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

TL;DR

This paper proves the non-existence of elliptic points on Shimura curves of b3_0(p)-type under mild assumptions, extending previous classifications and providing explicit examples.

Contribution

It establishes the non-existence of elliptic points on these Shimura curves under mild conditions, advancing the understanding of their algebraic points.

Findings

No elliptic points exist under certain conditions

Explicit example of such a Shimura curve

Extension of previous classification results

Abstract

In previous articles, we classified the characters associated to algebraic points on Shimura curves of $\Gamma_0(p)$-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 $\Gamma_0(p)$-type under a mild assumption. We also give an explicit example.