On a generalization of Hecke \(\vartheta\)-functions and the analytic proof of higher reciprocity laws (Q1801585)

Let \(K\) be a totally real algebraic number field of degree \(n\) over \(\mathbb{Q}\). Let \(\alpha\in K\), \(\alpha^{(j)}\) \((j=1,2,\dots,n)\) its conjugates in \(K\) and \(t_ 1,t_ 2,\dots,t_ n\in\mathbb{R}\) with \(t_ j>0\) for every \(j=1,2,\dots,n\). Hecke defined the positive quadratic form \(Q(x_ 1,x_ 2,\dots,x_ n)=\sum_{i=1}^ n t_ i(\alpha_ 1^{(i)} x_ 1+\cdots+ \alpha_ n^{(i)} x_ n)^ 2\) and considered the corresponding theta function of \(Q\) whose functional equation gives the quadratic reciprocity law in \(K\). The author of this paper considers the generalised form \(M(x_ 1,x_ 2,\dots,x_ n):= \sum_{i=1}^ n t_ i(\alpha_ 1^{(i)} x_ 1+ \cdots+ \alpha_ n^{(i)} x_ n)^ m\) with \(m\in\mathbb{N}\), \(m\) even and calculates the Fourier expansion of the corresponding exponential sum (called by him pseudo \(\vartheta\)-function). This expansion is very complicated and so it was not possible to give a proof of the reciprocity law in \(K\) for the \(m\)th power residue symbol.