted torsion.

Contribution

It provides a classification of characters associated to algebraic points on Shimura curves of b0b0-type and establishes finiteness results over quadratic fields for large primes.

Findings

Finiteness of algebraic points over quadratic fields for large primes p.

Classification of characters related to algebraic points on Shimura curves.

Application to a conjecture on abelian varieties with prime power torsion.

Abstract

In this article, we classify the characters associated to algebraic points on Shimura curves of $\Gamma_0(p)$-type, and over a quadratic field we show that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number $p$. This is an analogue of the study of rational points or points over a quadratic field on the modular curve $X_0(p)$ by Mazur and one of the author (Momose). We also apply the result to a finiteness conjecture on abelian varieties with constrained prime power torsion by Rasmussen-Tamagawa.