145
Views
2
CrossRef citations to date
0
Altmetric
Original Articles

Binary Hermitian Lattices over Number Fields

&

References

  • [Bayer-Fluckiger 02] E. Bayer-Fluckiger. “Ideal Lattices.” In A Panorama of Number Theory or the View from Baker’s Garden (Zürich, 1999), edited by G. Wüstholz, pp. 168–184. Cambridge: Cambridge University Press, 2002.
  • [Bosma et al. 97] W. Bosma, J. Cannon, and C. Playoust. “The Magma Algebra System. I. The User Language.” J. Symb. Comput. 24:3-4 (1997), 235–265.
  • [Cohn and Kumar 09] H. Cohn and A. Kumar. “Optimality and Uniqueness of the Leech Lattice among Lattices.” Ann. Math. 170:3 (2009), 1003–1050.
  • [Conway 69] J. H. Conway. “A Group of Order 8,315,553,613,086,720,000.” Bull. Lond. Math. Soc. 1 (1969), 79–88.
  • [Conway and Sloane 99] J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups, Volume 290 of Grundlehren der Mathematischen Wissenschaften, Third edition. New York: Springer, 1999.
  • [Dieudonné 69] J. Dieudonné. Linear Algebra and Geometry. Paris: Hermann, 1969.
  • [Ebeling 13] W. Ebeling. Lattices and Codes. Advanced Lectures in Mathematics, Third edition. Wiesbaden: Springer Spektrum, 2013. A Course Partially Based on Lectures. by Friedrich Hirzebruch.
  • [Eichler 55] M. Eichler. “Zur Zahlentheorie der Quaternionen-Algebren.” J. Reine u. Angew. Math. 195 (1955), 127–151. Correction in: J. Reine u. Angew. Math. 197 (1957), p. 220.
  • [Fincke and Pohst 85] U. Fincke and M. Pohst. “Improved Methods for Calculating Vectors of Short Length in a Lattice, Including a Complexity Analysis.” Math. Comput. 44:170 (1985), 463–471.
  • [Gan and Yu 00] W. T. Gan and J.-K. Yu. “Group Schemes and Local Densities.” Duke Math. J. 105:3 (2000), 497–524.
  • [Hasse 24] H. Hasse. “Äquivalenz quadratischer Formen in einem beliebigen Zahlkörper.” J. Reine Angew. Math. 153 (1924), 158–162.
  • [Hasse 52] H. Hasse. Über die Klassenzahl Abelscher Zahlkörper. Berlin: Akademie-Verlag, 1952.
  • [Johnson 68] A. A. Johnson. “Integral Representations of Hermitian Forms over Local Fields.” J. Reine Angew. Math. 229 (1968), 57–80.
  • [Jürgens 15] M. Jürgens. Nicht-Existenz und Konstruktion extremaler Gitter. PhD thesis, Technische Universität Dortmund, 2015. See also www.mathematik.tu-dortmund.de/sites/michael-juergens/extremal-lattices.
  • [Kirschmer and Nebe 18] M. Kirschmer and G. Nebe. Quaternary Quadratic Lattices over Number Fields. arXiv preprint arXiv:1705.06525 (2018) doi:https://doi.org/10.1142/S1793042119500131,
  • [Kirschmer and Voight 10] M. Kirschmer and J. Voight. “Algorithmic Enumeration of Ideal Classes for Quaternion Orders.” SIAM J. Comput. 39:5 (2010), 1714–1747. See also https://arxiv.org/pdf/0808.3833.pdf.
  • [Kneser 02] M. Kneser. Quadratische Formen. Berlin: Springer, 2002. Revised and Edited in Collaboration with Rudolf Scharlau.
  • [Leech 64] J. Leech. “Some Sphere Packings in Higher Space.” Can. J. Math. 16 (1964), 657–682.
  • [Minkowski 07] H. Minkowski. Diophantische Approximationen; Eine Einführung in die Zahlentheorie. Leipzig: Teubner, 1907.
  • [Nebe 98a] G. Nebe. “Finite Quaternionic Matrix Groups.” Represent. Theory 2:05 (1998a), 106–223.
  • [Nebe 98b] G. Nebe. “Some Cyclo-quaternionic Lattices.” J. Algebra 199:2 (1998b), 472–498.
  • [Nebe 12] G. Nebe. “An Even Unimodular 72-dimensional Lattice of Minimum 8.” J. Reine Angew. Math. 673 (2012), 237–247.
  • [Nebe 13] G. Nebe. “On Automorphisms of Extremal Even Unimodular Lattices.” Int. J. Number Theory 09:08 (2013), 1933–1959.
  • [Nebe 14] G. Nebe. “A Fourth Extremal Even Unimodular Lattice of Dimension 48.” Discrete Math. 331 (2014), 133–136.
  • [Nebe 16] G. Nebe. “Automorphisms of Extremal Unimodular Lattices in Dimension 72.” J. Number Theory 161 (2016), 362–383.
  • [Neukirch 92] J. Neukirch. Algebraische Zahlentheorie. Berlin: Springer, 1992.
  • [Niemeier 73] H.-V. Niemeier. “Definite quadratische Formen der Dimension 24 und Diskriminante 1.” J. Number Theory. 5:2 (1973), 142–178.
  • [O’Meara 73] O. T. O’Meara. Introduction to Quadratic Forms. Berlin: Springer, 1973.
  • [Quebbemann 95] H.-G. Quebbemann. “Modular Lattices in Euclidean Spaces.” J. Number Theory 54:2 (1995), 190–202.
  • [Reiner 03] I. Reiner. Maximal Orders. Oxford Science Publications. New York: Clarendon Press, 2003.
  • [Scharlau 85] W. Scharlau. Quadratic and Hermitian Forms, Volume 270 of Grundlehren Der Mathematischen Wissenschaften. Berlin: Springer, 1985.
  • [Schiemann 98] A. Schiemann. “Classification of Hermitian Forms with the Neighbor Method.” J. Symb. Comput. 26:4 (1998), 487–508.
  • [Shimura 64] G. Shimura. “Arithmetic of Unitary Groups.” Ann. Math. 79 (1964), 269–409.
  • [Siegel 69] C. L. Siegel. “Berechnung von Zetafunktionen an ganzzahligen Stellen.” Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II. 1969 (1969), 87–102.
  • [Vignéras 76] M.-F. Vignéras. “Simplification pour les ordres des corps de quaternions totalement définis.” J. Reine Angew. Math. 286/287 (1976), 257–277.
  • [Vignéras 80] M.-F. Vignéras. Arithmétique Des Algebrès de Quaternions, Volume 800 of Lecture Notes in Mathematics. Berlin: Springer, 1980.
  • [Washington 97] L. C. Washington. Introduction to Cyclotomic Fields, Volume 83 of Graduate Texts in Mathematics, Second edition. New York: Springer, 1997.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.