Algorithmic Enumeration of Ideal Classes for Quaternion Orders

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 two-sided 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. 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 ClZF and unit group ZF . This problem appears in general to be quite difficult. The best known (probabilistic) algorithm is due to Buchmann [7]: for a field F of degree n and absolute discriminant dF , it runs in time d 1/2 F (log dF ) O(n) without any hypothesis [32], and assuming the Generalized Riemann Hypothesis (GRH), it runs in expected time exp ( O ( (log dF ) (log log dF ) 1/2 )) , where the implied O-constant depends on n. According to the Brauer-Siegel theorem, already the case of imaginary quadratic fields shows that the class group is often roughly as large as d 1/2 F (log dF ) . Similarly, for the case of real quadratic fields, a fundamental unit is conjectured to have height often as large as d 1/2 F (log dF ) , so even to write down the output in a näıve way requires exponential time (but see Remark 1.2). The problem of simply computing the class number h(F ) = # ClZF , or for that matter determining whether or not a given ideal of ZF is principal, appears in general to be no easier than solving Problem (ClassUnitGroup). In this article, we consider a noncommutative generalization of the above problem. We refer to §1 for precise definitions and specification of the input and output. Problem (ClassNumber(O)). Given an Eichler order O in a quaternion algebra over a number field F , compute the class number h(O). Problem (ClassSet(O)). Given an Eichler order O in a quaternion algebra over a number field F , compute a set of representatives for the set of invertible right O-ideal classes ClO. The main results of this article are embodied in the following two theorems, which provide algorithms to solve these two problems depending on whether the order is definite or indefinite.