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.
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Acknowledgements
This work was supported by JSPS KAKENHI Grant Numbers JP25800025, JP16K17578, JP21K03187 and Research Institute for Science and Technology of Tokyo Denki University Grant Numbers Q16K-06, Q20K-01 / Japan. The author would like to thank the anonymous referee for helpful comments.
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Arai, K. Points on Shimura curves rational over imaginary quadratic fields in the non-split case. Math. Z. 305, 60 (2023). https://doi.org/10.1007/s00209-023-03377-5
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DOI: https://doi.org/10.1007/s00209-023-03377-5