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See detailPresentations of groups acting discontinuously on direct products of hyperbolic spaces
Jespers, E.; Kiefer, Ann UL; del Río, Á.

in Mathematics of Computation (2016), 85(301), 2515--2552

The problem of describing the group of units U(ZG) of the integral group ring ZG of a finite group G has attracted a lot of attention and providing presentations for such groups is a fundamental problem ... [more ▼]

The problem of describing the group of units U(ZG) of the integral group ring ZG of a finite group G has attracted a lot of attention and providing presentations for such groups is a fundamental problem. Within the context of orders, a central problem is to describe a presentation of the unit group of an order O in the simple epimorphic images A of the rational group algebra QG. Making use of the presentation part of Poincaré’s Polyhedron Theorem, Pita, del Río and Ruiz proposed such a method for a large family of finite groups G and consequently Jespers, Pita, del Río, Ruiz and Zalesskii described the structure of U(ZG) for a large family of finite groups G. In order to handle many more groups, one would like to extend Poincaré’s Method to discontinuous subgroups of the group of isometries of a direct product of hyperbolic spaces. If the algebra A has degree 2 then via the Galois embeddings of the centre of the algebra A one considers the group of reduced norm one elements of the order O as such a group and thus one would obtain a solution to the mentioned problem. This would provide presentations of the unit group of orders in the simple components of degree 2 of QG and in particular describe the unit group of ZG for every group G with irreducible character degrees less than or equal to 2. The aim of this paper is to initiate this approach by executing this method on the Hilbert modular group, i.e. the projective linear group of degree two over the ring of integers in a real quadratic extension of the rationals. This group acts discontinuously on a direct product of two hyperbolic spaces of dimension two. The fundamental domain constructed is an analogue of the Ford domain of a Fuchsian or a Kleinian group. [less ▲]

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See detailFrom the Poincaré theorem to generators of the unit group of integral group rings of finite groups
Jespers, E.; Juriaans, S. O.; Kiefer, Ann UL et al

in Mathematics of Computation (2015), 84(293), 1489--1520

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring ZG of a finite nilpotent group G, this provided the rational group ... [more ▼]

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring ZG of a finite nilpotent group G, this provided the rational group algebra QG does not have simple components that are division classical quaternion algebras or two-by-two matrices over a classical quaternion algebra with centre Q. The main difficulty is to deal with orders in quaternion algebras over the rationals or a quadratic imaginary extension of the rationals. In order to deal with these we give a finite and easy implementable algorithm to compute a fundamental domain in the hyperbolic three space H 3 (respectively hyperbolic two space H 2 ) for a discrete subgroup of PSL 2 (C) (respectively PSL 2 (R)) of finite covolume. Our results on group rings are a continuation of earlier work of Ritter and Sehgal, Jespers and Leal. [less ▲]

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See detailEmpirical evidence for the Birch and Swinnerton-Dyer conjectures for the modular Jacobians of genus 2 curves
Flynn, E. Victor; Leprévost, Franck UL; Schaefer, E. F. et al

in Mathematics of Computation (2001), 70(236), 1675-1697

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See detailComputing Discrete Logarithms in Real Quadratic Congruence Function Fields of Large Genus
Müller, Volker UL; Stein, Andreas; Thiel, Christoph

in Mathematics of Computation (1999), 68(226), 807-822

We present a sub-exponential algorithm for computing discrete loga­rithms in real quadratic congruence function fields of sufficiently large genus. We prove the correctness and the expected running time ... [more ▼]

We present a sub-exponential algorithm for computing discrete loga­rithms in real quadratic congruence function fields of sufficiently large genus. We prove the correctness and the expected running time of the algorithm. The algorithm is a gene­ralization of a similar algorithm for quadratic number fields. [less ▲]

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