@article{34916, abstract = {{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.}}, author = {{Kirschmer, Markus}}, issn = {{0022-314X}}, journal = {{Journal of Number Theory}}, keywords = {{Algebra and Number Theory}}, pages = {{121--134}}, publisher = {{Elsevier BV}}, title = {{{Automorphisms of even unimodular lattices over number fields}}}, doi = {{10.1016/j.jnt.2018.08.004}}, volume = {{197}}, year = {{2019}}, } @article{55284, author = {{Elsholtz, Ch. and Technau, Marc and Technau, N.}}, journal = {{Mathematika}}, number = {{4}}, pages = {{990–1009}}, title = {{{The maximal order of iterated multiplicative functions}}}, doi = {{10.1112/S0025579319000214}}, volume = {{64}}, year = {{2019}}, } @article{55285, author = {{Technau, Marc}}, journal = {{Notes Number Theory Discrete Math.}}, number = {{2}}, pages = {{127–135}}, title = {{{Generalised Beatty sets}}}, doi = {{10.7546/nntdm.2019.25.2.127-135}}, volume = {{25}}, year = {{2019}}, } @article{34915, abstract = {{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.}}, author = {{Kirschmer, Markus}}, issn = {{0003-889X}}, journal = {{Archiv der Mathematik}}, keywords = {{General Mathematics}}, number = {{4}}, pages = {{337--347}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Determinant groups of Hermitian lattices over local fields}}}, doi = {{10.1007/s00013-019-01348-z}}, volume = {{113}}, year = {{2019}}, } @phdthesis{55291, author = {{Technau, Marc}}, publisher = {{University of Würzburg}}, title = {{{On Beatty sets and some generalisations thereof}}}, doi = {{10.25972/WUP-978-3-95826-089-4}}, year = {{2018}}, } @article{34843, abstract = {{A polynomial time algorithm to find generators of the lattice of all subfields of a given number field was given in van Hoeij et al. (2013). This article reports on a massive speedup of this algorithm. This is primary achieved by our new concept of Galois-generating subfields. In general this is a very small set of subfields that determine all other subfields in a group-theoretic way. We compute them by targeted calls to the method from van Hoeij et al. (2013). For an early termination of these calls, we give a list of criteria that imply that further calls will not result in additional subfields. Finally, we explain how we use subfields to get a good starting group for the computation of Galois groups.}}, author = {{Elsenhans, Andreas-Stephan and Klüners, Jürgen}}, issn = {{0747-7171}}, journal = {{Journal of Symbolic Computation}}, keywords = {{Computational Mathematics, Algebra and Number Theory}}, pages = {{1--20}}, publisher = {{Elsevier BV}}, title = {{{Computing subfields of number fields and applications to Galois group computations}}}, doi = {{10.1016/j.jsc.2018.04.013}}, volume = {{93}}, year = {{2018}}, } @inbook{42788, abstract = {{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(L{p}) 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.}}, author = {{Kirschmer, Markus and Nebe, Gabriele}}, booktitle = {{Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory}}, isbn = {{9783319705651}}, publisher = {{Springer International Publishing}}, title = {{{One Class Genera of Lattice Chains Over Number Fields}}}, doi = {{10.1007/978-3-319-70566-8_22}}, year = {{2018}}, } @article{42790, abstract = {{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.}}, author = {{Kirschmer, Markus}}, issn = {{1246-7405}}, journal = {{Journal de Théorie des Nombres de Bordeaux}}, keywords = {{Algebra and Number Theory}}, number = {{3}}, pages = {{847--857}}, publisher = {{Cellule MathDoc/CEDRAM}}, title = {{{One-class genera of exceptional groups over number fields}}}, doi = {{10.5802/jtnb.1052}}, volume = {{30}}, year = {{2018}}, } @techreport{55293, author = {{Barth, D. and Beck, M. and Dose, T. and Glaßer, Ch. and Michler, L. and Technau, Marc}}, title = {{{Emptiness problems for integer circuits}}}, year = {{2017}}, } @inproceedings{55292, author = {{Barth, D. and Beck, M. and Dose, T. and Glaßer, Ch. and Michler, L. and Technau, Marc}}, booktitle = {{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}}, editor = {{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}}, pages = {{33:1–33:14}}, publisher = {{Schloss Dagstuhl–Leibniz-Zentrum für Informatik}}, title = {{{Emptiness problems for integer circuits}}}, doi = {{10.4230/LIPIcs.MFCS.2017.33}}, volume = {{83}}, year = {{2017}}, } @article{55275, author = {{Technau, Marc and Technau, N.}}, journal = {{Comput. Methods Funct. Theory}}, number = {{2}}, pages = {{255–272}}, title = {{{A Loewner equation for infinitely many slits}}}, doi = {{10.1007/s40315-016-0179-6}}, volume = {{17}}, year = {{2017}}, } @article{42791, abstract = {{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.}}, author = {{Kirschmer, Markus and Rüther, Marion G.}}, issn = {{0021-8693}}, journal = {{Journal of Algebra}}, keywords = {{Algebra and Number Theory}}, pages = {{519--548}}, publisher = {{Elsevier BV}}, title = {{{The constructive membership problem for discrete two-generator subgroups of SL(2,R)}}}, doi = {{10.1016/j.jalgebra.2017.02.029}}, volume = {{480}}, year = {{2017}}, } @article{55281, author = {{Steuding, J. and Technau, Marc}}, journal = {{J. Number Theory}}, pages = {{144–159}}, title = {{{The least prime number in a Beatty sequence}}}, doi = {{10.1016/j.jnt.2016.05.022}}, volume = {{169}}, year = {{2016}}, } @article{34844, abstract = {{Let k be a number field, K/k a finite Galois extension with Galois group G, χ a faithful character of G. We prove that the Artin L-function L(s,χ,K/k) determines the Galois closure of K over $\ℚ$. In the special case $k=\ℚ$ it also determines the character χ. }}, author = {{Klüners, Jürgen and Nicolae, Florin}}, issn = {{0022-314X}}, journal = {{Journal of Number Theory}}, keywords = {{Algebra and Number Theory}}, pages = {{161--168}}, publisher = {{Elsevier BV}}, title = {{{Are number fields determined by Artin L-functions?}}}, doi = {{10.1016/j.jnt.2016.03.023}}, volume = {{167}}, year = {{2016}}, } @article{42792, abstract = {{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.}}, author = {{Kirschmer, Markus and Lorch, David}}, issn = {{0022-314X}}, journal = {{Journal of Number Theory}}, keywords = {{Algebra and Number Theory}}, pages = {{343--361}}, publisher = {{Elsevier BV}}, title = {{{Ternary quadratic forms over number fields with small class number}}}, doi = {{10.1016/j.jnt.2014.11.001}}, volume = {{161}}, year = {{2016}}, } @misc{43454, abstract = {{Die Gitter von Klassenzahl eins oder zwei sind hier verfügbar: http://www.math.rwth-aachen.de/~Markus.Kirschmer/forms/}}, author = {{Kirschmer, Markus}}, pages = {{166}}, title = {{{Definite quadratic and hermitian forms with small class number (Habilitation)}}}, year = {{2016}}, } @article{34845, abstract = {{Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar’s method. Computationally, one of the key challenges in the application of Stauduhar’s method is to find, for a given pair of groups H