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Algorithmic enumeration of ideal classes for quaternion orders
 SIAM J. Comput. (SICOMP
"... Abstract. We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of twosided ideal ..."
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Abstract. We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of twosided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We conclude by giving the complete list of definite Eichler orders with class number at most 2. Key words. quaternion algebras, maximal orders, ideal classes, number theory AMS subject classifications. 11R52 Since the very first calculations of Gauss for imaginary quadratic fields, the problem of computing the class group of a number field F has seen broad interest. Due to the evident close association between the class number and regulator (embodied in the Dirichlet class number formula), one often computes the class group and unit group in tandem as follows. Problem (ClassUnitGroup(ZF)). Given the ring of integers ZF of a number field F, compute the class group Cl ZF and unit group Z ∗ F.
Computing fundamental domains for Fuchsian Groups
"... We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group Γ with cofinite area. As a consequence, we compute the invariants of Γ, including an explicit finite presentation for Γ. ..."
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Cited by 16 (3 self)
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We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group Γ with cofinite area. As a consequence, we compute the invariants of Γ, including an explicit finite presentation for Γ.
Identifying the Matrix Ring: ALGORITHMS FOR QUATERNION ALGEBRAS AND QUADRATIC FORMS
, 2010
"... We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2matrix ring M2(R) and, if so, to compute such an embedding. We d ..."
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Cited by 13 (1 self)
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We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders.
Computing Hilbert modular forms over fields with nontrivial class group, in Algorithmic Number Theory
 Lecture notes in computer science
, 2008
"... Abstract. In previous work, the first author developed an algorithm for the computation of Hilbert modular forms. In this paper, we extend this to all totally real number fields of even degree and nontrivial class group. Using the algorithm over Q( 10) and Q( ..."
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Cited by 11 (1 self)
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Abstract. In previous work, the first author developed an algorithm for the computation of Hilbert modular forms. In this paper, we extend this to all totally real number fields of even degree and nontrivial class group. Using the algorithm over Q( 10) and Q(
Computing CM points on Shimura curves arising from cocompact arithmetic triangle groups
"... Let PSL2(R) be a cocompact arithmetic triangle group, i.e. a triangle Fuchsiangroupthatarisesfromtheunitgroupofaquaternion algebra over ..."
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Cited by 5 (4 self)
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Let PSL2(R) be a cocompact arithmetic triangle group, i.e. a triangle Fuchsiangroupthatarisesfromtheunitgroupofaquaternion algebra over