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# Publikationen von Prof. Dr. Jürgen Klüners

## Preprints:

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Chow groups of orders in number fields

M. Kirschmer, J. Klüners, in: arXiv:2208.14688, 2022

We discuss various connections between ideal classes, divisors, Picard and Chow groups of orders in number fields. As a result of these, we give a method to compute Chow groups of such orders and show that there are infinitely many number fields which contain orders with trivial Chow groups.

Computing quadratic subfields of number fields

A. Elsenhans, J. Klüners, in: arXiv:1907.13383, 2019

Given a number field, it is an important question in algorithmic number theory to determine all its subfields. If the search is restricted to abelian subfields, one can try to determine them by using class field theory. For this, it is necessary to know the ramified primes. We show that the ramified primes of the subfield can be computed efficiently. Using this information we give algorithms to determine all the quadratic and the cyclic cubic subfields of the initial field. The approach generalises to cyclic subfields of prime degree. In the case of quadratic subfields, our approach is much faster than other methods.

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## Papers:

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ℓ-torsion bounds for the class group of number fields with an ℓ-group as Galois group

J. Klüners, J. Wang, Proceedings of the American Mathematical Society (2022), 150(7), pp. 2793-2805

We describe the relations among the ℓ-torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent in some sense. Altogether, the three conjectures are equivalent for the class of solvable groups. We then prove the ℓ-torsion conjecture for ℓ-groups and the other two conjectures for nilpotent groups.

The asymptotics of nilpotent Galois groups

J. Klüners, Acta Arithmetica (2022), 204(2), pp. 165-184

We prove an upper bound for the asymptotics of counting functions of number fields with nilpotent Galois groups.

Imaginary multiquadratic number fields with class group of exponent \$3\$ and \$5\$

J. Klüners, T. Komatsu, Mathematics of Computation (2021), 90(329), pp. 1483-1497

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.

Imaginary quadratic number fields with class groups of small exponent

A. Elsenhans, J. Klüners, F. Nicolae, Acta Arithmetica (2020), 193(3), pp. 217-233

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.

The conductor density of local function fields with abelian Galois group

J. Klüners, R. Müller, Journal of Number Theory (2020), 212, pp. 311-322

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.

Computing subfields of number fields and applications to Galois group computations

A. Elsenhans, J. Klüners, Journal of Symbolic Computation (2018), 93, pp. 1-20

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.

Are number fields determined by Artin L-functions?

J. Klüners, F. Nicolae, Journal of Number Theory (2016), 167, pp. 161-168

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.

Computation of Galois groups of rational polynomials

C. Fieker, J. Klüners, LMS Journal of Computation and Mathematics (2014), 17(1), pp. 141-158

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<G, a G-relative H-invariant, that is a multivariate polynomial F that is H-invariant, but not G-invariant. While generic, theoretical methods are known to find such F, in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.

The Distribution of Number Fields with Wreath Products as Galois Groups

J. Klüners, International Journal of Number Theory (2012), 08(03), pp. 845-858

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.

Weighted Distribution of the 4-rank of Class Groups and Applications

Fouvry, J. Klüners, International Mathematics Research Notices (2011), 2011(16), pp. 3618-3656

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.

Generating subfields

M. van Hoeij, J. Klüners, A. Novocin, Journal of Symbolic Computation (2011), 52, pp. 17-34

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.

On the negative Pell equation

Fouvry, J. Klüners, Annals of Mathematics (2010), 172(3), pp. 2035-2104

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.

The parity of the period of the continued fraction of d

Fouvry, J. Klüners, Proceedings of the London Mathematical Society (2010), 101(2), pp. 337-391

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.

On the Spiegelungssatz for the 4-rank

Fouvry, J. Klüners, Algebra &amp; Number Theory (2010), 4(5), pp. 493-508

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.

Factoring polynomials over global fields

K. Belabas, M. van Hoeij, J. Klüners, A. Steel, Journal de Théorie des Nombres de Bordeaux (2009), 21(1), pp. 15-39

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.

The van Hoeij Algorithm for Factoring Polynomials

J. Klüners, in: The LLL Algorithm, Springer Berlin Heidelberg, 2009

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.

Counting nilpotent Galois extensions

J. Klüners, G. Malle, Journal für die reine und angewandte Mathematik (Crelles Journal) (2006), 2004(572), pp. 1-26

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.

Asymptotics of number fields and the Cohen–Lenstra heuristics

J. Klüners, Journal de Théorie des Nombres de Bordeaux (2006), 18(3), pp. 607-615

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.

On the 4-rank of class groups of quadratic number fields

Fouvry, J. Klüners, Inventiones mathematicae (2006), 167(3), pp. 455-513

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.

Cohen–Lenstra Heuristics of Quadratic Number Fields

Fouvry, J. Klüners, in: Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2006

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.

The number of S₄-fields with given discriminant

J. Klüners, Acta Arithmetica (2006), 122(2), pp. 185-194

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).

A counter example to Malle's conjecture on the asymptotics of discriminants

J. Klüners, Comptes Rendus Mathematique (2005), 340(6), pp. 411-414

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.

Computing residue class rings and Picard groups of orders

J. Klüners, S. Pauli, Journal of Algebra (2005), 292(1), pp. 47-64

Let K be a global field and O be an order of K. We develop algorithms for the computation of the unit group of residue class rings for ideals O in . As an application we show how to compute the unit group and the Picard group of O provided that we are able to compute the unit group and class group of the maximal order O of K.

Minimal discriminants for fields with small Frobenius groups as Galois groups

C. Fieker, J. Klüners, Journal of Number Theory (2003), 99(2), pp. 318-337

We apply class field theory to the computation of the minimal discriminants for certain solvable groups. In particular, we apply our techniques to small Frobenius groups and all imprimitive degree 8 groups such that the corresponding fields have only a degree 2 and no degree 4 subfield.

Algorithms for function fields

J. Klüners, Experiment. Math. (2002), 11(2), pp. 171-181

Let {\ASIE K}\,/{\small \ℚ}({\ASIE t \!}) be a finite extension. We describe algorithms for computing subfields and automorphisms of {\ASIE K}\,/{\small \ℚ}({\ASIE t }\!). As an application we give an algorithm for finding decompositions of rational functions in {\small \ℚ(α)}. We also present an algorithm which decides if an extension {\ASIE L}\,/{\small \ℚ}({\ASIE t \!}) is a subfield of {\ASIE K}. In case [{\ASIE K : \;}{\small\ℚ}({\ASIE t \!})] = [{\ASIE L : \;}{\small \ℚ}({\ASIE t \!})] we obtain a {\small \ℚ}({\ASIE t \!})-isomorphism test. Furthermore, we describe an algorithm which computes subfields of the normal closure of {\ASIE K}\,/{\small \ℚ}({\ASIE t \!}).

A Database for Field Extensions of the Rationals

J. Klüners, G. Malle, LMS Journal of Computation and Mathematics (2001), 4, pp. 182-196

This paper announces the creation of a database for number fields. It describes the contents and the methods of access, indicates the origin of the polynomials, and formulates the aims of this collection of fields.

Galois Group Computation for Rational Polynomials

K. Geissler, J. Klüners, Journal of Symbolic Computation (2000), 30(6), pp. 653-674

We describe methods for the computation of Galois groups of univariate polynomials over the rationals which we have implemented up to degree 15. These methods are based on Stauduhar’s algorithm. All computations are done in unramified p -adic extensions. For imprimitive groups we give an improvement using subfields. In the primitive case we use known subgroups of the Galois group together with a combination of Stauduhar’s method and the absolute resolvent method.

Computing Local Artin Maps, and Solvability of Norm Equations

V. Acciaro, J. Klüners, Journal of Symbolic Computation (2000), 30(3), pp. 239-252

Let L = K(α) be an Abelian extension of degree n of a number field K, given by the minimal polynomial of α over K. We describe an algorithm for computing the local Artin map associated with the extension L / K at a finite or infinite prime v of K. We apply this algorithm to decide if a nonzero a ∈ K is a norm from L, assuming that L / K is cyclic.

Explicit Galois Realization of Transitive Groups of Degree up to 15

J. Klüners, G. Malle, Journal of Symbolic Computation (2000), 30(6), pp. 675-716

We describe methods for the construction of polynomials with certain types of Galois groups. As an application we deduce that all transitive groups G up to degree 15 occur as Galois groups of regular extensions of ℚ (t), and in each case compute a polynomial f ∈ ℚ [ x ] with Gal(f) = G.

A Polynomial with Galois GroupSL2(11)

J. Klüners, Journal of Symbolic Computation (2000), 30(6), pp. 733-737

We compute a polynomial with Galois group SL₂(11) over ℚ. Furthermore we prove that SL₂(11) is the Galois group of a regular extension of ℚ (t).

On Polynomial Decompositions

J. Klüners, Journal of Symbolic Computation (1999), 27(3), pp. 261-269

We present a new polynomial decomposition which generalizes the functional and homogeneous bivariate decomposition of irreducible monic polynomials in one variable over the rationals. With these decompositions it is possible to calculate the roots of an imprimitive polynomial by solving polynomial equations of lower degree.

Computing Automorphisms of Abelian Number Fields

J. Klüners, V. Acciaro, Mathematics of Computation (1999), 68(227), pp. 1179-1186

Let L = ℚ(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x) of α over L. In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of α, which is based on p-adic techniques. Given mα(x) and a rational prime p which does not divide the discriminant disc(mα(x)) of mα(x), the algorithm computes the Frobenius automorphism of p in time polynomial in the size of p and in the size of mα(x). By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of α.

On computing subfields. A detailed description of the algorithm

J. Klüners, Journal de Theorie des Nombres de Bordeaux (1998), 10(2), pp. 243-271

Let ℚ(α) be an algebraic number field given by the minimal polynomial f of α. We want to determine all subfields ℚ(β) ⊂ Q(α) of given degree. It is convenient to describe each subfield by a pair (g, h) ∈ Z [t] x ℚ[t] such that g is the minimal polynomial of β = h(α) . There is a bijection between the block systems of the Galois group of f and the subfields of ℚ(α). These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding subfield using p- adic methods. We give a detailed description for all parts of the algorithm.

KANT V4

M. DABERKOW, C. FIEKER, J. Klüners, M. POHST, K. ROEGNER, M. SCHÖRNIG, K. WILDANGER, Journal of Symbolic Computation (1997), 24(3-4), pp. 267-283

The software packageKANT V4for computations in algebraic number fields is now available in version 4. In addition a new user interface has been released. We will outline the features of this new software package.

On Computing Subfields

J. Klüners, M. Pohst, Journal of Symbolic Computation (1997), 24(3-4), pp. 385-397

The purpose of this article is to determine all subfields ℚ(β) of fixed degree of a given algebraic number field ℚ(α). It is convenient to describe each subfield by a pair (h,g) of polynomials in ℚ[t] resp. Z[t] such thatgis the minimal polynomial of β = h(α). The computations are done in unramifiedp-adic extensions and use information concerning subgroups of the Galois group of the normal closure of ℚ(α) obtained from the van der Waerden criterion.

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