---
_id: '55554'
abstract:
- lang: eng
  text: "We discuss various connections between ideal classes, divisors, Picard and\r\nChow
    groups of one-dimensional noetherian domains. As a result of these, we\r\ngive
    a method to compute Chow groups of orders in global fields and show that\r\nthere
    are infinitely many number fields which contain orders with trivial Chow\r\ngroups."
article_number: '89'
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Kirschmer M, Klüners J. Chow groups of one-dimensional noetherian domains.
    <i>Research in Number Theory</i>. 2024;10(4). doi:<a href="https://doi.org/10.1007/s40993-024-00579-6">10.1007/s40993-024-00579-6</a>
  apa: Kirschmer, M., &#38; Klüners, J. (2024). Chow groups of one-dimensional noetherian
    domains. <i>Research in Number Theory</i>, <i>10</i>(4), Article 89. <a href="https://doi.org/10.1007/s40993-024-00579-6">https://doi.org/10.1007/s40993-024-00579-6</a>
  bibtex: '@article{Kirschmer_Klüners_2024, title={Chow groups of one-dimensional
    noetherian domains}, volume={10}, DOI={<a href="https://doi.org/10.1007/s40993-024-00579-6">10.1007/s40993-024-00579-6</a>},
    number={489}, journal={Research in Number Theory}, publisher={Springer Science
    and Business Media LLC}, author={Kirschmer, Markus and Klüners, Jürgen}, year={2024}
    }'
  chicago: Kirschmer, Markus, and Jürgen Klüners. “Chow Groups of One-Dimensional
    Noetherian Domains.” <i>Research in Number Theory</i> 10, no. 4 (2024). <a href="https://doi.org/10.1007/s40993-024-00579-6">https://doi.org/10.1007/s40993-024-00579-6</a>.
  ieee: 'M. Kirschmer and J. Klüners, “Chow groups of one-dimensional noetherian domains,”
    <i>Research in Number Theory</i>, vol. 10, no. 4, Art. no. 89, 2024, doi: <a href="https://doi.org/10.1007/s40993-024-00579-6">10.1007/s40993-024-00579-6</a>.'
  mla: Kirschmer, Markus, and Jürgen Klüners. “Chow Groups of One-Dimensional Noetherian
    Domains.” <i>Research in Number Theory</i>, vol. 10, no. 4, 89, Springer Science
    and Business Media LLC, 2024, doi:<a href="https://doi.org/10.1007/s40993-024-00579-6">10.1007/s40993-024-00579-6</a>.
  short: M. Kirschmer, J. Klüners, Research in Number Theory 10 (2024).
date_created: 2024-08-06T07:03:20Z
date_updated: 2024-11-05T09:46:04Z
doi: 10.1007/s40993-024-00579-6
external_id:
  arxiv:
  - '2208.14688'
intvolume: '        10'
issue: '4'
language:
- iso: eng
publication: Research in Number Theory
publication_identifier:
  issn:
  - 2522-0160
  - 2363-9555
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Chow groups of one-dimensional noetherian domains
type: journal_article
user_id: '82981'
volume: 10
year: '2024'
...
---
_id: '45854'
abstract:
- lang: eng
  text: In a previous paper the authors developed an algorithm to classify certain
    quaternary quadratic lattices over totally real fields. The present article applies
    this algorithm to the classification of binary Hermitian lattices over totally
    imaginary fields. We use it in particular to classify the 48-dimensional extremal
    even unimodular lattices over the integers that admit a semilarge automorphism.
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: Gabriele
  full_name: Nebe, Gabriele
  last_name: Nebe
citation:
  ama: Kirschmer M, Nebe G. Binary Hermitian Lattices over Number Fields. <i>Experimental
    Mathematics</i>. 2022;31(1):280-301. doi:<a href="https://doi.org/10.1080/10586458.2019.1618756">10.1080/10586458.2019.1618756</a>
  apa: Kirschmer, M., &#38; Nebe, G. (2022). Binary Hermitian Lattices over Number
    Fields. <i>Experimental Mathematics</i>, <i>31</i>(1), 280–301. <a href="https://doi.org/10.1080/10586458.2019.1618756">https://doi.org/10.1080/10586458.2019.1618756</a>
  bibtex: '@article{Kirschmer_Nebe_2022, title={Binary Hermitian Lattices over Number
    Fields}, volume={31}, DOI={<a href="https://doi.org/10.1080/10586458.2019.1618756">10.1080/10586458.2019.1618756</a>},
    number={1}, journal={Experimental Mathematics}, publisher={Informa UK Limited},
    author={Kirschmer, Markus and Nebe, Gabriele}, year={2022}, pages={280–301} }'
  chicago: 'Kirschmer, Markus, and Gabriele Nebe. “Binary Hermitian Lattices over
    Number Fields.” <i>Experimental Mathematics</i> 31, no. 1 (2022): 280–301. <a
    href="https://doi.org/10.1080/10586458.2019.1618756">https://doi.org/10.1080/10586458.2019.1618756</a>.'
  ieee: 'M. Kirschmer and G. Nebe, “Binary Hermitian Lattices over Number Fields,”
    <i>Experimental Mathematics</i>, vol. 31, no. 1, pp. 280–301, 2022, doi: <a href="https://doi.org/10.1080/10586458.2019.1618756">10.1080/10586458.2019.1618756</a>.'
  mla: Kirschmer, Markus, and Gabriele Nebe. “Binary Hermitian Lattices over Number
    Fields.” <i>Experimental Mathematics</i>, vol. 31, no. 1, Informa UK Limited,
    2022, pp. 280–301, doi:<a href="https://doi.org/10.1080/10586458.2019.1618756">10.1080/10586458.2019.1618756</a>.
  short: M. Kirschmer, G. Nebe, Experimental Mathematics 31 (2022) 280–301.
date_created: 2023-07-04T08:28:04Z
date_updated: 2023-07-04T08:29:22Z
department:
- _id: '102'
doi: 10.1080/10586458.2019.1618756
intvolume: '        31'
issue: '1'
keyword:
- General Mathematics
language:
- iso: eng
page: 280-301
publication: Experimental Mathematics
publication_identifier:
  issn:
  - 1058-6458
  - 1944-950X
publication_status: published
publisher: Informa UK Limited
status: public
title: Binary Hermitian Lattices over Number Fields
type: journal_article
user_id: '93826'
volume: 31
year: '2022'
...
---
_id: '34912'
abstract:
- lang: eng
  text: 'Let E be an ordinary elliptic curve over a finite field and g be a positive
    integer. Under some technical assumptions, we give an algorithm to span the isomorphism
    classes of principally polarized abelian varieties in the isogeny class of E⁹
    . The varieties are first described as hermitian lattices over (not necessarily
    maximal) quadratic orders and then geometrically in terms of their algebraic theta
    null point. We also show how to algebraically compute Siegel modular forms of
    even weight given as polynomials in the theta constants by a careful choice of
    an affine lift of the theta null point. We then use these results to give an algebraic
    computation of Serre’s obstruction for principally polarized abelian threefolds
    isogenous to E³ and of the Igusa modular form in dimension 4. We illustrate our
    algorithms with examples of curves with many rational points over finite fields. '
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: Fabien
  full_name: Narbonne, Fabien
  last_name: Narbonne
- first_name: Christophe
  full_name: Ritzenthaler, Christophe
  last_name: Ritzenthaler
- first_name: Damien
  full_name: Robert, Damien
  last_name: Robert
citation:
  ama: Kirschmer M, Narbonne F, Ritzenthaler C, Robert D. Spanning the isogeny class
    of a power of an elliptic curve. <i>Mathematics of Computation</i>. 2021;91(333):401-449.
    doi:<a href="https://doi.org/10.1090/mcom/3672">10.1090/mcom/3672</a>
  apa: Kirschmer, M., Narbonne, F., Ritzenthaler, C., &#38; Robert, D. (2021). Spanning
    the isogeny class of a power of an elliptic curve. <i>Mathematics of Computation</i>,
    <i>91</i>(333), 401–449. <a href="https://doi.org/10.1090/mcom/3672">https://doi.org/10.1090/mcom/3672</a>
  bibtex: '@article{Kirschmer_Narbonne_Ritzenthaler_Robert_2021, title={Spanning the
    isogeny class of a power of an elliptic curve}, volume={91}, DOI={<a href="https://doi.org/10.1090/mcom/3672">10.1090/mcom/3672</a>},
    number={333}, journal={Mathematics of Computation}, publisher={American Mathematical
    Society (AMS)}, author={Kirschmer, Markus and Narbonne, Fabien and Ritzenthaler,
    Christophe and Robert, Damien}, year={2021}, pages={401–449} }'
  chicago: 'Kirschmer, Markus, Fabien Narbonne, Christophe Ritzenthaler, and Damien
    Robert. “Spanning the Isogeny Class of a Power of an Elliptic Curve.” <i>Mathematics
    of Computation</i> 91, no. 333 (2021): 401–49. <a href="https://doi.org/10.1090/mcom/3672">https://doi.org/10.1090/mcom/3672</a>.'
  ieee: 'M. Kirschmer, F. Narbonne, C. Ritzenthaler, and D. Robert, “Spanning the
    isogeny class of a power of an elliptic curve,” <i>Mathematics of Computation</i>,
    vol. 91, no. 333, pp. 401–449, 2021, doi: <a href="https://doi.org/10.1090/mcom/3672">10.1090/mcom/3672</a>.'
  mla: Kirschmer, Markus, et al. “Spanning the Isogeny Class of a Power of an Elliptic
    Curve.” <i>Mathematics of Computation</i>, vol. 91, no. 333, American Mathematical
    Society (AMS), 2021, pp. 401–49, doi:<a href="https://doi.org/10.1090/mcom/3672">10.1090/mcom/3672</a>.
  short: M. Kirschmer, F. Narbonne, C. Ritzenthaler, D. Robert, Mathematics of Computation
    91 (2021) 401–449.
date_created: 2022-12-23T11:02:02Z
date_updated: 2023-04-04T07:52:43Z
department:
- _id: '102'
doi: 10.1090/mcom/3672
intvolume: '        91'
issue: '333'
keyword:
- Applied Mathematics
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 401-449
publication: Mathematics of Computation
publication_identifier:
  issn:
  - 0025-5718
  - 1088-6842
publication_status: published
publisher: American Mathematical Society (AMS)
status: public
title: Spanning the isogeny class of a power of an elliptic curve
type: journal_article
user_id: '93826'
volume: 91
year: '2021'
...
---
_id: '34917'
abstract:
- lang: eng
  text: We relate proper isometry classes of maximal lattices in a totally definite
    quaternary quadratic space (V,q) with trivial discriminant to certain equivalence
    classes of ideals in the quaternion algebra representing the Clifford invariant
    of (V,q). This yields a good algorithm to enumerate a system of representatives
    of proper isometry classes of lattices in genera of maximal lattices in (V,q).
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: Gabriele
  full_name: Nebe, Gabriele
  last_name: Nebe
citation:
  ama: Kirschmer M, Nebe G. Quaternary quadratic lattices over number fields. <i>International
    Journal of Number Theory</i>. 2019;15(02):309-325. doi:<a href="https://doi.org/10.1142/s1793042119500131">10.1142/s1793042119500131</a>
  apa: Kirschmer, M., &#38; Nebe, G. (2019). Quaternary quadratic lattices over number
    fields. <i>International Journal of Number Theory</i>, <i>15</i>(02), 309–325.
    <a href="https://doi.org/10.1142/s1793042119500131">https://doi.org/10.1142/s1793042119500131</a>
  bibtex: '@article{Kirschmer_Nebe_2019, title={Quaternary quadratic lattices over
    number fields}, volume={15}, DOI={<a href="https://doi.org/10.1142/s1793042119500131">10.1142/s1793042119500131</a>},
    number={02}, journal={International Journal of Number Theory}, publisher={World
    Scientific Pub Co Pte Lt}, author={Kirschmer, Markus and Nebe, Gabriele}, year={2019},
    pages={309–325} }'
  chicago: 'Kirschmer, Markus, and Gabriele Nebe. “Quaternary Quadratic Lattices over
    Number Fields.” <i>International Journal of Number Theory</i> 15, no. 02 (2019):
    309–25. <a href="https://doi.org/10.1142/s1793042119500131">https://doi.org/10.1142/s1793042119500131</a>.'
  ieee: 'M. Kirschmer and G. Nebe, “Quaternary quadratic lattices over number fields,”
    <i>International Journal of Number Theory</i>, vol. 15, no. 02, pp. 309–325, 2019,
    doi: <a href="https://doi.org/10.1142/s1793042119500131">10.1142/s1793042119500131</a>.'
  mla: Kirschmer, Markus, and Gabriele Nebe. “Quaternary Quadratic Lattices over Number
    Fields.” <i>International Journal of Number Theory</i>, vol. 15, no. 02, World
    Scientific Pub Co Pte Lt, 2019, pp. 309–25, doi:<a href="https://doi.org/10.1142/s1793042119500131">10.1142/s1793042119500131</a>.
  short: M. Kirschmer, G. Nebe, International Journal of Number Theory 15 (2019) 309–325.
date_created: 2022-12-23T11:05:09Z
date_updated: 2023-12-06T10:05:59Z
department:
- _id: '102'
doi: 10.1142/s1793042119500131
intvolume: '        15'
issue: '02'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 309-325
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: Quaternary quadratic lattices over number fields
type: journal_article
user_id: '82258'
volume: 15
year: '2019'
...
---
_id: '34916'
abstract:
- lang: eng
  text: We describe the powers of irreducible polynomials occurring as characteristic
    polynomials of automorphisms of even unimodular lattices over number fields. This
    generalizes results of Gross & McMullen and Bayer-Fluckiger & Taelman.
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
citation:
  ama: Kirschmer M. Automorphisms of even unimodular lattices over number fields.
    <i>Journal of Number Theory</i>. 2019;197:121-134. doi:<a href="https://doi.org/10.1016/j.jnt.2018.08.004">10.1016/j.jnt.2018.08.004</a>
  apa: Kirschmer, M. (2019). Automorphisms of even unimodular lattices over number
    fields. <i>Journal of Number Theory</i>, <i>197</i>, 121–134. <a href="https://doi.org/10.1016/j.jnt.2018.08.004">https://doi.org/10.1016/j.jnt.2018.08.004</a>
  bibtex: '@article{Kirschmer_2019, title={Automorphisms of even unimodular lattices
    over number fields}, volume={197}, DOI={<a href="https://doi.org/10.1016/j.jnt.2018.08.004">10.1016/j.jnt.2018.08.004</a>},
    journal={Journal of Number Theory}, publisher={Elsevier BV}, author={Kirschmer,
    Markus}, year={2019}, pages={121–134} }'
  chicago: 'Kirschmer, Markus. “Automorphisms of Even Unimodular Lattices over Number
    Fields.” <i>Journal of Number Theory</i> 197 (2019): 121–34. <a href="https://doi.org/10.1016/j.jnt.2018.08.004">https://doi.org/10.1016/j.jnt.2018.08.004</a>.'
  ieee: 'M. Kirschmer, “Automorphisms of even unimodular lattices over number fields,”
    <i>Journal of Number Theory</i>, vol. 197, pp. 121–134, 2019, doi: <a href="https://doi.org/10.1016/j.jnt.2018.08.004">10.1016/j.jnt.2018.08.004</a>.'
  mla: Kirschmer, Markus. “Automorphisms of Even Unimodular Lattices over Number Fields.”
    <i>Journal of Number Theory</i>, vol. 197, Elsevier BV, 2019, pp. 121–34, doi:<a
    href="https://doi.org/10.1016/j.jnt.2018.08.004">10.1016/j.jnt.2018.08.004</a>.
  short: M. Kirschmer, Journal of Number Theory 197 (2019) 121–134.
date_created: 2022-12-23T11:04:34Z
date_updated: 2023-12-06T10:07:17Z
department:
- _id: '102'
doi: 10.1016/j.jnt.2018.08.004
intvolume: '       197'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 121-134
publication: Journal of Number Theory
publication_identifier:
  issn:
  - 0022-314X
publication_status: published
publisher: Elsevier BV
status: public
title: Automorphisms of even unimodular lattices over number fields
type: journal_article
user_id: '82258'
volume: 197
year: '2019'
...
---
_id: '34915'
abstract:
- lang: eng
  text: We describe the determinants of the automorphism groups of Hermitian lattices
    over local fields. Using a result of G. Shimura, this yields an explicit method
    to compute the special genera in a given genus of Hermitian lattices over a number
    field.
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
citation:
  ama: Kirschmer M. Determinant groups of Hermitian lattices over local fields. <i>Archiv
    der Mathematik</i>. 2019;113(4):337-347. doi:<a href="https://doi.org/10.1007/s00013-019-01348-z">10.1007/s00013-019-01348-z</a>
  apa: Kirschmer, M. (2019). Determinant groups of Hermitian lattices over local fields.
    <i>Archiv Der Mathematik</i>, <i>113</i>(4), 337–347. <a href="https://doi.org/10.1007/s00013-019-01348-z">https://doi.org/10.1007/s00013-019-01348-z</a>
  bibtex: '@article{Kirschmer_2019, title={Determinant groups of Hermitian lattices
    over local fields}, volume={113}, DOI={<a href="https://doi.org/10.1007/s00013-019-01348-z">10.1007/s00013-019-01348-z</a>},
    number={4}, journal={Archiv der Mathematik}, publisher={Springer Science and Business
    Media LLC}, author={Kirschmer, Markus}, year={2019}, pages={337–347} }'
  chicago: 'Kirschmer, Markus. “Determinant Groups of Hermitian Lattices over Local
    Fields.” <i>Archiv Der Mathematik</i> 113, no. 4 (2019): 337–47. <a href="https://doi.org/10.1007/s00013-019-01348-z">https://doi.org/10.1007/s00013-019-01348-z</a>.'
  ieee: 'M. Kirschmer, “Determinant groups of Hermitian lattices over local fields,”
    <i>Archiv der Mathematik</i>, vol. 113, no. 4, pp. 337–347, 2019, doi: <a href="https://doi.org/10.1007/s00013-019-01348-z">10.1007/s00013-019-01348-z</a>.'
  mla: Kirschmer, Markus. “Determinant Groups of Hermitian Lattices over Local Fields.”
    <i>Archiv Der Mathematik</i>, vol. 113, no. 4, Springer Science and Business Media
    LLC, 2019, pp. 337–47, doi:<a href="https://doi.org/10.1007/s00013-019-01348-z">10.1007/s00013-019-01348-z</a>.
  short: M. Kirschmer, Archiv Der Mathematik 113 (2019) 337–347.
date_created: 2022-12-23T11:03:41Z
date_updated: 2023-04-04T09:05:04Z
department:
- _id: '102'
doi: 10.1007/s00013-019-01348-z
intvolume: '       113'
issue: '4'
keyword:
- General Mathematics
language:
- iso: eng
page: 337-347
publication: Archiv der Mathematik
publication_identifier:
  issn:
  - 0003-889X
  - 1420-8938
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Determinant groups of Hermitian lattices over local fields
type: journal_article
user_id: '93826'
volume: 113
year: '2019'
...
---
_id: '42788'
abstract:
- lang: eng
  text: We classify all one-class genera of admissible lattice chains of length at
    least 2 in hermitian spaces over number fields. If L is a lattice in the chain
    and p the prime ideal dividing the index of the lattices in the chain, then the
    {p}-arithmetic group Aut(L{p}) acts chamber transitively on the corresponding
    Bruhat-Tits building. So our classification provides a step forward to a complete
    classification of these chamber transitive groups which has been announced 1987
    (without a detailed proof) by Kantor, Liebler and Tits. In fact we find all their
    groups over number fields and one additional building with a discrete chamber
    transitive group.
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: Gabriele
  full_name: Nebe, Gabriele
  last_name: Nebe
citation:
  ama: 'Kirschmer M, Nebe G. One Class Genera of Lattice Chains Over Number Fields.
    In: <i>Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory</i>.
    Springer International Publishing; 2018. doi:<a href="https://doi.org/10.1007/978-3-319-70566-8_22">10.1007/978-3-319-70566-8_22</a>'
  apa: Kirschmer, M., &#38; Nebe, G. (2018). One Class Genera of Lattice Chains Over
    Number Fields. In <i>Algorithmic and Experimental Methods in Algebra, Geometry,
    and Number Theory</i>. Springer International Publishing. <a href="https://doi.org/10.1007/978-3-319-70566-8_22">https://doi.org/10.1007/978-3-319-70566-8_22</a>
  bibtex: '@inbook{Kirschmer_Nebe_2018, place={Cham}, title={One Class Genera of Lattice
    Chains Over Number Fields}, DOI={<a href="https://doi.org/10.1007/978-3-319-70566-8_22">10.1007/978-3-319-70566-8_22</a>},
    booktitle={Algorithmic and Experimental Methods in Algebra, Geometry, and Number
    Theory}, publisher={Springer International Publishing}, author={Kirschmer, Markus
    and Nebe, Gabriele}, year={2018} }'
  chicago: 'Kirschmer, Markus, and Gabriele Nebe. “One Class Genera of Lattice Chains
    Over Number Fields.” In <i>Algorithmic and Experimental Methods in Algebra, Geometry,
    and Number Theory</i>. Cham: Springer International Publishing, 2018. <a href="https://doi.org/10.1007/978-3-319-70566-8_22">https://doi.org/10.1007/978-3-319-70566-8_22</a>.'
  ieee: 'M. Kirschmer and G. Nebe, “One Class Genera of Lattice Chains Over Number
    Fields,” in <i>Algorithmic and Experimental Methods in Algebra, Geometry, and
    Number Theory</i>, Cham: Springer International Publishing, 2018.'
  mla: Kirschmer, Markus, and Gabriele Nebe. “One Class Genera of Lattice Chains Over
    Number Fields.” <i>Algorithmic and Experimental Methods in Algebra, Geometry,
    and Number Theory</i>, Springer International Publishing, 2018, doi:<a href="https://doi.org/10.1007/978-3-319-70566-8_22">10.1007/978-3-319-70566-8_22</a>.
  short: 'M. Kirschmer, G. Nebe, in: Algorithmic and Experimental Methods in Algebra,
    Geometry, and Number Theory, Springer International Publishing, Cham, 2018.'
date_created: 2023-03-07T08:23:48Z
date_updated: 2023-04-04T09:08:19Z
department:
- _id: '102'
doi: 10.1007/978-3-319-70566-8_22
extern: '1'
language:
- iso: eng
place: Cham
publication: Algorithmic and Experimental Methods in Algebra, Geometry, and Number
  Theory
publication_identifier:
  isbn:
  - '9783319705651'
  - '9783319705668'
publication_status: published
publisher: Springer International Publishing
status: public
title: One Class Genera of Lattice Chains Over Number Fields
type: book_chapter
user_id: '93826'
year: '2018'
...
---
_id: '42790'
abstract:
- lang: eng
  text: We show that exceptional algebraic groups over number fields do not admit
    one-class genera of parahoric groups, except in the case G₂ . For the group G₂,
    we enumerate all such one-class genera for the usual seven-dimensional representation.
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
citation:
  ama: Kirschmer M. One-class genera of exceptional groups over number fields. <i>Journal
    de Théorie des Nombres de Bordeaux</i>. 2018;30(3):847-857. doi:<a href="https://doi.org/10.5802/jtnb.1052">10.5802/jtnb.1052</a>
  apa: Kirschmer, M. (2018). One-class genera of exceptional groups over number fields.
    <i>Journal de Théorie Des Nombres de Bordeaux</i>, <i>30</i>(3), 847–857. <a href="https://doi.org/10.5802/jtnb.1052">https://doi.org/10.5802/jtnb.1052</a>
  bibtex: '@article{Kirschmer_2018, title={One-class genera of exceptional groups
    over number fields}, volume={30}, DOI={<a href="https://doi.org/10.5802/jtnb.1052">10.5802/jtnb.1052</a>},
    number={3}, journal={Journal de Théorie des Nombres de Bordeaux}, publisher={Cellule
    MathDoc/CEDRAM}, author={Kirschmer, Markus}, year={2018}, pages={847–857} }'
  chicago: 'Kirschmer, Markus. “One-Class Genera of Exceptional Groups over Number
    Fields.” <i>Journal de Théorie Des Nombres de Bordeaux</i> 30, no. 3 (2018): 847–57.
    <a href="https://doi.org/10.5802/jtnb.1052">https://doi.org/10.5802/jtnb.1052</a>.'
  ieee: 'M. Kirschmer, “One-class genera of exceptional groups over number fields,”
    <i>Journal de Théorie des Nombres de Bordeaux</i>, vol. 30, no. 3, pp. 847–857,
    2018, doi: <a href="https://doi.org/10.5802/jtnb.1052">10.5802/jtnb.1052</a>.'
  mla: Kirschmer, Markus. “One-Class Genera of Exceptional Groups over Number Fields.”
    <i>Journal de Théorie Des Nombres de Bordeaux</i>, vol. 30, no. 3, Cellule MathDoc/CEDRAM,
    2018, pp. 847–57, doi:<a href="https://doi.org/10.5802/jtnb.1052">10.5802/jtnb.1052</a>.
  short: M. Kirschmer, Journal de Théorie Des Nombres de Bordeaux 30 (2018) 847–857.
date_created: 2023-03-07T08:27:36Z
date_updated: 2023-04-04T09:07:32Z
department:
- _id: '102'
doi: 10.5802/jtnb.1052
extern: '1'
intvolume: '        30'
issue: '3'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 847-857
publication: Journal de Théorie des Nombres de Bordeaux
publication_identifier:
  issn:
  - 1246-7405
  - 2118-8572
publication_status: published
publisher: Cellule MathDoc/CEDRAM
status: public
title: One-class genera of exceptional groups over number fields
type: journal_article
user_id: '93826'
volume: 30
year: '2018'
...
---
_id: '42791'
abstract:
- lang: eng
  text: We describe a practical algorithm to solve the constructive membership problem
    for discrete two-generator subgroups of SL₂(R) or PSL₂(R). This algorithm has
    been implemented in Magma for groups defined over real algebraic number fields.
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: Marion G.
  full_name: Rüther, Marion G.
  last_name: Rüther
citation:
  ama: Kirschmer M, Rüther MG. The constructive membership problem for discrete two-generator
    subgroups of SL(2,R). <i>Journal of Algebra</i>. 2017;480:519-548. doi:<a href="https://doi.org/10.1016/j.jalgebra.2017.02.029">10.1016/j.jalgebra.2017.02.029</a>
  apa: Kirschmer, M., &#38; Rüther, M. G. (2017). The constructive membership problem
    for discrete two-generator subgroups of SL(2,R). <i>Journal of Algebra</i>, <i>480</i>,
    519–548. <a href="https://doi.org/10.1016/j.jalgebra.2017.02.029">https://doi.org/10.1016/j.jalgebra.2017.02.029</a>
  bibtex: '@article{Kirschmer_Rüther_2017, title={The constructive membership problem
    for discrete two-generator subgroups of SL(2,R)}, volume={480}, DOI={<a href="https://doi.org/10.1016/j.jalgebra.2017.02.029">10.1016/j.jalgebra.2017.02.029</a>},
    journal={Journal of Algebra}, publisher={Elsevier BV}, author={Kirschmer, Markus
    and Rüther, Marion G.}, year={2017}, pages={519–548} }'
  chicago: 'Kirschmer, Markus, and Marion G. Rüther. “The Constructive Membership
    Problem for Discrete Two-Generator Subgroups of SL(2,R).” <i>Journal of Algebra</i>
    480 (2017): 519–48. <a href="https://doi.org/10.1016/j.jalgebra.2017.02.029">https://doi.org/10.1016/j.jalgebra.2017.02.029</a>.'
  ieee: 'M. Kirschmer and M. G. Rüther, “The constructive membership problem for discrete
    two-generator subgroups of SL(2,R),” <i>Journal of Algebra</i>, vol. 480, pp.
    519–548, 2017, doi: <a href="https://doi.org/10.1016/j.jalgebra.2017.02.029">10.1016/j.jalgebra.2017.02.029</a>.'
  mla: Kirschmer, Markus, and Marion G. Rüther. “The Constructive Membership Problem
    for Discrete Two-Generator Subgroups of SL(2,R).” <i>Journal of Algebra</i>, vol.
    480, Elsevier BV, 2017, pp. 519–48, doi:<a href="https://doi.org/10.1016/j.jalgebra.2017.02.029">10.1016/j.jalgebra.2017.02.029</a>.
  short: M. Kirschmer, M.G. Rüther, Journal of Algebra 480 (2017) 519–548.
date_created: 2023-03-07T08:28:11Z
date_updated: 2023-04-04T09:10:14Z
department:
- _id: '102'
doi: 10.1016/j.jalgebra.2017.02.029
extern: '1'
intvolume: '       480'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 519-548
publication: Journal of Algebra
publication_identifier:
  issn:
  - 0021-8693
publication_status: published
publisher: Elsevier BV
status: public
title: The constructive membership problem for discrete two-generator subgroups of
  SL(2,R)
type: journal_article
user_id: '93826'
volume: 480
year: '2017'
...
---
_id: '42792'
abstract:
- lang: eng
  text: We enumerate all positive definite ternary quadratic forms over number fields
    with class number at most 2. This is done by constructing all definite quaternion
    orders of type number at most 2 over number fields. Finally, we list all definite
    quaternion orders of ideal class number 1 or 2.
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: David
  full_name: Lorch, David
  last_name: Lorch
citation:
  ama: Kirschmer M, Lorch D. Ternary quadratic forms over number fields with small
    class number. <i>Journal of Number Theory</i>. 2016;161:343-361. doi:<a href="https://doi.org/10.1016/j.jnt.2014.11.001">10.1016/j.jnt.2014.11.001</a>
  apa: Kirschmer, M., &#38; Lorch, D. (2016). Ternary quadratic forms over number
    fields with small class number. <i>Journal of Number Theory</i>, <i>161</i>, 343–361.
    <a href="https://doi.org/10.1016/j.jnt.2014.11.001">https://doi.org/10.1016/j.jnt.2014.11.001</a>
  bibtex: '@article{Kirschmer_Lorch_2016, title={Ternary quadratic forms over number
    fields with small class number}, volume={161}, DOI={<a href="https://doi.org/10.1016/j.jnt.2014.11.001">10.1016/j.jnt.2014.11.001</a>},
    journal={Journal of Number Theory}, publisher={Elsevier BV}, author={Kirschmer,
    Markus and Lorch, David}, year={2016}, pages={343–361} }'
  chicago: 'Kirschmer, Markus, and David Lorch. “Ternary Quadratic Forms over Number
    Fields with Small Class Number.” <i>Journal of Number Theory</i> 161 (2016): 343–61.
    <a href="https://doi.org/10.1016/j.jnt.2014.11.001">https://doi.org/10.1016/j.jnt.2014.11.001</a>.'
  ieee: 'M. Kirschmer and D. Lorch, “Ternary quadratic forms over number fields with
    small class number,” <i>Journal of Number Theory</i>, vol. 161, pp. 343–361, 2016,
    doi: <a href="https://doi.org/10.1016/j.jnt.2014.11.001">10.1016/j.jnt.2014.11.001</a>.'
  mla: Kirschmer, Markus, and David Lorch. “Ternary Quadratic Forms over Number Fields
    with Small Class Number.” <i>Journal of Number Theory</i>, vol. 161, Elsevier
    BV, 2016, pp. 343–61, doi:<a href="https://doi.org/10.1016/j.jnt.2014.11.001">10.1016/j.jnt.2014.11.001</a>.
  short: M. Kirschmer, D. Lorch, Journal of Number Theory 161 (2016) 343–361.
date_created: 2023-03-07T08:28:46Z
date_updated: 2023-04-04T09:10:42Z
department:
- _id: '102'
doi: 10.1016/j.jnt.2014.11.001
extern: '1'
intvolume: '       161'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 343-361
publication: Journal of Number Theory
publication_identifier:
  issn:
  - 0022-314X
publication_status: published
publisher: Elsevier BV
status: public
title: Ternary quadratic forms over number fields with small class number
type: journal_article
user_id: '93826'
volume: 161
year: '2016'
...
---
_id: '43454'
abstract:
- lang: eng
  text: 'Die Gitter von Klassenzahl eins oder zwei sind hier verfügbar: http://www.math.rwth-aachen.de/~Markus.Kirschmer/forms/'
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
citation:
  ama: Kirschmer M. <i>Definite Quadratic and Hermitian Forms with Small Class Number
    (Habilitation)</i>.; 2016.
  apa: Kirschmer, M. (2016). <i>Definite quadratic and hermitian forms with small
    class number (Habilitation)</i>.
  bibtex: '@book{Kirschmer_2016, place={RWTH Aachen University}, title={Definite quadratic
    and hermitian forms with small class number (Habilitation)}, author={Kirschmer,
    Markus}, year={2016} }'
  chicago: Kirschmer, Markus. <i>Definite Quadratic and Hermitian Forms with Small
    Class Number (Habilitation)</i>. RWTH Aachen University, 2016.
  ieee: M. Kirschmer, <i>Definite quadratic and hermitian forms with small class number
    (Habilitation)</i>. RWTH Aachen University, 2016.
  mla: Kirschmer, Markus. <i>Definite Quadratic and Hermitian Forms with Small Class
    Number (Habilitation)</i>. 2016.
  short: M. Kirschmer, Definite Quadratic and Hermitian Forms with Small Class Number
    (Habilitation), RWTH Aachen University, 2016.
date_created: 2023-04-11T08:06:35Z
date_updated: 2023-04-11T08:11:20Z
department:
- _id: '102'
extern: '1'
language:
- iso: eng
page: '166'
place: RWTH Aachen University
status: public
title: Definite quadratic and hermitian forms with small class number (Habilitation)
type: misc
user_id: '93826'
year: '2016'
...
---
_id: '42793'
abstract:
- lang: eng
  text: Suppose Q is a definite quadratic form on a vector space V over some totally
    real field K ≠ Q. Then the maximal integral Zₖ-lattices in (V,Q) are locally isometric
    everywhere and hence form a single genus. We enumerate all orthogonal spaces (V,Q)
    of dimension at least 3, where the corresponding genus of maximal integral lattices
    consists of a single isometry class. It turns out, there are 471 such genera.
    Moreover, the dimension of V and the degree of K are bounded by 6 and 5 respectively.
    This classification also yields all maximal quaternion orders of type number one.
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
citation:
  ama: Kirschmer M. One-class genera of maximal integral quadratic forms. <i>Journal
    of Number Theory</i>. 2014;136:375-393. doi:<a href="https://doi.org/10.1016/j.jnt.2013.10.007">10.1016/j.jnt.2013.10.007</a>
  apa: Kirschmer, M. (2014). One-class genera of maximal integral quadratic forms.
    <i>Journal of Number Theory</i>, <i>136</i>, 375–393. <a href="https://doi.org/10.1016/j.jnt.2013.10.007">https://doi.org/10.1016/j.jnt.2013.10.007</a>
  bibtex: '@article{Kirschmer_2014, title={One-class genera of maximal integral quadratic
    forms}, volume={136}, DOI={<a href="https://doi.org/10.1016/j.jnt.2013.10.007">10.1016/j.jnt.2013.10.007</a>},
    journal={Journal of Number Theory}, publisher={Elsevier BV}, author={Kirschmer,
    Markus}, year={2014}, pages={375–393} }'
  chicago: 'Kirschmer, Markus. “One-Class Genera of Maximal Integral Quadratic Forms.”
    <i>Journal of Number Theory</i> 136 (2014): 375–93. <a href="https://doi.org/10.1016/j.jnt.2013.10.007">https://doi.org/10.1016/j.jnt.2013.10.007</a>.'
  ieee: 'M. Kirschmer, “One-class genera of maximal integral quadratic forms,” <i>Journal
    of Number Theory</i>, vol. 136, pp. 375–393, 2014, doi: <a href="https://doi.org/10.1016/j.jnt.2013.10.007">10.1016/j.jnt.2013.10.007</a>.'
  mla: Kirschmer, Markus. “One-Class Genera of Maximal Integral Quadratic Forms.”
    <i>Journal of Number Theory</i>, vol. 136, Elsevier BV, 2014, pp. 375–93, doi:<a
    href="https://doi.org/10.1016/j.jnt.2013.10.007">10.1016/j.jnt.2013.10.007</a>.
  short: M. Kirschmer, Journal of Number Theory 136 (2014) 375–393.
date_created: 2023-03-07T08:29:34Z
date_updated: 2023-04-04T09:13:29Z
department:
- _id: '102'
doi: 10.1016/j.jnt.2013.10.007
extern: '1'
intvolume: '       136'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 375-393
publication: Journal of Number Theory
publication_identifier:
  issn:
  - 0022-314X
publication_status: published
publisher: Elsevier BV
status: public
title: One-class genera of maximal integral quadratic forms
type: journal_article
user_id: '93826'
volume: 136
year: '2014'
...
---
_id: '42801'
abstract:
- lang: eng
  text: We exhibit a practical algorithm for solving the constructive membership problem
    for discrete free subgroups of rank 2 in PSL₂(R) or SL₂(R). This algorithm, together
    with methods for checking whether a two-generator subgroup of PSL₂(R) or SL₂(R)
    is discrete and free, have been implemented in Magma for groups defined over real
    algebraic number fields.
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: CHARLES
  full_name: LEEDHAM-GREEN, CHARLES
  last_name: LEEDHAM-GREEN
citation:
  ama: Kirschmer M, LEEDHAM-GREEN C. Computing with subgroups of the modular group
    . <i>Glasgow Mathematical Journal</i>. 2014;57(1):173-180. doi:<a href="https://doi.org/10.1017/s0017089514000202">10.1017/s0017089514000202</a>
  apa: Kirschmer, M., &#38; LEEDHAM-GREEN, C. (2014). Computing with subgroups of
    the modular group . <i>Glasgow Mathematical Journal</i>, <i>57</i>(1), 173–180.
    <a href="https://doi.org/10.1017/s0017089514000202">https://doi.org/10.1017/s0017089514000202</a>
  bibtex: '@article{Kirschmer_LEEDHAM-GREEN_2014, title={Computing with subgroups
    of the modular group }, volume={57}, DOI={<a href="https://doi.org/10.1017/s0017089514000202">10.1017/s0017089514000202</a>},
    number={1}, journal={Glasgow Mathematical Journal}, publisher={Cambridge University
    Press (CUP)}, author={Kirschmer, Markus and LEEDHAM-GREEN, CHARLES}, year={2014},
    pages={173–180} }'
  chicago: 'Kirschmer, Markus, and CHARLES LEEDHAM-GREEN. “Computing with Subgroups
    of the Modular Group .” <i>Glasgow Mathematical Journal</i> 57, no. 1 (2014):
    173–80. <a href="https://doi.org/10.1017/s0017089514000202">https://doi.org/10.1017/s0017089514000202</a>.'
  ieee: 'M. Kirschmer and C. LEEDHAM-GREEN, “Computing with subgroups of the modular
    group ,” <i>Glasgow Mathematical Journal</i>, vol. 57, no. 1, pp. 173–180, 2014,
    doi: <a href="https://doi.org/10.1017/s0017089514000202">10.1017/s0017089514000202</a>.'
  mla: Kirschmer, Markus, and CHARLES LEEDHAM-GREEN. “Computing with Subgroups of
    the Modular Group .” <i>Glasgow Mathematical Journal</i>, vol. 57, no. 1, Cambridge
    University Press (CUP), 2014, pp. 173–80, doi:<a href="https://doi.org/10.1017/s0017089514000202">10.1017/s0017089514000202</a>.
  short: M. Kirschmer, C. LEEDHAM-GREEN, Glasgow Mathematical Journal 57 (2014) 173–180.
date_created: 2023-03-07T08:47:42Z
date_updated: 2023-04-04T07:55:16Z
department:
- _id: '102'
doi: 10.1017/s0017089514000202
extern: '1'
intvolume: '        57'
issue: '1'
keyword:
- General Mathematics
language:
- iso: eng
page: 173-180
publication: Glasgow Mathematical Journal
publication_identifier:
  issn:
  - 0017-0895
  - 1469-509X
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: 'Computing with subgroups of the modular group '
type: journal_article
user_id: '93826'
volume: 57
year: '2014'
...
---
_id: '42794'
abstract:
- lang: eng
  text: We exhibit a practical algorithm for solving the constructive membership problem
    for discrete free subgroups of rank 2 in PSL₂(R) or SL₂(R). This algorithm, together
    with methods for checking whether a two-generator subgroup of PSL₂(R) or SL₂(R)
    is discrete and free, have been implemented in Magma for groups defined over real
    algebraic number fields.
author:
- first_name: B.
  full_name: Eick, B.
  last_name: Eick
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: C.
  full_name: Leedham-Green, C.
  last_name: Leedham-Green
citation:
  ama: Eick B, Kirschmer M, Leedham-Green C. The constructive membership problem for
    discrete free subgroups of rank 2 of SL₂(R). <i>LMS Journal of Computation and
    Mathematics</i>. 2014;17(1):345-359. doi:<a href="https://doi.org/10.1112/s1461157014000047">10.1112/s1461157014000047</a>
  apa: Eick, B., Kirschmer, M., &#38; Leedham-Green, C. (2014). The constructive membership
    problem for discrete free subgroups of rank 2 of SL₂(R). <i>LMS Journal of Computation
    and Mathematics</i>, <i>17</i>(1), 345–359. <a href="https://doi.org/10.1112/s1461157014000047">https://doi.org/10.1112/s1461157014000047</a>
  bibtex: '@article{Eick_Kirschmer_Leedham-Green_2014, title={The constructive membership
    problem for discrete free subgroups of rank 2 of SL₂(R)}, volume={17}, DOI={<a
    href="https://doi.org/10.1112/s1461157014000047">10.1112/s1461157014000047</a>},
    number={1}, journal={LMS Journal of Computation and Mathematics}, publisher={Wiley},
    author={Eick, B. and Kirschmer, Markus and Leedham-Green, C.}, year={2014}, pages={345–359}
    }'
  chicago: 'Eick, B., Markus Kirschmer, and C. Leedham-Green. “The Constructive Membership
    Problem for Discrete Free Subgroups of Rank 2 of SL₂(R).” <i>LMS Journal of Computation
    and Mathematics</i> 17, no. 1 (2014): 345–59. <a href="https://doi.org/10.1112/s1461157014000047">https://doi.org/10.1112/s1461157014000047</a>.'
  ieee: 'B. Eick, M. Kirschmer, and C. Leedham-Green, “The constructive membership
    problem for discrete free subgroups of rank 2 of SL₂(R),” <i>LMS Journal of Computation
    and Mathematics</i>, vol. 17, no. 1, pp. 345–359, 2014, doi: <a href="https://doi.org/10.1112/s1461157014000047">10.1112/s1461157014000047</a>.'
  mla: Eick, B., et al. “The Constructive Membership Problem for Discrete Free Subgroups
    of Rank 2 of SL₂(R).” <i>LMS Journal of Computation and Mathematics</i>, vol.
    17, no. 1, Wiley, 2014, pp. 345–59, doi:<a href="https://doi.org/10.1112/s1461157014000047">10.1112/s1461157014000047</a>.
  short: B. Eick, M. Kirschmer, C. Leedham-Green, LMS Journal of Computation and Mathematics
    17 (2014) 345–359.
date_created: 2023-03-07T08:30:15Z
date_updated: 2023-04-04T09:31:17Z
department:
- _id: '102'
doi: 10.1112/s1461157014000047
extern: '1'
intvolume: '        17'
issue: '1'
keyword:
- Computational Theory and Mathematics
- General Mathematics
language:
- iso: eng
page: 345-359
publication: LMS Journal of Computation and Mathematics
publication_identifier:
  issn:
  - 1461-1570
publication_status: published
publisher: Wiley
status: public
title: The constructive membership problem for discrete free subgroups of rank 2 of
  SL₂(R)
type: journal_article
user_id: '93826'
volume: 17
year: '2014'
...
---
_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: '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: '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: '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: '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'
...