Points on Shimura curves rational over imaginary quadratic fields in the non-split case

TL;DR

This paper proves finiteness results for rational points on Shimura curves over imaginary quadratic fields with non-split quaternion algebras, identifying a finite set of primes related to the discriminant and providing an effective bound.

Contribution

It establishes a finiteness theorem for rational points on Shimura curves in the non-split case and explicitly describes the set of primes involved, extending previous work by Jordan.

Findings

Finite set of primes P(k) depending on the field k

Effective computability of the constant C(k)

Finiteness of k-rational points on Shimura curves in the non-split case

Abstract

For an imaginary quadratic field $k$ of class number $>1$, we prove that there are only finitely many isomorphism classes of rational indefinite quaternion division algebras $B$ such that the associated Shimura curve $M^B$ has $k$-rational points. In other words, the main result asserts that there is a finite set $P(k)$ of prime numbers depending on $k$ such that: if there is a prime divisor of the discriminant of $B$ which is not in $P(k)$, then $M^B$ has no $k$-rational points. Moreover, we can take $P(k)$ to satisfy the following: There is an effectively computable constant $C(k)$ depending on $k$ such that $p\in P(k)$ implies $p<C(k)$ with at most one possible exception. The case where $k$ splits $B$ was done by Jordan. In the non-split case, the proof is done by studying a canonical isogeny character and its composition with the transfer map.