---
_id: '42805'
abstract:
- lang: eng
  text: 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:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: Michael H.
  full_name: Mertens, Michael H.
  last_name: Mertens
citation:
  ama: 'Kirschmer M, Mertens MH. On an analogue to the Lucas-Lehmer-Riesel test using
    elliptic curves. In: <i>Integers</i>. DE GRUYTER; 2013. doi:<a href="https://doi.org/10.1515/9783110298161.212">10.1515/9783110298161.212</a>'
  apa: Kirschmer, M., &#38; Mertens, M. H. (2013). On an analogue to the Lucas-Lehmer-Riesel
    test using elliptic curves. In <i>Integers</i>. DE GRUYTER. <a href="https://doi.org/10.1515/9783110298161.212">https://doi.org/10.1515/9783110298161.212</a>
  bibtex: '@inbook{Kirschmer_Mertens_2013, title={On an analogue to the Lucas-Lehmer-Riesel
    test using elliptic curves}, DOI={<a href="https://doi.org/10.1515/9783110298161.212">10.1515/9783110298161.212</a>},
    booktitle={Integers}, publisher={DE GRUYTER}, author={Kirschmer, Markus and Mertens,
    Michael H.}, year={2013} }'
  chicago: Kirschmer, Markus, and Michael H. Mertens. “On an Analogue to the Lucas-Lehmer-Riesel
    Test Using Elliptic Curves.” In <i>Integers</i>. DE GRUYTER, 2013. <a href="https://doi.org/10.1515/9783110298161.212">https://doi.org/10.1515/9783110298161.212</a>.
  ieee: M. Kirschmer and M. H. Mertens, “On an analogue to the Lucas-Lehmer-Riesel
    test using elliptic curves,” in <i>Integers</i>, DE GRUYTER, 2013.
  mla: Kirschmer, Markus, and Michael H. Mertens. “On an Analogue to the Lucas-Lehmer-Riesel
    Test Using Elliptic Curves.” <i>Integers</i>, DE GRUYTER, 2013, doi:<a href="https://doi.org/10.1515/9783110298161.212">10.1515/9783110298161.212</a>.
  short: 'M. Kirschmer, M.H. Mertens, in: Integers, DE GRUYTER, 2013.'
date_created: 2023-03-07T08:51:46Z
date_updated: 2023-04-04T09:17:32Z
department:
- _id: '102'
doi: 10.1515/9783110298161.212
extern: '1'
language:
- iso: eng
publication: Integers
publication_identifier:
  isbn:
  - '9783110298116'
publication_status: published
publisher: DE GRUYTER
status: public
title: On an analogue to the Lucas-Lehmer-Riesel test using elliptic curves
type: book_chapter
user_id: '93826'
year: '2013'
...
---
_id: '42796'
abstract:
- lang: eng
  text: "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.\r\n\r\nWe
    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:
- first_name: David
  full_name: Lorch, David
  last_name: Lorch
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
citation:
  ama: Lorch D, Kirschmer M. Single-class genera of positive integral lattices. <i>LMS
    Journal of Computation and Mathematics</i>. 2013;16:172-186. doi:<a href="https://doi.org/10.1112/s1461157013000107">10.1112/s1461157013000107</a>
  apa: Lorch, D., &#38; Kirschmer, M. (2013). Single-class genera of positive integral
    lattices. <i>LMS Journal of Computation and Mathematics</i>, <i>16</i>, 172–186.
    <a href="https://doi.org/10.1112/s1461157013000107">https://doi.org/10.1112/s1461157013000107</a>
  bibtex: '@article{Lorch_Kirschmer_2013, title={Single-class genera of positive integral
    lattices}, volume={16}, DOI={<a href="https://doi.org/10.1112/s1461157013000107">10.1112/s1461157013000107</a>},
    journal={LMS Journal of Computation and Mathematics}, publisher={Wiley}, author={Lorch,
    David and Kirschmer, Markus}, year={2013}, pages={172–186} }'
  chicago: 'Lorch, David, and Markus Kirschmer. “Single-Class Genera of Positive Integral
    Lattices.” <i>LMS Journal of Computation and Mathematics</i> 16 (2013): 172–86.
    <a href="https://doi.org/10.1112/s1461157013000107">https://doi.org/10.1112/s1461157013000107</a>.'
  ieee: 'D. Lorch and M. Kirschmer, “Single-class genera of positive integral lattices,”
    <i>LMS Journal of Computation and Mathematics</i>, vol. 16, pp. 172–186, 2013,
    doi: <a href="https://doi.org/10.1112/s1461157013000107">10.1112/s1461157013000107</a>.'
  mla: Lorch, David, and Markus Kirschmer. “Single-Class Genera of Positive Integral
    Lattices.” <i>LMS Journal of Computation and Mathematics</i>, vol. 16, Wiley,
    2013, pp. 172–86, doi:<a href="https://doi.org/10.1112/s1461157013000107">10.1112/s1461157013000107</a>.
  short: D. Lorch, M. Kirschmer, LMS Journal of Computation and Mathematics 16 (2013)
    172–186.
date_created: 2023-03-07T08:34:28Z
date_updated: 2023-04-04T07:57:04Z
department:
- _id: '102'
doi: 10.1112/s1461157013000107
extern: '1'
intvolume: '        16'
keyword:
- Computational Theory and Mathematics
- General Mathematics
language:
- iso: eng
page: 172-186
publication: LMS Journal of Computation and Mathematics
publication_identifier:
  issn:
  - 1461-1570
publication_status: published
publisher: Wiley
status: public
title: Single-class genera of positive integral lattices
type: journal_article
user_id: '93826'
volume: 16
year: '2013'
...
---
_id: '34847'
abstract:
- lang: eng
  text: '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:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. The Distribution of Number Fields with Wreath Products as Galois
    Groups . <i>International Journal of Number Theory</i>. 2012;08(03):845-858. doi:<a
    href="https://doi.org/10.1142/s1793042112500492">10.1142/s1793042112500492</a>
  apa: Klüners, J. (2012). The Distribution of Number Fields with Wreath Products
    as Galois Groups . <i>International Journal of Number Theory</i>, <i>08</i>(03),
    845–858. <a href="https://doi.org/10.1142/s1793042112500492">https://doi.org/10.1142/s1793042112500492</a>
  bibtex: '@article{Klüners_2012, title={The Distribution of Number Fields with Wreath
    Products as Galois Groups }, volume={08}, DOI={<a href="https://doi.org/10.1142/s1793042112500492">10.1142/s1793042112500492</a>},
    number={03}, journal={International Journal of Number Theory}, publisher={World
    Scientific Pub Co Pte Lt}, author={Klüners, Jürgen}, year={2012}, pages={845–858}
    }'
  chicago: 'Klüners, Jürgen. “The Distribution of Number Fields with Wreath Products
    as Galois Groups .” <i>International Journal of Number Theory</i> 08, no. 03 (2012):
    845–58. <a href="https://doi.org/10.1142/s1793042112500492">https://doi.org/10.1142/s1793042112500492</a>.'
  ieee: 'J. Klüners, “The Distribution of Number Fields with Wreath Products as Galois
    Groups ,” <i>International Journal of Number Theory</i>, vol. 08, no. 03, pp.
    845–858, 2012, doi: <a href="https://doi.org/10.1142/s1793042112500492">10.1142/s1793042112500492</a>.'
  mla: Klüners, Jürgen. “The Distribution of Number Fields with Wreath Products as
    Galois Groups .” <i>International Journal of Number Theory</i>, vol. 08, no. 03,
    World Scientific Pub Co Pte Lt, 2012, pp. 845–58, doi:<a href="https://doi.org/10.1142/s1793042112500492">10.1142/s1793042112500492</a>.
  short: J. Klüners, International Journal of Number Theory 08 (2012) 845–858.
date_created: 2022-12-22T10:55:47Z
date_updated: 2023-03-02T14:10:38Z
department:
- _id: '102'
doi: 10.1142/s1793042112500492
external_id:
  arxiv:
  - '1108.5597 '
intvolume: '         8'
issue: '03'
language:
- iso: eng
page: 845-858
publication: International Journal of Number Theory
publication_identifier:
  issn:
  - 1793-0421
  - 1793-7310
publication_status: published
publisher: World Scientific Pub Co Pte Lt
status: public
title: 'The Distribution of Number Fields with Wreath Products as Galois Groups '
type: journal_article
user_id: '21202'
volume: '08'
year: '2012'
...
---
_id: '42797'
abstract:
- lang: eng
  text: '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:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
citation:
  ama: Kirschmer M. A normal form for definite quadratic forms over $\mathbb{F}_{q}[t]$.
    <i>Mathematics of Computation</i>. 2012;81(279):1619-1634. doi:<a href="https://doi.org/10.1090/s0025-5718-2011-02570-6">10.1090/s0025-5718-2011-02570-6</a>
  apa: Kirschmer, M. (2012). A normal form for definite quadratic forms over $\mathbb{F}_{q}[t]$.
    <i>Mathematics of Computation</i>, <i>81</i>(279), 1619–1634. <a href="https://doi.org/10.1090/s0025-5718-2011-02570-6">https://doi.org/10.1090/s0025-5718-2011-02570-6</a>
  bibtex: '@article{Kirschmer_2012, title={A normal form for definite quadratic forms
    over $\mathbb{F}_{q}[t]$}, volume={81}, DOI={<a href="https://doi.org/10.1090/s0025-5718-2011-02570-6">10.1090/s0025-5718-2011-02570-6</a>},
    number={279}, journal={Mathematics of Computation}, publisher={American Mathematical
    Society (AMS)}, author={Kirschmer, Markus}, year={2012}, pages={1619–1634} }'
  chicago: 'Kirschmer, Markus. “A Normal Form for Definite Quadratic Forms over $\mathbb{F}_{q}[t]$.”
    <i>Mathematics of Computation</i> 81, no. 279 (2012): 1619–34. <a href="https://doi.org/10.1090/s0025-5718-2011-02570-6">https://doi.org/10.1090/s0025-5718-2011-02570-6</a>.'
  ieee: 'M. Kirschmer, “A normal form for definite quadratic forms over $\mathbb{F}_{q}[t]$,”
    <i>Mathematics of Computation</i>, vol. 81, no. 279, pp. 1619–1634, 2012, doi:
    <a href="https://doi.org/10.1090/s0025-5718-2011-02570-6">10.1090/s0025-5718-2011-02570-6</a>.'
  mla: Kirschmer, Markus. “A Normal Form for Definite Quadratic Forms over $\mathbb{F}_{q}[t]$.”
    <i>Mathematics of Computation</i>, vol. 81, no. 279, American Mathematical Society
    (AMS), 2012, pp. 1619–34, doi:<a href="https://doi.org/10.1090/s0025-5718-2011-02570-6">10.1090/s0025-5718-2011-02570-6</a>.
  short: M. Kirschmer, Mathematics of Computation 81 (2012) 1619–1634.
date_created: 2023-03-07T08:35:56Z
date_updated: 2023-04-04T09:22:22Z
department:
- _id: '102'
doi: 10.1090/s0025-5718-2011-02570-6
extern: '1'
intvolume: '        81'
issue: '279'
keyword:
- Applied Mathematics
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 1619-1634
publication: Mathematics of Computation
publication_identifier:
  issn:
  - 0025-5718
  - 1088-6842
publication_status: published
publisher: American Mathematical Society (AMS)
status: public
title: A normal form for definite quadratic forms over $\mathbb{F}_{q}[t]$
type: journal_article
user_id: '93826'
volume: 81
year: '2012'
...
---
_id: '34885'
abstract:
- lang: eng
  text: 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:
- first_name: Étienne
  full_name: Fouvry, Étienne
  last_name: Fouvry
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Fouvry É, Klüners J. Weighted Distribution of the 4-rank of Class Groups and
    Applications. <i>International Mathematics Research Notices</i>. 2011;2011(16):3618-3656.
    doi:<a href="https://doi.org/10.1093/imrn/rnq223">10.1093/imrn/rnq223</a>
  apa: Fouvry, É., &#38; Klüners, J. (2011). Weighted Distribution of the 4-rank of
    Class Groups and Applications. <i>International Mathematics Research Notices</i>,
    <i>2011</i>(16), 3618–3656. <a href="https://doi.org/10.1093/imrn/rnq223">https://doi.org/10.1093/imrn/rnq223</a>
  bibtex: '@article{Fouvry_Klüners_2011, title={Weighted Distribution of the 4-rank
    of Class Groups and Applications}, volume={2011}, DOI={<a href="https://doi.org/10.1093/imrn/rnq223">10.1093/imrn/rnq223</a>},
    number={16}, journal={International Mathematics Research Notices}, publisher={Oxford
    University Press (OUP)}, author={Fouvry, Étienne and Klüners, Jürgen}, year={2011},
    pages={3618–3656} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “Weighted Distribution of the 4-Rank
    of Class Groups and Applications.” <i>International Mathematics Research Notices</i>
    2011, no. 16 (2011): 3618–56. <a href="https://doi.org/10.1093/imrn/rnq223">https://doi.org/10.1093/imrn/rnq223</a>.'
  ieee: 'É. Fouvry and J. Klüners, “Weighted Distribution of the 4-rank of Class Groups
    and Applications,” <i>International Mathematics Research Notices</i>, vol. 2011,
    no. 16, pp. 3618–3656, 2011, doi: <a href="https://doi.org/10.1093/imrn/rnq223">10.1093/imrn/rnq223</a>.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “Weighted Distribution of the 4-Rank of
    Class Groups and Applications.” <i>International Mathematics Research Notices</i>,
    vol. 2011, no. 16, Oxford University Press (OUP), 2011, pp. 3618–56, doi:<a href="https://doi.org/10.1093/imrn/rnq223">10.1093/imrn/rnq223</a>.
  short: É. Fouvry, J. Klüners, International Mathematics Research Notices 2011 (2011)
    3618–3656.
date_created: 2022-12-23T09:08:00Z
date_updated: 2023-03-06T09:07:46Z
department:
- _id: '102'
doi: 10.1093/imrn/rnq223
intvolume: '      2011'
issue: '16'
keyword:
- General Mathematics
language:
- iso: eng
page: 3618-3656
publication: International Mathematics Research Notices
publication_identifier:
  issn:
  - 1687-0247
  - 1073-7928
publication_status: published
publisher: Oxford University Press (OUP)
status: public
title: Weighted Distribution of the 4-rank of Class Groups and Applications
type: journal_article
user_id: '93826'
volume: 2011
year: '2011'
...
---
_id: '34846'
abstract:
- lang: eng
  text: 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:
- first_name: Mark
  full_name: van Hoeij, Mark
  last_name: van Hoeij
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Andrew
  full_name: Novocin, Andrew
  last_name: Novocin
citation:
  ama: van Hoeij M, Klüners J, Novocin A. Generating subfields. <i>Journal of Symbolic
    Computation</i>. 2011;52:17-34. doi:<a href="https://doi.org/10.1016/j.jsc.2012.05.010">10.1016/j.jsc.2012.05.010</a>
  apa: van Hoeij, M., Klüners, J., &#38; Novocin, A. (2011). Generating subfields.
    <i>Journal of Symbolic Computation</i>, <i>52</i>, 17–34. <a href="https://doi.org/10.1016/j.jsc.2012.05.010">https://doi.org/10.1016/j.jsc.2012.05.010</a>
  bibtex: '@article{van Hoeij_Klüners_Novocin_2011, title={Generating subfields},
    volume={52}, DOI={<a href="https://doi.org/10.1016/j.jsc.2012.05.010">10.1016/j.jsc.2012.05.010</a>},
    journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={van
    Hoeij, Mark and Klüners, Jürgen and Novocin, Andrew}, year={2011}, pages={17–34}
    }'
  chicago: 'Hoeij, Mark van, Jürgen Klüners, and Andrew Novocin. “Generating Subfields.”
    <i>Journal of Symbolic Computation</i> 52 (2011): 17–34. <a href="https://doi.org/10.1016/j.jsc.2012.05.010">https://doi.org/10.1016/j.jsc.2012.05.010</a>.'
  ieee: 'M. van Hoeij, J. Klüners, and A. Novocin, “Generating subfields,” <i>Journal
    of Symbolic Computation</i>, vol. 52, pp. 17–34, 2011, doi: <a href="https://doi.org/10.1016/j.jsc.2012.05.010">10.1016/j.jsc.2012.05.010</a>.'
  mla: van Hoeij, Mark, et al. “Generating Subfields.” <i>Journal of Symbolic Computation</i>,
    vol. 52, Elsevier BV, 2011, pp. 17–34, doi:<a href="https://doi.org/10.1016/j.jsc.2012.05.010">10.1016/j.jsc.2012.05.010</a>.
  short: M. van Hoeij, J. Klüners, A. Novocin, Journal of Symbolic Computation 52
    (2011) 17–34.
date_created: 2022-12-22T10:54:15Z
date_updated: 2023-03-06T09:46:15Z
department:
- _id: '102'
doi: 10.1016/j.jsc.2012.05.010
intvolume: '        52'
keyword:
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 17-34
publication: Journal of Symbolic Computation
publication_identifier:
  issn:
  - 0747-7171
publication_status: published
publisher: Elsevier BV
status: public
title: Generating subfields
type: journal_article
user_id: '93826'
volume: 52
year: '2011'
...
---
_id: '42798'
abstract:
- lang: eng
  text: This paper classifies the maximal finite subgroups of SP₂ₙ(Q) for 1⩽n⩽11 up
    to GL₂ₙ(Q) conjugacy in .
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
citation:
  ama: Kirschmer M. Finite Symplectic Matrix Groups. <i>Experimental Mathematics</i>.
    2011;20(2):217-228. doi:<a href="https://doi.org/10.1080/10586458.2011.564964">10.1080/10586458.2011.564964</a>
  apa: Kirschmer, M. (2011). Finite Symplectic Matrix Groups. <i>Experimental Mathematics</i>,
    <i>20</i>(2), 217–228. <a href="https://doi.org/10.1080/10586458.2011.564964">https://doi.org/10.1080/10586458.2011.564964</a>
  bibtex: '@article{Kirschmer_2011, title={Finite Symplectic Matrix Groups}, volume={20},
    DOI={<a href="https://doi.org/10.1080/10586458.2011.564964">10.1080/10586458.2011.564964</a>},
    number={2}, journal={Experimental Mathematics}, publisher={Informa UK Limited},
    author={Kirschmer, Markus}, year={2011}, pages={217–228} }'
  chicago: 'Kirschmer, Markus. “Finite Symplectic Matrix Groups.” <i>Experimental
    Mathematics</i> 20, no. 2 (2011): 217–28. <a href="https://doi.org/10.1080/10586458.2011.564964">https://doi.org/10.1080/10586458.2011.564964</a>.'
  ieee: 'M. Kirschmer, “Finite Symplectic Matrix Groups,” <i>Experimental Mathematics</i>,
    vol. 20, no. 2, pp. 217–228, 2011, doi: <a href="https://doi.org/10.1080/10586458.2011.564964">10.1080/10586458.2011.564964</a>.'
  mla: Kirschmer, Markus. “Finite Symplectic Matrix Groups.” <i>Experimental Mathematics</i>,
    vol. 20, no. 2, Informa UK Limited, 2011, pp. 217–28, doi:<a href="https://doi.org/10.1080/10586458.2011.564964">10.1080/10586458.2011.564964</a>.
  short: M. Kirschmer, Experimental Mathematics 20 (2011) 217–228.
date_created: 2023-03-07T08:36:46Z
date_updated: 2023-04-04T09:24:42Z
department:
- _id: '102'
doi: 10.1080/10586458.2011.564964
extern: '1'
intvolume: '        20'
issue: '2'
keyword:
- General Mathematics
language:
- iso: eng
page: 217-228
publication: Experimental Mathematics
publication_identifier:
  issn:
  - 1058-6458
  - 1944-950X
publication_status: published
publisher: Informa UK Limited
status: public
title: Finite Symplectic Matrix Groups
type: journal_article
user_id: '93826'
volume: 20
year: '2011'
...
---
_id: '34886'
abstract:
- lang: eng
  text: 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:
- first_name: Étienne
  full_name: Fouvry, Étienne
  last_name: Fouvry
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Fouvry É, Klüners J. On the negative Pell equation. <i>Annals of Mathematics</i>.
    2010;172(3):2035-2104. doi:<a href="https://doi.org/10.4007/annals.2010.172.2035">10.4007/annals.2010.172.2035</a>
  apa: Fouvry, É., &#38; Klüners, J. (2010). On the negative Pell equation. <i>Annals
    of Mathematics</i>, <i>172</i>(3), 2035–2104. <a href="https://doi.org/10.4007/annals.2010.172.2035">https://doi.org/10.4007/annals.2010.172.2035</a>
  bibtex: '@article{Fouvry_Klüners_2010, title={On the negative Pell equation}, volume={172},
    DOI={<a href="https://doi.org/10.4007/annals.2010.172.2035">10.4007/annals.2010.172.2035</a>},
    number={3}, journal={Annals of Mathematics}, publisher={Annals of Mathematics},
    author={Fouvry, Étienne and Klüners, Jürgen}, year={2010}, pages={2035–2104} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “On the Negative Pell Equation.”
    <i>Annals of Mathematics</i> 172, no. 3 (2010): 2035–2104. <a href="https://doi.org/10.4007/annals.2010.172.2035">https://doi.org/10.4007/annals.2010.172.2035</a>.'
  ieee: 'É. Fouvry and J. Klüners, “On the negative Pell equation,” <i>Annals of Mathematics</i>,
    vol. 172, no. 3, pp. 2035–2104, 2010, doi: <a href="https://doi.org/10.4007/annals.2010.172.2035">10.4007/annals.2010.172.2035</a>.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “On the Negative Pell Equation.” <i>Annals
    of Mathematics</i>, vol. 172, no. 3, Annals of Mathematics, 2010, pp. 2035–104,
    doi:<a href="https://doi.org/10.4007/annals.2010.172.2035">10.4007/annals.2010.172.2035</a>.
  short: É. Fouvry, J. Klüners, Annals of Mathematics 172 (2010) 2035–2104.
date_created: 2022-12-23T09:09:02Z
date_updated: 2023-03-06T09:50:37Z
department:
- _id: '102'
doi: 10.4007/annals.2010.172.2035
intvolume: '       172'
issue: '3'
keyword:
- Statistics
- Probability and Uncertainty
- Mathematics (miscellaneous)
language:
- iso: eng
page: 2035-2104
publication: Annals of Mathematics
publication_identifier:
  issn:
  - 0003-486X
publication_status: published
publisher: Annals of Mathematics
status: public
title: On the negative Pell equation
type: journal_article
user_id: '93826'
volume: 172
year: '2010'
...
---
_id: '34888'
abstract:
- lang: eng
  text: 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:
- first_name: Étienne
  full_name: Fouvry, Étienne
  last_name: Fouvry
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Fouvry É, Klüners J. The parity of the period of the continued fraction of
    d. <i>Proceedings of the London Mathematical Society</i>. 2010;101(2):337-391.
    doi:<a href="https://doi.org/10.1112/plms/pdp057">10.1112/plms/pdp057</a>
  apa: Fouvry, É., &#38; Klüners, J. (2010). The parity of the period of the continued
    fraction of d. <i>Proceedings of the London Mathematical Society</i>, <i>101</i>(2),
    337–391. <a href="https://doi.org/10.1112/plms/pdp057">https://doi.org/10.1112/plms/pdp057</a>
  bibtex: '@article{Fouvry_Klüners_2010, title={The parity of the period of the continued
    fraction of d}, volume={101}, DOI={<a href="https://doi.org/10.1112/plms/pdp057">10.1112/plms/pdp057</a>},
    number={2}, journal={Proceedings of the London Mathematical Society}, publisher={Wiley},
    author={Fouvry, Étienne and Klüners, Jürgen}, year={2010}, pages={337–391} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “The Parity of the Period of the
    Continued Fraction of d.” <i>Proceedings of the London Mathematical Society</i>
    101, no. 2 (2010): 337–91. <a href="https://doi.org/10.1112/plms/pdp057">https://doi.org/10.1112/plms/pdp057</a>.'
  ieee: 'É. Fouvry and J. Klüners, “The parity of the period of the continued fraction
    of d,” <i>Proceedings of the London Mathematical Society</i>, vol. 101, no. 2,
    pp. 337–391, 2010, doi: <a href="https://doi.org/10.1112/plms/pdp057">10.1112/plms/pdp057</a>.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “The Parity of the Period of the Continued
    Fraction of d.” <i>Proceedings of the London Mathematical Society</i>, vol. 101,
    no. 2, Wiley, 2010, pp. 337–91, doi:<a href="https://doi.org/10.1112/plms/pdp057">10.1112/plms/pdp057</a>.
  short: É. Fouvry, J. Klüners, Proceedings of the London Mathematical Society 101
    (2010) 337–391.
date_created: 2022-12-23T09:22:49Z
date_updated: 2023-03-06T10:16:54Z
department:
- _id: '102'
doi: 10.1112/plms/pdp057
intvolume: '       101'
issue: '2'
keyword:
- General Mathematics
language:
- iso: eng
page: 337-391
publication: Proceedings of the London Mathematical Society
publication_identifier:
  issn:
  - 0024-6115
publication_status: published
publisher: Wiley
related_material:
  link:
  - relation: confirmation
    url: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=fc7ab412993fd2bf0069de42fbef1ecc69137755
status: public
title: The parity of the period of the continued fraction of d
type: journal_article
user_id: '93826'
volume: 101
year: '2010'
...
---
_id: '34887'
abstract:
- lang: eng
  text: '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:
- first_name: Étienne
  full_name: Fouvry, Étienne
  last_name: Fouvry
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Fouvry É, Klüners J. On the Spiegelungssatz for the 4-rank. <i>Algebra &#38;amp;
    Number Theory</i>. 2010;4(5):493-508. doi:<a href="https://doi.org/10.2140/ant.2010.4.493">10.2140/ant.2010.4.493</a>
  apa: Fouvry, É., &#38; Klüners, J. (2010). On the Spiegelungssatz for the 4-rank.
    <i>Algebra &#38;amp; Number Theory</i>, <i>4</i>(5), 493–508. <a href="https://doi.org/10.2140/ant.2010.4.493">https://doi.org/10.2140/ant.2010.4.493</a>
  bibtex: '@article{Fouvry_Klüners_2010, title={On the Spiegelungssatz for the 4-rank},
    volume={4}, DOI={<a href="https://doi.org/10.2140/ant.2010.4.493">10.2140/ant.2010.4.493</a>},
    number={5}, journal={Algebra &#38;amp; Number Theory}, publisher={Mathematical
    Sciences Publishers}, author={Fouvry, Étienne and Klüners, Jürgen}, year={2010},
    pages={493–508} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “On the Spiegelungssatz for the 4-Rank.”
    <i>Algebra &#38;amp; Number Theory</i> 4, no. 5 (2010): 493–508. <a href="https://doi.org/10.2140/ant.2010.4.493">https://doi.org/10.2140/ant.2010.4.493</a>.'
  ieee: 'É. Fouvry and J. Klüners, “On the Spiegelungssatz for the 4-rank,” <i>Algebra
    &#38;amp; Number Theory</i>, vol. 4, no. 5, pp. 493–508, 2010, doi: <a href="https://doi.org/10.2140/ant.2010.4.493">10.2140/ant.2010.4.493</a>.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “On the Spiegelungssatz for the 4-Rank.”
    <i>Algebra &#38;amp; Number Theory</i>, vol. 4, no. 5, Mathematical Sciences Publishers,
    2010, pp. 493–508, doi:<a href="https://doi.org/10.2140/ant.2010.4.493">10.2140/ant.2010.4.493</a>.
  short: É. Fouvry, J. Klüners, Algebra &#38;amp; Number Theory 4 (2010) 493–508.
date_created: 2022-12-23T09:10:12Z
date_updated: 2023-03-06T10:18:14Z
department:
- _id: '102'
doi: 10.2140/ant.2010.4.493
intvolume: '         4'
issue: '5'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 493-508
publication: Algebra &amp; Number Theory
publication_identifier:
  issn:
  - 1937-0652
publication_status: published
publisher: Mathematical Sciences Publishers
status: public
title: On the Spiegelungssatz for the 4-rank
type: journal_article
user_id: '93826'
volume: 4
year: '2010'
...
---
_id: '42803'
abstract:
- lang: eng
  text: 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:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: John
  full_name: Voight, John
  last_name: Voight
citation:
  ama: Kirschmer M, Voight J. Algorithmic Enumeration of Ideal Classes for Quaternion
    Orders. <i>SIAM Journal on Computing</i>. 2010;39(5):1714-1747. doi:<a href="https://doi.org/10.1137/080734467">10.1137/080734467</a>
  apa: Kirschmer, M., &#38; Voight, J. (2010). Algorithmic Enumeration of Ideal Classes
    for Quaternion Orders. <i>SIAM Journal on Computing</i>, <i>39</i>(5), 1714–1747.
    <a href="https://doi.org/10.1137/080734467">https://doi.org/10.1137/080734467</a>
  bibtex: '@article{Kirschmer_Voight_2010, title={Algorithmic Enumeration of Ideal
    Classes for Quaternion Orders}, volume={39}, DOI={<a href="https://doi.org/10.1137/080734467">10.1137/080734467</a>},
    number={5}, journal={SIAM Journal on Computing}, publisher={Society for Industrial
    &#38; Applied Mathematics (SIAM)}, author={Kirschmer, Markus and Voight, John},
    year={2010}, pages={1714–1747} }'
  chicago: 'Kirschmer, Markus, and John Voight. “Algorithmic Enumeration of Ideal
    Classes for Quaternion Orders.” <i>SIAM Journal on Computing</i> 39, no. 5 (2010):
    1714–47. <a href="https://doi.org/10.1137/080734467">https://doi.org/10.1137/080734467</a>.'
  ieee: 'M. Kirschmer and J. Voight, “Algorithmic Enumeration of Ideal Classes for
    Quaternion Orders,” <i>SIAM Journal on Computing</i>, vol. 39, no. 5, pp. 1714–1747,
    2010, doi: <a href="https://doi.org/10.1137/080734467">10.1137/080734467</a>.'
  mla: Kirschmer, Markus, and John Voight. “Algorithmic Enumeration of Ideal Classes
    for Quaternion Orders.” <i>SIAM Journal on Computing</i>, vol. 39, no. 5, Society
    for Industrial &#38; Applied Mathematics (SIAM), 2010, pp. 1714–47, doi:<a href="https://doi.org/10.1137/080734467">10.1137/080734467</a>.
  short: M. Kirschmer, J. Voight, SIAM Journal on Computing 39 (2010) 1714–1747.
date_created: 2023-03-07T08:49:35Z
date_updated: 2023-04-04T09:25:08Z
department:
- _id: '102'
doi: 10.1137/080734467
extern: '1'
intvolume: '        39'
issue: '5'
keyword:
- General Mathematics
- General Computer Science
language:
- iso: eng
page: 1714-1747
publication: SIAM Journal on Computing
publication_identifier:
  issn:
  - 0097-5397
  - 1095-7111
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: Algorithmic Enumeration of Ideal Classes for Quaternion Orders
type: journal_article
user_id: '93826'
volume: 39
year: '2010'
...
---
_id: '34889'
abstract:
- lang: eng
  text: 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:
- first_name: Karim
  full_name: Belabas, Karim
  last_name: Belabas
- first_name: Mark
  full_name: van Hoeij, Mark
  last_name: van Hoeij
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Allan
  full_name: Steel, Allan
  last_name: Steel
citation:
  ama: Belabas K, van Hoeij M, Klüners J, Steel A. Factoring polynomials over global
    fields. <i>Journal de Théorie des Nombres de Bordeaux</i>. 2009;21(1):15-39. doi:<a
    href="https://doi.org/10.5802/jtnb.655">10.5802/jtnb.655</a>
  apa: Belabas, K., van Hoeij, M., Klüners, J., &#38; Steel, A. (2009). Factoring
    polynomials over global fields. <i>Journal de Théorie Des Nombres de Bordeaux</i>,
    <i>21</i>(1), 15–39. <a href="https://doi.org/10.5802/jtnb.655">https://doi.org/10.5802/jtnb.655</a>
  bibtex: '@article{Belabas_van Hoeij_Klüners_Steel_2009, title={Factoring polynomials
    over global fields}, volume={21}, DOI={<a href="https://doi.org/10.5802/jtnb.655">10.5802/jtnb.655</a>},
    number={1}, journal={Journal de Théorie des Nombres de Bordeaux}, publisher={Cellule
    MathDoc/CEDRAM}, author={Belabas, Karim and van Hoeij, Mark and Klüners, Jürgen
    and Steel, Allan}, year={2009}, pages={15–39} }'
  chicago: 'Belabas, Karim, Mark van Hoeij, Jürgen Klüners, and Allan Steel. “Factoring
    Polynomials over Global Fields.” <i>Journal de Théorie Des Nombres de Bordeaux</i>
    21, no. 1 (2009): 15–39. <a href="https://doi.org/10.5802/jtnb.655">https://doi.org/10.5802/jtnb.655</a>.'
  ieee: 'K. Belabas, M. van Hoeij, J. Klüners, and A. Steel, “Factoring polynomials
    over global fields,” <i>Journal de Théorie des Nombres de Bordeaux</i>, vol. 21,
    no. 1, pp. 15–39, 2009, doi: <a href="https://doi.org/10.5802/jtnb.655">10.5802/jtnb.655</a>.'
  mla: Belabas, Karim, et al. “Factoring Polynomials over Global Fields.” <i>Journal
    de Théorie Des Nombres de Bordeaux</i>, vol. 21, no. 1, Cellule MathDoc/CEDRAM,
    2009, pp. 15–39, doi:<a href="https://doi.org/10.5802/jtnb.655">10.5802/jtnb.655</a>.
  short: K. Belabas, M. van Hoeij, J. Klüners, A. Steel, Journal de Théorie Des Nombres
    de Bordeaux 21 (2009) 15–39.
date_created: 2022-12-23T09:33:37Z
date_updated: 2023-03-06T09:09:56Z
department:
- _id: '102'
doi: 10.5802/jtnb.655
external_id:
  arxiv:
  - 'math/0409510 '
intvolume: '        21'
issue: '1'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 15-39
publication: Journal de Théorie des Nombres de Bordeaux
publication_identifier:
  issn:
  - 1246-7405
publication_status: published
publisher: Cellule MathDoc/CEDRAM
status: public
title: Factoring polynomials over global fields
type: journal_article
user_id: '93826'
volume: 21
year: '2009'
...
---
_id: '35959'
abstract:
- lang: eng
  text: 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:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: 'Klüners J. The van Hoeij Algorithm for Factoring Polynomials. In: <i>The LLL
    Algorithm</i>. Springer Berlin Heidelberg; 2009. doi:<a href="https://doi.org/10.1007/978-3-642-02295-1_8">10.1007/978-3-642-02295-1_8</a>'
  apa: Klüners, J. (2009). The van Hoeij Algorithm for Factoring Polynomials. In <i>The
    LLL Algorithm</i>. Springer Berlin Heidelberg. <a href="https://doi.org/10.1007/978-3-642-02295-1_8">https://doi.org/10.1007/978-3-642-02295-1_8</a>
  bibtex: '@inbook{Klüners_2009, place={Berlin, Heidelberg}, title={The van Hoeij
    Algorithm for Factoring Polynomials}, DOI={<a href="https://doi.org/10.1007/978-3-642-02295-1_8">10.1007/978-3-642-02295-1_8</a>},
    booktitle={The LLL Algorithm}, publisher={Springer Berlin Heidelberg}, author={Klüners,
    Jürgen}, year={2009} }'
  chicago: 'Klüners, Jürgen. “The van Hoeij Algorithm for Factoring Polynomials.”
    In <i>The LLL Algorithm</i>. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009.
    <a href="https://doi.org/10.1007/978-3-642-02295-1_8">https://doi.org/10.1007/978-3-642-02295-1_8</a>.'
  ieee: 'J. Klüners, “The van Hoeij Algorithm for Factoring Polynomials,” in <i>The
    LLL Algorithm</i>, Berlin, Heidelberg: Springer Berlin Heidelberg, 2009.'
  mla: Klüners, Jürgen. “The van Hoeij Algorithm for Factoring Polynomials.” <i>The
    LLL Algorithm</i>, Springer Berlin Heidelberg, 2009, doi:<a href="https://doi.org/10.1007/978-3-642-02295-1_8">10.1007/978-3-642-02295-1_8</a>.
  short: 'J. Klüners, in: The LLL Algorithm, Springer Berlin Heidelberg, Berlin, Heidelberg,
    2009.'
date_created: 2023-01-11T09:48:17Z
date_updated: 2023-03-06T09:10:34Z
department:
- _id: '102'
doi: 10.1007/978-3-642-02295-1_8
language:
- iso: eng
place: Berlin, Heidelberg
publication: The LLL Algorithm
publication_identifier:
  isbn:
  - '9783642022944'
  - '9783642022951'
  issn:
  - 1619-7100
publication_status: published
publisher: Springer Berlin Heidelberg
related_material:
  link:
  - relation: confirmation
    url: https://www.researchgate.net/profile/Juergen-Klueners/publication/226764840_The_van_Hoeij_Algorithm_for_Factoring_Polynomials/links/00463532f2216a64ae000000/The-van-Hoeij-Algorithm-for-Factoring-Polynomials.pdf?origin=publication_detail
status: public
title: The van Hoeij Algorithm for Factoring Polynomials
type: book_chapter
user_id: '93826'
year: '2009'
...
---
_id: '43453'
abstract:
- lang: eng
  text: Die invarianten Formen aus dem Anhang sind hier verfügbar:http://www.math.rwth-aachen.de/homes/Markus.Kirschmer/symplectic/
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
citation:
  ama: Kirschmer M. <i>Finite Symplectic Matrix Groups (Dissertation)</i>.; 2009.
  apa: Kirschmer, M. (2009). <i>Finite symplectic matrix groups (Dissertation)</i>.
  bibtex: '@book{Kirschmer_2009, place={RWTH Aachen University}, title={Finite symplectic
    matrix groups (Dissertation)}, author={Kirschmer, Markus}, year={2009} }'
  chicago: Kirschmer, Markus. <i>Finite Symplectic Matrix Groups (Dissertation)</i>.
    RWTH Aachen University, 2009.
  ieee: M. Kirschmer, <i>Finite symplectic matrix groups (Dissertation)</i>. RWTH
    Aachen University, 2009.
  mla: Kirschmer, Markus. <i>Finite Symplectic Matrix Groups (Dissertation)</i>. 2009.
  short: M. Kirschmer, Finite Symplectic Matrix Groups (Dissertation), RWTH Aachen
    University, 2009.
date_created: 2023-04-11T08:03:29Z
date_updated: 2023-04-11T08:14:10Z
department:
- _id: '102'
extern: '1'
language:
- iso: eng
page: '149'
place: RWTH Aachen University
status: public
title: Finite symplectic matrix groups (Dissertation)
type: dissertation
user_id: '93826'
year: '2009'
...
---
_id: '34895'
abstract:
- lang: eng
  text: '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:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: G.
  full_name: Malle, G.
  last_name: Malle
citation:
  ama: Klüners J, Malle G. Counting nilpotent Galois extensions. <i>Journal für die
    reine und angewandte Mathematik (Crelles Journal)</i>. 2006;2004(572):1-26. doi:<a
    href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>
  apa: Klüners, J., &#38; Malle, G. (2006). Counting nilpotent Galois extensions.
    <i>Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)</i>, <i>2004</i>(572),
    1–26. <a href="https://doi.org/10.1515/crll.2004.050">https://doi.org/10.1515/crll.2004.050</a>
  bibtex: '@article{Klüners_Malle_2006, title={Counting nilpotent Galois extensions},
    volume={2004}, DOI={<a href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>},
    number={572}, journal={Journal für die reine und angewandte Mathematik (Crelles
    Journal)}, publisher={Walter de Gruyter GmbH}, author={Klüners, Jürgen and Malle,
    G.}, year={2006}, pages={1–26} }'
  chicago: 'Klüners, Jürgen, and G. Malle. “Counting Nilpotent Galois Extensions.”
    <i>Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)</i> 2004,
    no. 572 (2006): 1–26. <a href="https://doi.org/10.1515/crll.2004.050">https://doi.org/10.1515/crll.2004.050</a>.'
  ieee: 'J. Klüners and G. Malle, “Counting nilpotent Galois extensions,” <i>Journal
    für die reine und angewandte Mathematik (Crelles Journal)</i>, vol. 2004, no.
    572, pp. 1–26, 2006, doi: <a href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>.'
  mla: Klüners, Jürgen, and G. Malle. “Counting Nilpotent Galois Extensions.” <i>Journal
    Für Die Reine Und Angewandte Mathematik (Crelles Journal)</i>, vol. 2004, no.
    572, Walter de Gruyter GmbH, 2006, pp. 1–26, doi:<a href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>.
  short: J. Klüners, G. Malle, Journal Für Die Reine Und Angewandte Mathematik (Crelles
    Journal) 2004 (2006) 1–26.
date_created: 2022-12-23T09:50:49Z
date_updated: 2023-03-06T09:11:16Z
department:
- _id: '102'
doi: 10.1515/crll.2004.050
external_id:
  arxiv:
  - math/0112318
intvolume: '      2004'
issue: '572'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
page: 1-26
publication: Journal für die reine und angewandte Mathematik (Crelles Journal)
publication_identifier:
  issn:
  - 0075-4102
  - 1435-5345
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: Counting nilpotent Galois extensions
type: journal_article
user_id: '93826'
volume: 2004
year: '2006'
...
---
_id: '34891'
abstract:
- lang: eng
  text: '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:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. Asymptotics of number fields and the Cohen–Lenstra heuristics. <i>Journal
    de Théorie des Nombres de Bordeaux</i>. 2006;18(3):607-615. doi:<a href="https://doi.org/10.5802/jtnb.561">10.5802/jtnb.561</a>
  apa: Klüners, J. (2006). Asymptotics of number fields and the Cohen–Lenstra heuristics.
    <i>Journal de Théorie Des Nombres de Bordeaux</i>, <i>18</i>(3), 607–615. <a href="https://doi.org/10.5802/jtnb.561">https://doi.org/10.5802/jtnb.561</a>
  bibtex: '@article{Klüners_2006, title={Asymptotics of number fields and the Cohen–Lenstra
    heuristics}, volume={18}, DOI={<a href="https://doi.org/10.5802/jtnb.561">10.5802/jtnb.561</a>},
    number={3}, journal={Journal de Théorie des Nombres de Bordeaux}, publisher={Cellule
    MathDoc/CEDRAM}, author={Klüners, Jürgen}, year={2006}, pages={607–615} }'
  chicago: 'Klüners, Jürgen. “Asymptotics of Number Fields and the Cohen–Lenstra Heuristics.”
    <i>Journal de Théorie Des Nombres de Bordeaux</i> 18, no. 3 (2006): 607–15. <a
    href="https://doi.org/10.5802/jtnb.561">https://doi.org/10.5802/jtnb.561</a>.'
  ieee: 'J. Klüners, “Asymptotics of number fields and the Cohen–Lenstra heuristics,”
    <i>Journal de Théorie des Nombres de Bordeaux</i>, vol. 18, no. 3, pp. 607–615,
    2006, doi: <a href="https://doi.org/10.5802/jtnb.561">10.5802/jtnb.561</a>.'
  mla: Klüners, Jürgen. “Asymptotics of Number Fields and the Cohen–Lenstra Heuristics.”
    <i>Journal de Théorie Des Nombres de Bordeaux</i>, vol. 18, no. 3, Cellule MathDoc/CEDRAM,
    2006, pp. 607–15, doi:<a href="https://doi.org/10.5802/jtnb.561">10.5802/jtnb.561</a>.
  short: J. Klüners, Journal de Théorie Des Nombres de Bordeaux 18 (2006) 607–615.
date_created: 2022-12-23T09:37:01Z
date_updated: 2023-03-06T09:12:04Z
department:
- _id: '102'
doi: 10.5802/jtnb.561
external_id:
  arxiv:
  - 'math/0512260 '
intvolume: '        18'
issue: '3'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 607-615
publication: Journal de Théorie des Nombres de Bordeaux
publication_identifier:
  issn:
  - 1246-7405
publication_status: published
publisher: Cellule MathDoc/CEDRAM
status: public
title: Asymptotics of number fields and the Cohen–Lenstra heuristics
type: journal_article
user_id: '93826'
volume: 18
year: '2006'
...
---
_id: '34890'
abstract:
- lang: eng
  text: '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:
- first_name: Étienne
  full_name: Fouvry, Étienne
  last_name: Fouvry
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Fouvry É, Klüners J. On the 4-rank of class groups of quadratic number fields.
    <i>Inventiones mathematicae</i>. 2006;167(3):455-513. doi:<a href="https://doi.org/10.1007/s00222-006-0021-2">10.1007/s00222-006-0021-2</a>
  apa: Fouvry, É., &#38; Klüners, J. (2006). On the 4-rank of class groups of quadratic
    number fields. <i>Inventiones Mathematicae</i>, <i>167</i>(3), 455–513. <a href="https://doi.org/10.1007/s00222-006-0021-2">https://doi.org/10.1007/s00222-006-0021-2</a>
  bibtex: '@article{Fouvry_Klüners_2006, title={On the 4-rank of class groups of quadratic
    number fields}, volume={167}, DOI={<a href="https://doi.org/10.1007/s00222-006-0021-2">10.1007/s00222-006-0021-2</a>},
    number={3}, journal={Inventiones mathematicae}, publisher={Springer Science and
    Business Media LLC}, author={Fouvry, Étienne and Klüners, Jürgen}, year={2006},
    pages={455–513} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “On the 4-Rank of Class Groups of
    Quadratic Number Fields.” <i>Inventiones Mathematicae</i> 167, no. 3 (2006): 455–513.
    <a href="https://doi.org/10.1007/s00222-006-0021-2">https://doi.org/10.1007/s00222-006-0021-2</a>.'
  ieee: 'É. Fouvry and J. Klüners, “On the 4-rank of class groups of quadratic number
    fields,” <i>Inventiones mathematicae</i>, vol. 167, no. 3, pp. 455–513, 2006,
    doi: <a href="https://doi.org/10.1007/s00222-006-0021-2">10.1007/s00222-006-0021-2</a>.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “On the 4-Rank of Class Groups of Quadratic
    Number Fields.” <i>Inventiones Mathematicae</i>, vol. 167, no. 3, Springer Science
    and Business Media LLC, 2006, pp. 455–513, doi:<a href="https://doi.org/10.1007/s00222-006-0021-2">10.1007/s00222-006-0021-2</a>.
  short: É. Fouvry, J. Klüners, Inventiones Mathematicae 167 (2006) 455–513.
date_created: 2022-12-23T09:36:15Z
date_updated: 2023-03-06T09:12:30Z
department:
- _id: '102'
doi: 10.1007/s00222-006-0021-2
intvolume: '       167'
issue: '3'
keyword:
- General Mathematics
language:
- iso: eng
page: 455-513
publication: Inventiones mathematicae
publication_identifier:
  issn:
  - 0020-9910
  - 1432-1297
publication_status: published
publisher: Springer Science and Business Media LLC
related_material:
  link:
  - relation: confirmation
    url: https://math.uni-paderborn.de/fileadmin/mathematik/AG-Computeralgebra/Publications-klueners/ranks.pdf
status: public
title: On the 4-rank of class groups of quadratic number fields
type: journal_article
user_id: '93826'
volume: 167
year: '2006'
...
---
_id: '35958'
abstract:
- lang: eng
  text: 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:
- first_name: Étienne
  full_name: Fouvry, Étienne
  last_name: Fouvry
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: 'Fouvry É, Klüners J. Cohen–Lenstra Heuristics of Quadratic Number Fields.
    In: <i>Lecture Notes in Computer Science</i>. Springer Berlin Heidelberg; 2006.
    doi:<a href="https://doi.org/10.1007/11792086_4">10.1007/11792086_4</a>'
  apa: Fouvry, É., &#38; Klüners, J. (2006). Cohen–Lenstra Heuristics of Quadratic
    Number Fields. In <i>Lecture Notes in Computer Science</i>. Springer Berlin Heidelberg.
    <a href="https://doi.org/10.1007/11792086_4">https://doi.org/10.1007/11792086_4</a>
  bibtex: '@inbook{Fouvry_Klüners_2006, place={Berlin, Heidelberg}, title={Cohen–Lenstra
    Heuristics of Quadratic Number Fields}, DOI={<a href="https://doi.org/10.1007/11792086_4">10.1007/11792086_4</a>},
    booktitle={Lecture Notes in Computer Science}, publisher={Springer Berlin Heidelberg},
    author={Fouvry, Étienne and Klüners, Jürgen}, year={2006} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “Cohen–Lenstra Heuristics of Quadratic
    Number Fields.” In <i>Lecture Notes in Computer Science</i>. Berlin, Heidelberg:
    Springer Berlin Heidelberg, 2006. <a href="https://doi.org/10.1007/11792086_4">https://doi.org/10.1007/11792086_4</a>.'
  ieee: 'É. Fouvry and J. Klüners, “Cohen–Lenstra Heuristics of Quadratic Number Fields,”
    in <i>Lecture Notes in Computer Science</i>, Berlin, Heidelberg: Springer Berlin
    Heidelberg, 2006.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “Cohen–Lenstra Heuristics of Quadratic
    Number Fields.” <i>Lecture Notes in Computer Science</i>, Springer Berlin Heidelberg,
    2006, doi:<a href="https://doi.org/10.1007/11792086_4">10.1007/11792086_4</a>.
  short: 'É. Fouvry, J. Klüners, in: Lecture Notes in Computer Science, Springer Berlin
    Heidelberg, Berlin, Heidelberg, 2006.'
date_created: 2023-01-11T09:46:47Z
date_updated: 2023-03-06T09:13:15Z
department:
- _id: '102'
doi: 10.1007/11792086_4
language:
- iso: eng
place: Berlin, Heidelberg
publication: Lecture Notes in Computer Science
publication_identifier:
  isbn:
  - '9783540360759'
  - '9783540360766'
  issn:
  - 0302-9743
  - 1611-3349
publication_status: published
publisher: Springer Berlin Heidelberg
related_material:
  link:
  - relation: confirmation
    url: https://www.researchgate.net/profile/Juergen-Klueners/publication/221451567_Cohen-Lenstra_Heuristics_of_Quadratic_Number_Fields/links/0a85e53298eaed8777000000/Cohen-Lenstra-Heuristics-of-Quadratic-Number-Fields.pdf?origin=publication_detail
status: public
title: Cohen–Lenstra Heuristics of Quadratic Number Fields
type: book_chapter
user_id: '93826'
year: '2006'
...
---
_id: '34892'
abstract:
- lang: eng
  text: '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:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. The number of S₄-fields with given discriminant. <i>Acta Arithmetica</i>.
    2006;122(2):185-194. doi:<a href="https://doi.org/10.4064/aa122-2-3">10.4064/aa122-2-3</a>
  apa: Klüners, J. (2006). The number of S₄-fields with given discriminant. <i>Acta
    Arithmetica</i>, <i>122</i>(2), 185–194. <a href="https://doi.org/10.4064/aa122-2-3">https://doi.org/10.4064/aa122-2-3</a>
  bibtex: '@article{Klüners_2006, title={The number of S₄-fields with given discriminant},
    volume={122}, DOI={<a href="https://doi.org/10.4064/aa122-2-3">10.4064/aa122-2-3</a>},
    number={2}, journal={Acta Arithmetica}, publisher={Institute of Mathematics, Polish
    Academy of Sciences}, author={Klüners, Jürgen}, year={2006}, pages={185–194} }'
  chicago: 'Klüners, Jürgen. “The Number of S₄-Fields with given Discriminant.” <i>Acta
    Arithmetica</i> 122, no. 2 (2006): 185–94. <a href="https://doi.org/10.4064/aa122-2-3">https://doi.org/10.4064/aa122-2-3</a>.'
  ieee: 'J. Klüners, “The number of S₄-fields with given discriminant,” <i>Acta Arithmetica</i>,
    vol. 122, no. 2, pp. 185–194, 2006, doi: <a href="https://doi.org/10.4064/aa122-2-3">10.4064/aa122-2-3</a>.'
  mla: Klüners, Jürgen. “The Number of S₄-Fields with given Discriminant.” <i>Acta
    Arithmetica</i>, vol. 122, no. 2, Institute of Mathematics, Polish Academy of
    Sciences, 2006, pp. 185–94, doi:<a href="https://doi.org/10.4064/aa122-2-3">10.4064/aa122-2-3</a>.
  short: J. Klüners, Acta Arithmetica 122 (2006) 185–194.
date_created: 2022-12-23T09:40:25Z
date_updated: 2023-03-06T09:52:41Z
department:
- _id: '102'
doi: 10.4064/aa122-2-3
external_id:
  arxiv:
  - math/0411484
intvolume: '       122'
issue: '2'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 185-194
publication: Acta Arithmetica
publication_identifier:
  issn:
  - 0065-1036
  - 1730-6264
publication_status: published
publisher: Institute of Mathematics, Polish Academy of Sciences
status: public
title: The number of S₄-fields with given discriminant
type: journal_article
user_id: '93826'
volume: 122
year: '2006'
...
---
_id: '34894'
abstract:
- lang: eng
  text: '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:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. A counter example to Malle’s conjecture on the asymptotics of discriminants.
    <i>Comptes Rendus Mathematique</i>. 2005;340(6):411-414. doi:<a href="https://doi.org/10.1016/j.crma.2005.02.010">10.1016/j.crma.2005.02.010</a>
  apa: Klüners, J. (2005). A counter example to Malle’s conjecture on the asymptotics
    of discriminants. <i>Comptes Rendus Mathematique</i>, <i>340</i>(6), 411–414.
    <a href="https://doi.org/10.1016/j.crma.2005.02.010">https://doi.org/10.1016/j.crma.2005.02.010</a>
  bibtex: '@article{Klüners_2005, title={A counter example to Malle’s conjecture on
    the asymptotics of discriminants}, volume={340}, DOI={<a href="https://doi.org/10.1016/j.crma.2005.02.010">10.1016/j.crma.2005.02.010</a>},
    number={6}, journal={Comptes Rendus Mathematique}, publisher={Elsevier BV}, author={Klüners,
    Jürgen}, year={2005}, pages={411–414} }'
  chicago: 'Klüners, Jürgen. “A Counter Example to Malle’s Conjecture on the Asymptotics
    of Discriminants.” <i>Comptes Rendus Mathematique</i> 340, no. 6 (2005): 411–14.
    <a href="https://doi.org/10.1016/j.crma.2005.02.010">https://doi.org/10.1016/j.crma.2005.02.010</a>.'
  ieee: 'J. Klüners, “A counter example to Malle’s conjecture on the asymptotics of
    discriminants,” <i>Comptes Rendus Mathematique</i>, vol. 340, no. 6, pp. 411–414,
    2005, doi: <a href="https://doi.org/10.1016/j.crma.2005.02.010">10.1016/j.crma.2005.02.010</a>.'
  mla: Klüners, Jürgen. “A Counter Example to Malle’s Conjecture on the Asymptotics
    of Discriminants.” <i>Comptes Rendus Mathematique</i>, vol. 340, no. 6, Elsevier
    BV, 2005, pp. 411–14, doi:<a href="https://doi.org/10.1016/j.crma.2005.02.010">10.1016/j.crma.2005.02.010</a>.
  short: J. Klüners, Comptes Rendus Mathematique 340 (2005) 411–414.
date_created: 2022-12-23T09:44:05Z
date_updated: 2023-03-06T09:15:10Z
department:
- _id: '102'
doi: 10.1016/j.crma.2005.02.010
external_id:
  arxiv:
  - 'math/0411486 '
intvolume: '       340'
issue: '6'
keyword:
- General Mathematics
language:
- iso: eng
page: 411-414
publication: Comptes Rendus Mathematique
publication_identifier:
  issn:
  - 1631-073X
publication_status: published
publisher: Elsevier BV
status: public
title: A counter example to Malle's conjecture on the asymptotics of discriminants
type: journal_article
user_id: '93826'
volume: 340
year: '2005'
...