Proceedings of the Japan Academy, Series A, Mathematical Sciences

Construction of a Hermitian lattice without a basis of minimal vectors

Poo-Sung Park

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Abstract

It is known that infinitely many imaginary quadratic fields allow Hermitian lattices which are generated by minimal vectors but have no basis of minimal vectors. In this article we construct systematically such Hermitian lattices over other imaginary quadratic fields. These lattices are binary and unimodular. This construction requires specific non-principal ideals.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 5 (2012), 75-77.

Dates
First available in Project Euclid: 7 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.pja/1336394841

Digital Object Identifier
doi:10.3792/pjaa.88.75

Mathematical Reviews number (MathSciNet)
MR2925286

Zentralblatt MATH identifier
1296.11024

Subjects
Primary: 11E39: Bilinear and Hermitian forms
Secondary: 11H50: Minima of forms

Keywords
Hermitian lattice minimal vector

Citation

Park, Poo-Sung. Construction of a Hermitian lattice without a basis of minimal vectors. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 5, 75--77. doi:10.3792/pjaa.88.75. https://projecteuclid.org/euclid.pja/1336394841


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References

  • J. H. Conway and N. J. A. Sloane, A lattice without a basis of minimal vectors, Mathematika 42 (1995), no. 1, 175–177.
  • C. Fieker and M. E. Pohst, On lattices over number fields, in Algorithmic number theory (Talence, 1996), 133–139, Lecture Notes in Comput. Sci., 1122, Springer, Berlin, 1996.
  • O. T. O'Meara, Introduction to quadratic forms, Spinger-Verlag, New York, 1973.
  • B. M. Kim and P.-S. Park, Hermitian lattices without a basis of minimal vectors, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3041–3044.
  • R. Sasaki, On a lower bound for the class number of an imaginary quadratic field, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), no. 1, 37–39.