A course in computational algebraic number theory
A course in computational algebraic number theory
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
Quadratic forms and quaternion algebras: algorithms and arithmetic
Quadratic forms and quaternion algebras: algorithms and arithmetic
Algorithmic Enumeration of Ideal Classes for Quaternion Orders
SIAM Journal on Computing
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Let $\Gamma \subset PSL_2({\mathbb R})$ be a cocompact arithmetic triangle group, i.e. a Fuchsian triangle group that arises from the unit group of a quaternion algebra over a totally real number field. The group Γ acts on the upper half-plane ${\mathfrak{H}}$; the quotient $X_{\mathbb C}=\Gamma \backslash {\mathfrak{H}}$ is a Shimura curve, and there is a map $j:X_{\mathbb C} \to {\mathbb P}^1_{\mathbb C}$. We algorithmically apply the Shimura reciprocity law to compute CM points $j(z_D) \in {\mathbb P}^1_{\mathbb C}$ and their Galois conjugates so as to recognize them as purported algebraic numbers. We conclude by giving some examples of how this method works in practice.