## Complete classification of binary normal regular Hermitian lattices.(English)Zbl 1248.11030

Let $${\mathcal O}$$ be the ring of algebraic integers in an imaginary quadratic number field $$K={\mathbb Q}(\sqrt{-m})$$, $$m\in {\mathbb Z}$$, $$m>0$$ and square free. An integral Hermitian lattice $$(H,L)$$ of rank $$n$$ is a finitely generated $${\mathcal O}$$-module $$L$$ inside $$K^n$$ such that $$KL=K^n$$, together with a Hermitian form $$H:L\times L\to {\mathcal O}$$. Note that for such lattices, $$H(v,v)\in {\mathbb Z}$$ for all $$v\in L$$, and we say that $$x\in {\mathbb Z}$$ is represented by $$(H,L)$$ if there exists $$v\in L$$ with $$H(v,v)=x$$. $$(H,L)$$ is called normal if the ideal in $${\mathcal O}$$ generated by $$\{ H(v,v)\,|\,v\in L\}$$ is equal to the ideal generated by $$\{ H(v,w)\,|\,v,w\in L\}$$, otherwise it is called subnormal. In this paper, only positive definite integral Hermitian lattices are considered, i.e. lattices for which $$H(v,v)>0$$ for all $$v\in L\setminus\{ 0\}$$. Such a positive definite lattice is called regular if any $$x\in {\mathbb Z}$$ that is represented by all localizations of $$L$$ at all places of $$K$$ is also represented globally. Regular quadratic or Hermitian lattices and their classification have been studied to quite some extent in the literature. The main result by the authors is the full classification of all positive definite binary normal regular Hermitian lattices over rings of algebraic integers in imaginary quadratic number fields up to similarity. There are altogether $$68$$ such lattices, and they are described explicitly in terms of Gram matrices. Those among them that are universal, i.e. that represent all positive integers, are highlighted. Such lattices only exist for $$m\in \{ 1,2,3,5,6,7,10,11,15,19,23,31\}$$. The authors also announce new results concerning the case of subnormal regular Hermitian lattices.

### MSC:

 1.1e+40 Bilinear and Hermitian forms 1.1e+21 General ternary and quaternary quadratic forms; forms of more than two variables 1.1e+42 Class numbers of quadratic and Hermitian forms
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### References:

  H. Brandt and O. Intrau, Tabellen reduzierter positiver ternärer quadratischer Formen, Abh. Sachs. Akad. Wiss. Math.-Nat. Kl., 45 , 1958. · Zbl 0082.03803  W. K. Chan, A. G. Earnest, M. I. Icaza and J. Y. Kim, Finiteness results for regular definite ternary quadratic forms over $$\mathbf{Q}(\sqrt{5})$$, Inter. J. Number Theory, 3 (2007), 541-556. · Zbl 1145.11028  W. K. Chan and A. Rokicki, Positive definite binary hermitian forms with finitely many exceptions, J. Number Theory, 124 (2007), 167-180. · Zbl 1131.11027  L. E. Dickson, Ternary quadratic forms and congruences, Ann. of Math., 28 (1927), 331-341. · JFM 53.0133.03  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. · Zbl 0833.11012  A. G. Earnest and A. Khosravani, Universal binary Hermitian forms, Math. Comp., 66 (1997), 1161-1168. · Zbl 0877.11028  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. · Zbl 0871.11028  M. I. Icaza, Sums of squares of integral linear forms, Acta Arith., 74 (1996), 231-240. · Zbl 0848.11015  H. Iwabuchi, Universal binary positive definite Hermitian lattices, Rocky Mountain J. Math., 30 (2000), 951-959. · Zbl 0972.11024  N. Jacobson, A note on hermitian forms, Amer. Math. Soc., 46 (1940), 264-268. · Zbl 0024.24503  W. C. Jagy, I. Kaplansky and A. Schiemann, There are 913 Regular Ternary Forms, Mathematika, 44 (1997), 332-341. · Zbl 0923.11060  B. M. Kim, Complete determination of regular positive diagonal quaternary integral quadratic forms,  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. · Zbl 1216.11046  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. · Zbl 1282.11021  J.-H. Kim and P.-S. Park, A few uncaught universal Hermitian forms, Proc. Amer. Math. Soc., 135 (2007), 47-49. · Zbl 1173.11021  B.-K. Oh, Regular positive ternary quadratic forms, · Zbl 1241.11044  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. · Zbl 0221.12007  A. Rokicki, Finiteness results for definite $$n$$-regular and almost $$n$$-regular hermitian forms, Ph.D. Thesis, Wesleyan University, (2005).  G. L. Watson, Some problems in the theory of numbers, Ph.D. Thesis, University of London, (1953). · Zbl 0053.13202  G. L. Watson, The representation of integers by positive ternary quadratic forms, Mathematika, 1 (1954), 104-110. · Zbl 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. · Zbl 0107.26901
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