On positive definite Hermitian forms. (English) Zbl 0729.11020
This work has originated from a question posed by Serre, in connection with his study of the maximal number of rational points on curves of genus 3 over finite fields, to Kneser io 1983. Over which orders in imaginary quadratic fields does an indecomposable positive definite unimodular Hermitian form of rank 3 exist? The corresponding question for rank 2 had been settled earlier by T. Hayashida and M. Nishi [J. Math. Soc. Japan 17, 1-16 (1965; Zbl 0132.417)]. The answers to both can be found in this paper: Up to 4 (resp. 3) exceptions for rank 3 (resp. 2), such a form always exists and is given explicitly.
The author applies Kneser’s method of constructing neighbour lattices. After developing the background of the method for maximal orders in greater generality, he mainly deals with the classification over such orders in imaginary quadratic fields. Representatives for the classes of unimodular lattices of rank 2 and 3 are listed for discriminant D down to -20, and the case \(D=-7\) is treated in detail.
The author applies Kneser’s method of constructing neighbour lattices. After developing the background of the method for maximal orders in greater generality, he mainly deals with the classification over such orders in imaginary quadratic fields. Representatives for the classes of unimodular lattices of rank 2 and 3 are listed for discriminant D down to -20, and the case \(D=-7\) is treated in detail.
Reviewer: H.G.Quebbemann (Oldenburg)
Keywords:
class numbers; unimodular Hermitian form of rank 3; Kneser’s method; neighbour lattices; orders; imaginary quadratic fieldsCitations:
Zbl 0132.417References:
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