@article{34835, abstract = {{We prove an upper bound for the asymptotics of counting functions of number fields with nilpotent Galois groups. }}, author = {{Klüners, Jürgen}}, issn = {{0065-1036}}, journal = {{Acta Arithmetica}}, keywords = {{Algebra and Number Theory}}, number = {{2}}, pages = {{165--184}}, publisher = {{Institute of Mathematics, Polish Academy of Sciences}}, title = {{{The asymptotics of nilpotent Galois groups}}}, doi = {{10.4064/aa211207-16-5}}, volume = {{204}}, year = {{2022}}, } @article{53192, abstract = {{The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of $\operatorname {GL}_{2n}$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive because we draw heavily upon the methods used in the recent and separate works of all three authors. By construction, our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Moreover, we work under a weaker Panchishkine-type condition on $\Pi _p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$-adic $L$-functions at all critical points. This has the striking consequence that, given a unitary $\Pi$ whose standard $L$-function admits at least two critical points, and given a prime $p$ such that $\Pi _p$ is ordinary, the central critical value $L(\frac {1}{2}, \Pi \otimes \chi )$ is non-zero for all except finitely many Dirichlet characters $\chi$ of $p$-power conductor.}}, author = {{Dimitrov, Mladen and Januszewski, Fabian and Raghuram, A.}}, issn = {{0010-437X}}, journal = {{Compositio Mathematica}}, keywords = {{Algebra and Number Theory}}, number = {{12}}, pages = {{2437--2468}}, publisher = {{Wiley}}, title = {{{L-functions of GL(2n): p-adic properties and non-vanishing of twists}}}, doi = {{10.1112/s0010437x20007551}}, volume = {{156}}, year = {{2021}}, } @article{34840, abstract = {{In this paper we obtain a complete list of imaginary n-quadratic fields with class groups of exponent 3 and 5 under ERH for every positive integer n where an n-quadratic field is a number field of degree 2ⁿ represented as the composite of n quadratic fields. }}, author = {{Klüners, Jürgen and Komatsu, Toru}}, issn = {{0025-5718}}, journal = {{Mathematics of Computation}}, keywords = {{Applied Mathematics, Computational Mathematics, Algebra and Number Theory}}, number = {{329}}, pages = {{1483--1497}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{Imaginary multiquadratic number fields with class group of exponent $3$ and $5$}}}, doi = {{10.1090/mcom/3609}}, volume = {{90}}, year = {{2021}}, } @article{34912, abstract = {{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. }}, author = {{Kirschmer, Markus and Narbonne, Fabien and Ritzenthaler, Christophe and Robert, Damien}}, issn = {{0025-5718}}, journal = {{Mathematics of Computation}}, keywords = {{Applied Mathematics, Computational Mathematics, Algebra and Number Theory}}, number = {{333}}, pages = {{401--449}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{Spanning the isogeny class of a power of an elliptic curve}}}, doi = {{10.1090/mcom/3672}}, volume = {{91}}, year = {{2021}}, } @article{33262, abstract = {{The authors of Berg et al. [J. Algebra 348 (2011) 446–461] provide an algorithm for finding a complete system of primitive orthogonal idempotents for CM, where M is any finite R-trivial monoid. Their method relies on a technical result stating that R-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an R-trivial monoid may be realized by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where L-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We first show that our algorithm works for ZM, after which we prove that it also works for RM where R is an arbitrary ring with a known complete system of primitive orthogonal idempotents. In particular, our algorithm works if R is any field. In this respect our result constitutes a considerable generalization of the results in Berg et al. [J. Algebra 348 (2011) 446–461]. Moreover, the system of idempotents for RM is obtained from the one our algorithm yields for ZM in a straightforward manner. In other words, for any finite R-trivial monoid M our algorithm only has to be performed for ZM, after which a system of idempotents follows for any ring with a given system of idempotents.}}, author = {{Nijholt, Eddie and Rink, Bob and Schwenker, Sören}}, issn = {{0219-4988}}, journal = {{Journal of Algebra and Its Applications}}, keywords = {{Applied Mathematics, Algebra and Number Theory}}, number = {{12}}, publisher = {{World Scientific Pub Co Pte Ltd}}, title = {{{A new algorithm for computing idempotents of ℛ-trivial monoids}}}, doi = {{10.1142/s0219498821502273}}, volume = {{20}}, year = {{2020}}, } @article{45955, author = {{Akrivis, Georgios and Feischl, Michael and Kovács, Balázs and Lubich, Christian}}, issn = {{0025-5718}}, journal = {{Mathematics of Computation}}, keywords = {{Applied Mathematics, Computational Mathematics, Algebra and Number Theory}}, number = {{329}}, pages = {{995--1038}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{Higher-order linearly implicit full discretization of the Landau–Lifshitz–Gilbert equation}}}, doi = {{10.1090/mcom/3597}}, volume = {{90}}, year = {{2020}}, } @article{34842, abstract = {{Let D<0 be a fundamental discriminant and denote by E(D) the exponent of the ideal class group Cl(D) of K=ℚ(√D). Under the assumption that no Siegel zeros exist we compute all such D with E(D) dividing 8. We compute all D with |D| ≤ 3.1⋅10²⁰ such that E(D) ≤ 8.}}, author = {{Elsenhans, Andreas-Stephan and Klüners, Jürgen and Nicolae, Florin}}, issn = {{0065-1036}}, journal = {{Acta Arithmetica}}, keywords = {{Algebra and Number Theory}}, number = {{3}}, pages = {{217--233}}, publisher = {{Institute of Mathematics, Polish Academy of Sciences}}, title = {{{Imaginary quadratic number fields with class groups of small exponent}}}, doi = {{10.4064/aa180220-20-3}}, volume = {{193}}, year = {{2020}}, } @inbook{35811, author = {{Biehler, Rolf and Durand-Guerrier, Viviane}}, booktitle = {{Proceedings of the Third Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2020, 12-19 September 2020)}}, editor = {{Hausberger, T. and Bosch, M. and Chelloughi, F.}}, keywords = {{Number Theory, Algebra, Discrete Mathematics, Logic, Research in University Mathematics Edcuation}}, pages = {{283--287}}, publisher = {{University of Carthage and INDRUM}}, title = {{{University Mathematics Didactic Research on Number Theory, Algebra, Discrete Mathematics, Logic}}}, year = {{2020}}, } @article{34841, abstract = {{We give an exact formula for the number of G-extensions of local function fields Fq((t)) for finite abelian groups G up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant. }}, author = {{Klüners, Jürgen and Müller, Raphael}}, issn = {{0022-314X}}, journal = {{Journal of Number Theory}}, keywords = {{Algebra and Number Theory}}, pages = {{311--322}}, publisher = {{Elsevier BV}}, title = {{{The conductor density of local function fields with abelian Galois group}}}, doi = {{10.1016/j.jnt.2019.11.007}}, volume = {{212}}, year = {{2020}}, } @article{34917, abstract = {{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).}}, author = {{Kirschmer, Markus and Nebe, Gabriele}}, issn = {{1793-0421}}, journal = {{International Journal of Number Theory}}, keywords = {{Algebra and Number Theory}}, number = {{02}}, pages = {{309--325}}, publisher = {{World Scientific Pub Co Pte Lt}}, title = {{{Quaternary quadratic lattices over number fields}}}, doi = {{10.1142/s1793042119500131}}, volume = {{15}}, year = {{2019}}, } @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{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}}, } @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}}, } @inproceedings{1590, abstract = {{We present the submatrix method, a highly parallelizable method for the approximate calculation of inverse p-th roots of large sparse symmetric matrices which are required in different scientific applications. Following the idea of Approximate Computing, we allow imprecision in the final result in order to utilize the sparsity of the input matrix and to allow massively parallel execution. For an n x n matrix, the proposed algorithm allows to distribute the calculations over n nodes with only little communication overhead. The result matrix exhibits the same sparsity pattern as the input matrix, allowing for efficient reuse of allocated data structures. We evaluate the algorithm with respect to the error that it introduces into calculated results, as well as its performance and scalability. We demonstrate that the error is relatively limited for well-conditioned matrices and that results are still valuable for error-resilient applications like preconditioning even for ill-conditioned matrices. We discuss the execution time and scaling of the algorithm on a theoretical level and present a distributed implementation of the algorithm using MPI and OpenMP. We demonstrate the scalability of this implementation by running it on a high-performance compute cluster comprised of 1024 CPU cores, showing a speedup of 665x compared to single-threaded execution.}}, author = {{Lass, Michael and Mohr, Stephan and Wiebeler, Hendrik and Kühne, Thomas and Plessl, Christian}}, booktitle = {{Proc. Platform for Advanced Scientific Computing (PASC) Conference}}, isbn = {{978-1-4503-5891-0/18/07}}, keywords = {{approximate computing, linear algebra, matrix inversion, matrix p-th roots, numeric algorithm, parallel computing}}, location = {{Basel, Switzerland}}, publisher = {{ACM}}, title = {{{A Massively Parallel Algorithm for the Approximate Calculation of Inverse p-th Roots of Large Sparse Matrices}}}, doi = {{10.1145/3218176.3218231}}, year = {{2018}}, } @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{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}}, } @article{42793, abstract = {{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.}}, author = {{Kirschmer, Markus}}, issn = {{0022-314X}}, journal = {{Journal of Number Theory}}, keywords = {{Algebra and Number Theory}}, pages = {{375--393}}, publisher = {{Elsevier BV}}, title = {{{One-class genera of maximal integral quadratic forms}}}, doi = {{10.1016/j.jnt.2013.10.007}}, volume = {{136}}, year = {{2014}}, } @article{37672, abstract = {{AbstractLet ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{C} }^{n} $. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $ \mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H} $ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski.}}, author = {{Rösler, Margit and Koornwinder, Tom and Voit, Michael}}, issn = {{0010-437X}}, journal = {{Compositio Mathematica}}, keywords = {{Algebra and Number Theory}}, number = {{8}}, pages = {{1381--1400}}, publisher = {{Wiley}}, title = {{{Limit transition between hypergeometric functions of type BC and type A}}}, doi = {{10.1112/s0010437x13007045}}, volume = {{149}}, year = {{2013}}, } @article{42797, abstract = {{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. }}, author = {{Kirschmer, Markus}}, issn = {{0025-5718}}, journal = {{Mathematics of Computation}}, keywords = {{Applied Mathematics, Computational Mathematics, Algebra and Number Theory}}, number = {{279}}, pages = {{1619--1634}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{A normal form for definite quadratic forms over $\mathbb{F}_{q}[t]$}}}, doi = {{10.1090/s0025-5718-2011-02570-6}}, volume = {{81}}, year = {{2012}}, }