Adele formula for the Gaussian integrals (Q1603062)

Let \(B=\left( b_{ij}\right)_{i,j=1}^n\) be a nondegenerate matrix with rational elements and \[ g_p(B)=\sqrt{|2^n\det B|_p}\lim\limits_{N\to \infty}\int\limits_{x\in \mathbb Q_p^n,|x_i|_p\leq N}\chi_p\left( \sum\limits_{i,j=1}^nb_{ij}x_ix_j\right) dx \] be an improper Gaussian integral (\(\chi_p\) is the canonical additive character on \(\mathbb Q_p\)). Denote by \(g_\infty (B)\) a similar integral over \(\mathbb R^n\); it can be calculated explicitly. The adele formula for the integrals \(g_p\) is the identity \[ g_\infty (B)\prod_pg_p(B)=1,\tag{1} \] where the product is taken over the set of all prime numbers. The formula (for the one-dimensional case, to which the general case can be easily reduced) was proved in the book by \textit{V. S. Vladimirov, I. V. Volovich} and \textit{E. I. Zelenov} [``\(p\)-adic analysis and mathematical physics'', Singapore: World Scientific (1994; Zbl 0812.46076)]. The author gives two new proofs. One of them is based on analysis over adeles, and the second establishes the equivalence of (1) with the reciprocity law for the Hilbert symbol.