Journal of the Mathematical Society of Japan

Complete classification of binary normal regular Hermitian lattices

Byeong Moon KIM, Ji Young KIM, and Poo-Sung PARK
Source: J. Math. Soc. Japan Volume 63, Number 3 (2011), 1001-1025.

Abstract

A positive definite Hermitian lattice is called regular if it represents all integers which can be represented locally by the lattice. We investigate binary regular Hermitian lattices over imaginary quadratic fields Q and provide a complete list of the normal binary regular Hermitian lattices.

First Page: Show Hide
Primary Subjects: 11E39
Secondary Subjects: 11E20, 11E41
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1312203809
Digital Object Identifier: doi:10.2969/jmsj/06331001
Zentralblatt MATH identifier: 05950730
Mathematical Reviews number (MathSciNet): MR2836753

References

H. Brandt and O. Intrau, Tabellen reduzierter positiver ternärer quadratischer Formen, Abh. Sachs. Akad. Wiss. Math.-Nat. Kl., 45, 1958.
Mathematical Reviews (MathSciNet): MR106204
W. K. Chan, A. G. Earnest, M. I. Icaza and J. Y. Kim, Finiteness results for regular definite ternary quadratic forms over $\Q{5}$, Inter. J. Number Theory, 3 (2007), 541–556.
Mathematical Reviews (MathSciNet): MR2371775
Zentralblatt MATH: 1145.11028
Digital Object Identifier: doi:10.1142/S1793042107001103
W. K. Chan and A. Rokicki, Positive definite binary hermitian forms with finitely many exceptions, J. Number Theory, 124 (2007), 167–180.
Mathematical Reviews (MathSciNet): MR2320998
Zentralblatt MATH: 1131.11027
Digital Object Identifier: doi:10.1016/j.jnt.2006.07.016
L. E. Dickson, Ternary quadratic forms and congruences, Ann. of Math., 28 (1927), 331–341.
A. G. Earnest, An application of character sum inequalities to quadratic forms, Number Theory, (Halifax, NS, 1994), CMS Conf. Proc., 15, Amer. Math. Soc. Providence, RI, 1995, pp.,155–158.
Mathematical Reviews (MathSciNet): MR1353928
Zentralblatt MATH: 0833.11012
A. G. Earnest and A. Khosravani, Universal binary Hermitian forms, Math. Comp., 66 (1997), 1161–1168.
Mathematical Reviews (MathSciNet): MR1422787
Zentralblatt MATH: 0877.11028
Digital Object Identifier: doi:10.1090/S0025-5718-97-00860-0
A. G. Earnest and A. Khosravani, Representation of integers by positive definite binary Hermitian lattices over imaginary quadratic fields, J. Number Theory, 62 (1997), 368–374.
Mathematical Reviews (MathSciNet): MR1432781
Zentralblatt MATH: 0871.11028
Digital Object Identifier: doi:10.1006/jnth.1997.2053
M. I. Icaza, Sums of squares of integral linear forms, Acta Arith., 74 (1996), 231–240.
Mathematical Reviews (MathSciNet): MR1373710
Zentralblatt MATH: 0848.11015
H. Iwabuchi, Universal binary positive definite Hermitian lattices, Rocky Mountain J. Math., 30 (2000), 951–959.
Mathematical Reviews (MathSciNet): MR1797825
Digital Object Identifier: doi:10.1216/rmjm/1021477254
Project Euclid: euclid.rmjm/1181070303
Zentralblatt MATH: 0972.11024
N. Jacobson, A note on hermitian forms, Amer. Math. Soc., 46 (1940), 264–268.
Mathematical Reviews (MathSciNet): MR1957
Digital Object Identifier: doi:10.1090/S0002-9904-1940-07187-3
Project Euclid: euclid.bams/1183502551
W. C. Jagy, I. Kaplansky and A. Schiemann, There are 913 Regular Ternary Forms, Mathematika, 44 (1997), 332–341.
Mathematical Reviews (MathSciNet): MR1600553
Digital Object Identifier: doi:10.1112/S002557930001264X
Zentralblatt MATH: 0923.11060
B. M. Kim, Complete determination of regular positive diagonal quaternary integral quadratic forms, preprint.
B. M. Kim, J. Y. Kim and P.-S. Park, The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields, Math. Comp., 79 (2010), 1123–1144.
Mathematical Reviews (MathSciNet): MR2600559
Zentralblatt MATH: 1216.11046
Digital Object Identifier: doi:10.1090/S0025-5718-09-02287-X
B. M. Kim, J. Y. Kim and P.-S. Park, Even universal binary Hermitian lattices over imaginary quadratic fields, Forum Math., to appear in print, ISSN (Online) 1435–5337, ISSN (Print) 0933-7741, DOI: 10.1515/FORM.2011.043.
J.-H. Kim and P.-S. Park, A few uncaught universal Hermitian forms, Proc. Amer. Math. Soc., 135 (2007), 47–49.
Mathematical Reviews (MathSciNet): MR2280173
Zentralblatt MATH: 1173.11021
Digital Object Identifier: doi:10.1090/S0002-9939-06-08457-7
B.-K. Oh, Regular positive ternary quadratic forms, preprint.
Mathematical Reviews (MathSciNet): MR2773202
Zentralblatt MATH: 05860372
Digital Object Identifier: doi:10.4064/aa147-3-3
O. T. O'Meara, Introduction to Quadratic Forms, Spinger-Verlag, New York, 1973.
G. Otremba, Zur Theorie der hermiteschen Formen in imaginär-quadratischen Zahlkörpern, J. Reine Angew. Math., 249 (1971), 1–19.
Mathematical Reviews (MathSciNet): MR318060
A. Rokicki, Finiteness results for definite $n$-regular and almost $n$-regular hermitian forms, Ph.D. Thesis, Wesleyan University, (2005).
Mathematical Reviews (MathSciNet): MR2707081
G. L. Watson, Some problems in the theory of numbers, Ph.D. Thesis, University of London, (1953).
G. L. Watson, The representation of integers by positive ternary quadratic forms, Mathematika, 1 (1954), 104–110.
Mathematical Reviews (MathSciNet): MR67162
Digital Object Identifier: doi:10.1112/S0025579300000589
Zentralblatt MATH: 0056.27201
G. L. Watson, Transformations of a quadratic form which do not increase the class-number, Proc. London Math. Soc. (3), 12 (1962), 577–587.
Mathematical Reviews (MathSciNet): MR142512
Zentralblatt MATH: 0107.26901
Digital Object Identifier: doi:10.1112/plms/s3-12.1.577

2013 © Mathematical Society of Japan

Journal of the Mathematical Society of Japan

Journal of the Mathematical Society of Japan

Turn MathJax Off
What is MathJax?