@inbook{42805, abstract = {{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.}}, author = {{Kirschmer, Markus and Mertens, Michael H.}}, booktitle = {{Integers}}, isbn = {{9783110298116}}, publisher = {{DE GRUYTER}}, title = {{{On an analogue to the Lucas-Lehmer-Riesel test using elliptic curves}}}, doi = {{10.1515/9783110298161.212}}, year = {{2013}}, } @article{42796, abstract = {{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.}}, author = {{Lorch, David and Kirschmer, Markus}}, issn = {{1461-1570}}, journal = {{LMS Journal of Computation and Mathematics}}, keywords = {{Computational Theory and Mathematics, General Mathematics}}, pages = {{172--186}}, publisher = {{Wiley}}, title = {{{Single-class genera of positive integral lattices}}}, doi = {{10.1112/s1461157013000107}}, volume = {{16}}, year = {{2013}}, } @article{34847, abstract = {{Let G be a wreath product of the form C₂ ≀ H, where C₂ is the cyclic group of order 2. Under mild conditions for H we determine the asymptotic behavior of the counting functions for number fields K/k with Galois group G and bounded discriminant. Those counting functions grow linearly with the norm of the discriminant and this result coincides with a conjecture of Malle. Up to a constant factor these groups have the same asymptotic behavior as the conjectured one for symmetric groups. }}, author = {{Klüners, Jürgen}}, issn = {{1793-0421}}, journal = {{International Journal of Number Theory}}, number = {{03}}, pages = {{845--858}}, publisher = {{World Scientific Pub Co Pte Lt}}, title = {{{The Distribution of Number Fields with Wreath Products as Galois Groups }}}, doi = {{10.1142/s1793042112500492}}, volume = {{08}}, year = {{2012}}, } @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}}, } @article{34885, abstract = {{We prove that the distribution of the values of the 4-rank of ideal class groups of quadratic fields is not affected when it is weighted by a divisor type function. We then give several applications concerning a new lower bound of the sums of class numbers of real quadratic fields with discriminant less than a bound tending to infinity and several questions of P. Sarnak concerning reciprocal geodesics.}}, author = {{Fouvry, Étienne and Klüners, Jürgen}}, issn = {{1687-0247}}, journal = {{International Mathematics Research Notices}}, keywords = {{General Mathematics}}, number = {{16}}, pages = {{3618--3656}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Weighted Distribution of the 4-rank of Class Groups and Applications}}}, doi = {{10.1093/imrn/rnq223}}, volume = {{2011}}, year = {{2011}}, } @article{34846, abstract = {{Given a field extension K/k of degree n we are interested in finding the subfields of K containing k. There can be more than polynomially many subfields. We introduce the notion of generating subfields, a set of up to n subfields whose intersections give the rest. We provide an efficient algorithm which uses linear algebra in k or lattice reduction along with factorization in any extension of K. Implementations show that previously difficult cases can now be handled.}}, author = {{van Hoeij, Mark and Klüners, Jürgen and Novocin, Andrew}}, issn = {{0747-7171}}, journal = {{Journal of Symbolic Computation}}, keywords = {{Computational Mathematics, Algebra and Number Theory}}, pages = {{17--34}}, publisher = {{Elsevier BV}}, title = {{{Generating subfields}}}, doi = {{10.1016/j.jsc.2012.05.010}}, volume = {{52}}, year = {{2011}}, } @article{42798, abstract = {{This paper classifies the maximal finite subgroups of SP₂ₙ(Q) for 1⩽n⩽11 up to GL₂ₙ(Q) conjugacy in .}}, author = {{Kirschmer, Markus}}, issn = {{1058-6458}}, journal = {{Experimental Mathematics}}, keywords = {{General Mathematics}}, number = {{2}}, pages = {{217--228}}, publisher = {{Informa UK Limited}}, title = {{{Finite Symplectic Matrix Groups}}}, doi = {{10.1080/10586458.2011.564964}}, volume = {{20}}, year = {{2011}}, } @article{34886, abstract = {{We give asymptotic upper and lower bounds for the number of squarefree d (0 < d ≤ X) such that the equation x² − dy²= −1 is solvable. These estimates, as usual, can equivalently be interpreted in terms of real quadratic fields with a fundamental unit with norm −1 and give strong evidence in the direction of a conjecture due to P. Stevenhagen.}}, author = {{Fouvry, Étienne and Klüners, Jürgen}}, issn = {{0003-486X}}, journal = {{Annals of Mathematics}}, keywords = {{Statistics, Probability and Uncertainty, Mathematics (miscellaneous)}}, number = {{3}}, pages = {{2035--2104}}, publisher = {{Annals of Mathematics}}, title = {{{On the negative Pell equation}}}, doi = {{10.4007/annals.2010.172.2035}}, volume = {{172}}, year = {{2010}}, } @article{34888, abstract = {{We call a positive square-free integer d special, if d is not divisible by primes congruent to 3 mod 4. We show that the period of the expansion of in continued fractions is asymptotically more often odd than even, when we restrict to special integers. We note that this period is always even for a non-special square-free integer d. It is well known that the above period is odd if and only if the negative Pell equation x²−dy²=−1 is solvable. The latter problem is solvable if and only if the narrow and the ordinary class groups of ℚ(√d) are equal. In a prior work we fully described the asymptotics of the 4-ranks of those class groups. Here we get the first non-trivial results about the asymptotic behavior of the 8-rank of the narrow class group. For example, we show that more than 76% of the quadratic fields ℚ(√d), where d is special, have the property that the 8-rank of the narrow class group is zero.}}, author = {{Fouvry, Étienne and Klüners, Jürgen}}, issn = {{0024-6115}}, journal = {{Proceedings of the London Mathematical Society}}, keywords = {{General Mathematics}}, number = {{2}}, pages = {{337--391}}, publisher = {{Wiley}}, title = {{{The parity of the period of the continued fraction of d}}}, doi = {{10.1112/plms/pdp057}}, volume = {{101}}, year = {{2010}}, } @article{34887, abstract = {{Let d be a nonsquare positive integer. We give the value of the natural probability that the narrow ideal class groups of the quadratic fields ℚ(√d) and ℚ(√−d) have the same 4-ranks. }}, author = {{Fouvry, Étienne and Klüners, Jürgen}}, issn = {{1937-0652}}, journal = {{Algebra & Number Theory}}, keywords = {{Algebra and Number Theory}}, number = {{5}}, pages = {{493--508}}, publisher = {{Mathematical Sciences Publishers}}, title = {{{On the Spiegelungssatz for the 4-rank}}}, doi = {{10.2140/ant.2010.4.493}}, volume = {{4}}, year = {{2010}}, } @article{42803, abstract = {{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.}}, author = {{Kirschmer, Markus and Voight, John}}, issn = {{0097-5397}}, journal = {{SIAM Journal on Computing}}, keywords = {{General Mathematics, General Computer Science}}, number = {{5}}, pages = {{1714--1747}}, publisher = {{Society for Industrial & Applied Mathematics (SIAM)}}, title = {{{Algorithmic Enumeration of Ideal Classes for Quaternion Orders}}}, doi = {{10.1137/080734467}}, volume = {{39}}, year = {{2010}}, } @article{34889, abstract = {{We prove that van Hoeij’s original algorithm to factor univariate polynomials over the rationals runs in polynomial time, as well as natural variants. In particular, our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.}}, author = {{Belabas, Karim and van Hoeij, Mark and Klüners, Jürgen and Steel, Allan}}, issn = {{1246-7405}}, journal = {{Journal de Théorie des Nombres de Bordeaux}}, keywords = {{Algebra and Number Theory}}, number = {{1}}, pages = {{15--39}}, publisher = {{Cellule MathDoc/CEDRAM}}, title = {{{Factoring polynomials over global fields}}}, doi = {{10.5802/jtnb.655}}, volume = {{21}}, year = {{2009}}, } @inbook{35959, abstract = {{In this survey, we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial problem that occurs in the Zassenhaus algorithm is reduced to a very special knapsack problem. In case of rational polynomials, this knapsack problem can be very efficiently solved by the LLL algorithm. This gives a polynomial time algorithm, which also works very well in practice.}}, author = {{Klüners, Jürgen}}, booktitle = {{The LLL Algorithm}}, isbn = {{9783642022944}}, issn = {{1619-7100}}, publisher = {{Springer Berlin Heidelberg}}, title = {{{The van Hoeij Algorithm for Factoring Polynomials}}}, doi = {{10.1007/978-3-642-02295-1_8}}, year = {{2009}}, } @phdthesis{43453, abstract = {{Die invarianten Formen aus dem Anhang sind hier verfügbar:http://www.math.rwth-aachen.de/homes/Markus.Kirschmer/symplectic/}}, author = {{Kirschmer, Markus}}, pages = {{149}}, title = {{{Finite symplectic matrix groups (Dissertation)}}}, year = {{2009}}, } @article{34895, abstract = {{We obtain strong information on the asymptotic behaviour of the counting function for nilpotent Galois extensions with bounded discriminant of arbitrary number fields. This extends previous investigations for the case of abelian groups. In particular, the result confirms a conjecture by the second author on this function for arbitrary groups in the nilpotent case. We further prove compatibility of the conjecture with taking wreath products with the cyclic group of order 2 and give examples in degree up to 8. }}, author = {{Klüners, Jürgen and Malle, G.}}, issn = {{0075-4102}}, journal = {{Journal für die reine und angewandte Mathematik (Crelles Journal)}}, keywords = {{Applied Mathematics, General Mathematics}}, number = {{572}}, pages = {{1--26}}, publisher = {{Walter de Gruyter GmbH}}, title = {{{Counting nilpotent Galois extensions}}}, doi = {{10.1515/crll.2004.050}}, volume = {{2004}}, year = {{2006}}, } @article{34891, abstract = {{We study the asymptotics conjecture of Malle for dihedral groups Dℓ of order 2ℓ, where ℓ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen--Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds. }}, author = {{Klüners, Jürgen}}, issn = {{1246-7405}}, journal = {{Journal de Théorie des Nombres de Bordeaux}}, keywords = {{Algebra and Number Theory}}, number = {{3}}, pages = {{607--615}}, publisher = {{Cellule MathDoc/CEDRAM}}, title = {{{Asymptotics of number fields and the Cohen–Lenstra heuristics}}}, doi = {{10.5802/jtnb.561}}, volume = {{18}}, year = {{2006}}, } @article{34890, abstract = {{We prove that the 4-rank of class groups of quadratic number fields behaves as predicted in an extension due to Gerth of the Cohen–Lenstra heuristics. }}, author = {{Fouvry, Étienne and Klüners, Jürgen}}, issn = {{0020-9910}}, journal = {{Inventiones mathematicae}}, keywords = {{General Mathematics}}, number = {{3}}, pages = {{455--513}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{On the 4-rank of class groups of quadratic number fields}}}, doi = {{10.1007/s00222-006-0021-2}}, volume = {{167}}, year = {{2006}}, } @inbook{35958, abstract = {{We establish a link between some heuristic asymptotic formulas (due to Cohen and Lenstra) concerning the moments of the p–part of the class groups of quadratic fields and formulas giving the frequency of the values of the p–rank of these class groups.}}, author = {{Fouvry, Étienne and Klüners, Jürgen}}, booktitle = {{Lecture Notes in Computer Science}}, isbn = {{9783540360759}}, issn = {{0302-9743}}, publisher = {{Springer Berlin Heidelberg}}, title = {{{Cohen–Lenstra Heuristics of Quadratic Number Fields}}}, doi = {{10.1007/11792086_4}}, year = {{2006}}, } @article{34892, abstract = {{We prove that the number of quartic S4--extensions of the rationals of given discriminant d is $O_\eps(d^{1/2+\eps})$ for all $\eps>0$. For a prime number p we derive that the dimension of the space of octahedral modular forms of weight 1 and conductor p or p² is bounded above by O(p¹/²log(p)²). }}, author = {{Klüners, Jürgen}}, issn = {{0065-1036}}, journal = {{Acta Arithmetica}}, keywords = {{Algebra and Number Theory}}, number = {{2}}, pages = {{185--194}}, publisher = {{Institute of Mathematics, Polish Academy of Sciences}}, title = {{{The number of S₄-fields with given discriminant}}}, doi = {{10.4064/aa122-2-3}}, volume = {{122}}, year = {{2006}}, } @article{34894, abstract = {{In this Note we give a counter example to a conjecture of Malle which predicts the asymptotic behavior of the counting functions for field extensions with given Galois group and bounded discriminant. }}, author = {{Klüners, Jürgen}}, issn = {{1631-073X}}, journal = {{Comptes Rendus Mathematique}}, keywords = {{General Mathematics}}, number = {{6}}, pages = {{411--414}}, publisher = {{Elsevier BV}}, title = {{{A counter example to Malle's conjecture on the asymptotics of discriminants}}}, doi = {{10.1016/j.crma.2005.02.010}}, volume = {{340}}, year = {{2005}}, }