Rational torsion points on Jacobians of Shimura curves

TL;DR

This paper investigates the existence of rational torsion points on Jacobians of Shimura curves associated with certain quaternion algebras, providing conditions for their non-existence and applications to isogeny kernels.

Contribution

It offers new criteria for the non-existence of rational torsion points on Jacobians of specific Shimura curves and applies these results to analyze isogeny kernels.

Findings

Established sufficient conditions for non-existence of rational $\, ext{ell}$-torsion points.

Identified non-trivial subgroups in the kernel of certain isogenies.

Enhanced understanding of torsion structures on Jacobians of Shimura curves.

Abstract

Let $p$ and $q$ be distinct primes. Consider the Shimura curve $\mathcal{X}$ associated to the indefinite quaternion algebra of discriminant $pq$ over $\mathbb{Q}$. Let $J$ be the Jacobian variety of $\mathcal{X}$, which is an abelian variety over $\mathbb{Q}$. For an odd prime $\ell$, we provide sufficient conditions for the non-existence of rational points of order $\ell$ on $J$. As an application, we find some non-trivial subgroups of the kernel of an isogeny from the new quotient of $J_0(pq)$ to $J$.