Chow groups of one-dimensional noetherian domains - Kirschmer, Markus; Klüners, Jürgen (Research in Number Theory)
We discuss various connections between ideal classes, divisors, Picard and Chow groups of one-dimensional noetherian domains. As a result of these, we give a method to compute Chow groups of orders in global fields and show that there are infinitely many number fields which contain orders with trivial Chow groups.

Binary Hermitian Lattices over Number Fields - Kirschmer, Markus; Nebe, Gabriele (Experimental Mathematics)
In a previous paper the authors developed an algorithm to classify certain quaternary quadratic lattices over totally real fields. The present article applies this algorithm to the classification of binary Hermitian lattices over totally imaginary fields. We use it in particular to classify the 48-dimensional extremal even unimodular lattices over the integers that admit a semilarge automorphism.
Keywords: General Mathematics

Spanning the isogeny class of a power of an elliptic curve - Kirschmer, Markus; Narbonne, Fabien; Ritzenthaler, Christophe; Robert, Damien (Mathematics of Computation)
Let E be an ordinary elliptic curve over a finite field and g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E⁹ . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E³ and of the Igusa modular form in dimension 4. We illustrate our algorithms with examples of curves with many rational points over finite fields.
Keywords: Applied Mathematics; Computational Mathematics; Algebra and Number Theory

Quaternary quadratic lattices over number fields - Kirschmer, Markus; Nebe, Gabriele (International Journal of Number Theory)
We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space (V,q) with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of (V,q). This yields a good algorithm to enumerate a system of representatives of proper isometry classes of lattices in genera of maximal lattices in (V,q).
Keywords: Algebra and Number Theory

Automorphisms of even unimodular lattices over number fields - Kirschmer, Markus (Journal of Number Theory)
We describe the powers of irreducible polynomials occurring as characteristic polynomials of automorphisms of even unimodular lattices over number fields. This generalizes results of Gross & McMullen and Bayer-Fluckiger & Taelman.
Keywords: Algebra and Number Theory

Determinant groups of Hermitian lattices over local fields - Kirschmer, Markus (Archiv der Mathematik)
We describe the determinants of the automorphism groups of Hermitian lattices over local fields. Using a result of G. Shimura, this yields an explicit method to compute the special genera in a given genus of Hermitian lattices over a number field.
Keywords: General Mathematics

One Class Genera of Lattice Chains Over Number Fields - Kirschmer, Markus; Nebe, Gabriele (Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory)
We classify all one-class genera of admissible lattice chains of length at least 2 in hermitian spaces over number fields. If L is a lattice in the chain and p the prime ideal dividing the index of the lattices in the chain, then the {p}-arithmetic group Aut(Lp) acts chamber transitively on the corresponding Bruhat-Tits building. So our classification provides a step forward to a complete classification of these chamber transitive groups which has been announced 1987 (without a detailed proof) by Kantor, Liebler and Tits. In fact we find all their groups over number fields and one additional building with a discrete chamber transitive group.

One-class genera of exceptional groups over number fields - Kirschmer, Markus (Journal de Théorie des Nombres de Bordeaux)
We show that exceptional algebraic groups over number fields do not admit one-class genera of parahoric groups, except in the case G₂ . For the group G₂, we enumerate all such one-class genera for the usual seven-dimensional representation.
Keywords: Algebra and Number Theory

The constructive membership problem for discrete two-generator subgroups of SL(2,R) - Kirschmer, Markus; Rüther, Marion G. (Journal of Algebra)
We describe a practical algorithm to solve the constructive membership problem for discrete two-generator subgroups of SL₂(R) or PSL₂(R). This algorithm has been implemented in Magma for groups defined over real algebraic number fields.
Keywords: Algebra and Number Theory

Ternary quadratic forms over number fields with small class number - Kirschmer, Markus; Lorch, David (Journal of Number Theory)
We enumerate all positive definite ternary quadratic forms over number fields with class number at most 2. This is done by constructing all definite quaternion orders of type number at most 2 over number fields. Finally, we list all definite quaternion orders of ideal class number 1 or 2.
Keywords: Algebra and Number Theory

Definite quadratic and hermitian forms with small class number (Habilitation) - Kirschmer, Markus
Die Gitter von Klassenzahl eins oder zwei sind hier verfügbar: http://www.math.rwth-aachen.de/~Markus.Kirschmer/forms/

One-class genera of maximal integral quadratic forms - Kirschmer, Markus (Journal of Number Theory)
Suppose Q is a definite quadratic form on a vector space V over some totally real field K ≠ Q. Then the maximal integral Zₖ-lattices in (V,Q) are locally isometric everywhere and hence form a single genus. We enumerate all orthogonal spaces (V,Q) of dimension at least 3, where the corresponding genus of maximal integral lattices consists of a single isometry class. It turns out, there are 471 such genera. Moreover, the dimension of V and the degree of K are bounded by 6 and 5 respectively. This classification also yields all maximal quaternion orders of type number one.
Keywords: Algebra and Number Theory

Computing with subgroups of the modular group - Kirschmer, Markus; LEEDHAM-GREEN, CHARLES (Glasgow Mathematical Journal)
We exhibit a practical algorithm for solving the constructive membership problem for discrete free subgroups of rank 2 in PSL₂(R) or SL₂(R). This algorithm, together with methods for checking whether a two-generator subgroup of PSL₂(R) or SL₂(R) is discrete and free, have been implemented in Magma for groups defined over real algebraic number fields.
Keywords: General Mathematics

The constructive membership problem for discrete free subgroups of rank 2 of SL₂(R) - Eick, B.; Kirschmer, Markus; Leedham-Green, C. (LMS Journal of Computation and Mathematics)
We exhibit a practical algorithm for solving the constructive membership problem for discrete free subgroups of rank 2 in PSL₂(R) or SL₂(R). This algorithm, together with methods for checking whether a two-generator subgroup of PSL₂(R) or SL₂(R) is discrete and free, have been implemented in Magma for groups defined over real algebraic number fields.
Keywords: Computational Theory and Mathematics; General Mathematics

On an analogue to the Lucas-Lehmer-Riesel test using elliptic curves - Kirschmer, Markus; Mertens, Michael H. (Integers)
Following an idea of B. H. Gross, who presented an elliptic curve test for Mersenneprimes Mₚ=2ᵖ−1, we propose a similar test with elliptic curves for generalizedThabit primesK(h, n) := h·2ⁿ−1 for any positive odd number h and any integer n> log₂(h)+2.

Single-class genera of positive integral lattices - Lorch, David; Kirschmer, Markus (LMS Journal of Computation and Mathematics)
We give an enumeration of all positive definite primitive Z-lattices in dimension n ≥ 3 whose genus consists of a single isometry class. This is achieved by using bounds obtained from the Smith–Minkowski–Siegel mass formula to computationally construct the square-free determinant lattices with this property, and then repeatedly calculating pre-images under a mapping first introduced by G. L. Watson. We hereby complete the classification of single-class genera in dimensions 4 and 5 and correct some mistakes in Watson’s classifications in other dimensions. A list of all single-class primitive Z-lattices has been compiled and incorporated into the Catalogue of Lattices.
Keywords: Computational Theory and Mathematics; General Mathematics

A normal form for definite quadratic forms over $\mathbb{F}_{q}[t]$ - Kirschmer, Markus (Mathematics of Computation)
An efficient algorithm to compute automorphism groups and isometries of definite Fq[t]-lattices for odd q is presented. The algorithm requires several square root computations in Fq₂ but no enumeration of orbits having more than eight elements.
Keywords: Applied Mathematics; Computational Mathematics; Algebra and Number Theory

Finite Symplectic Matrix Groups - Kirschmer, Markus (Experimental Mathematics)
This paper classifies the maximal finite subgroups of SP₂ₙ(Q) for 1⩽n⩽11 up to GL₂ₙ(Q) conjugacy in .
Keywords: General Mathematics

Algorithmic Enumeration of Ideal Classes for Quaternion Orders - Kirschmer, Markus; Voight, John (SIAM Journal on Computing)
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.
Keywords: General Mathematics; General Computer Science

Finite symplectic matrix groups (Dissertation) - Kirschmer, Markus
Die invarianten Formen aus dem Anhang sind hier verfügbar:http://www.math.rwth-aachen.de/homes/Markus.Kirschmer/symplectic/