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Zbl 1114.11069
Bailly, Catherine; Cabral, Maria de Jesus
(Gahide, Géraldine)
The regular octagon and the signature of nonsingular integral quadratic forms. With an appendice of Géraldine Gahide. (L'octogone régulier et la signature des formes quadratiques entières non singulières. Avec un appendice de Géraldine Gahide.) (French)
[J] Ann. Inst. Fourier 53, No. 3, 749-766 (2003). ISSN 0373-0956; ISSN 1777-5310/e

Let $L$ be a ${\Bbb Z}$-lattice of rank $n$ in an $n$-dimensional ${\Bbb Q}$-vector space $V$ equipped with a nondegenerate quadratic form $q:V\to {\Bbb Q}$ such that $q(L)\subset {\Bbb Z}$, and let $L^\#$ be the dual lattice in $V$. There is a well known Gaussian sum relation stating that $$\sum_{x\in L^\#/L}\exp(2\pi iq(x))=\sqrt{[L^\#:L]}\exp(2\pi i\sigma (q)/8)$$ where $\sigma(q)$ denotes the signature of $q$ [see, e.g. {\it W. Scharlau}\ 's book: Quadratic and Hermitian forms. Grundlehren der Mathematischen Wissenschaften, 270. Berlin etc.: Springer-Verlag (1985; Zbl 0584.10010), Ch. 5 \S 8]. \par In the present paper, the authors give a new elementary albeit somewhat lengthy and quite technical proof using as one of the ingredients the following geometric fact: Let $L$ be the length of a side of a regular octagon circumscribed around a circle of radius $R$, and let $l$ be the length of a side of a regular octagon inscribed in that same circle. Then $L^2<Rl$.
[Detlev Hoffmann (Nottingham)]

MSC 2000:: *11L07 Estimates on exponential sums
11E45 Analytic theory of forms
11E81 Algebraic theory of quadratic forms

Keywords: Gauss sum; lattice; dual lattice; octagon; signature of a quadratic form

Citations: Zbl 0584.10010

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