A quick proof of reciprocity for Hecke Gauss sums
From the text: In Chapter VIII of his book [Vorlesungen über die Theorie der algebraischen Zahlen. 2nd ed. Bronx, New York: Chelsea Publishing Company (1970; Zbl 0208.06101)], \textit{E. Hecke} introduced and studied certain Gauss sums associated to arbitrary number fields. In particular, he discovered a reciprocity law for these sums [loc. cit., Satz 163, p. 240], which he proved by analyzing the values of suitable theta functions in the cusps. The purpose of the present note is to give a short and elementary proof of Hecke's reciprocity law. Our proof is based on Milgram's formula [\textit{J. Milnor} and \textit{D. Husemoller}, Symmetric bilinear forms. Berlin: Springer (1973; Zbl 0292.10016), p. 127] \[ \frac 1{\sqrt{L^\sharp/L}} \sum_{x\in L^\sharp/L} e(B(x,x)/2)=e(s/8), \tag{1} \] where \((L,B)\) is an even integral lattice (i.e. \(L\) is a free \(\mathbb Z\)-module of finite rank and \(B\) a symmetric non-degenerate integer valued bilinear form on \(L\) such that \(B(x,x)\) is even for all \(x\) in \(L\)), \(L^\sharp\) denotes the dual lattice \(\{y\in L: B(y, L)\subseteq \mathbb Z\}\), \(s\) is the signature of \(L\), and \(e(x) = \exp(2\pi ix)\) as usual. Hecke's Gauss sum was defined by the formula \[ C(\omega) =\sum_{\mu\bmod \mathfrak a} e(\text{tr}(\mu\omega^2), \] where \(K\) is an arbitrary number field and \(\omega\) a non-zero element of \(K\). Here \(N\) and \(\text{tr}\) denote the (absolute) norm and the trace of \(K\) and \(\mathfrak a\) denotes the denominator of \(\omega\mathfrak d\), where \(\mathfrak d\) is the different of \(K\). The sum is to be taken over a complete set representatives for the ring \(\mathcal O\) of integers of \(K\) modulo \(\mathfrak a\).
scientific article; zbMATH DE number 6110183
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