The product of symbols of \(p^n\)th power residues as an abelian integral (Q2849055)

Let \((\frac{\alpha}{\beta})\) denote the \(p^n\)-th power residue symbol. The reciprocity law in the local cyclotomic field \(K = \mathbb Q_p(\zeta)\) of \(p^n\)-th roots of unity can be written in the simple form \((\frac{\alpha}{\beta})(\frac{\beta}{\alpha})^{-1} = (\frac{\alpha,\beta}{(\pi)})\) for \(\pi = \zeta-1\), but for applying the reciprocity law one needs an explicit formula for the norm residue symbol on the right. Already Hilbert pointed out that his product formula for the norm residue symbol is an analogue of Cauchy's integral formula, and \textit{I. R. Shafarevich} [Transl., Ser. 2, Am. Math. Soc. 4, 73--106 (1956); translation from Mat. Sb., N. Ser. 26(68), 113--146 (1950; Zbl 0036.15901)] interpreted the local norm residue symbol as an Abelian differential. In [\textit{S. V. Vostokov}, St. Petersbg. Math. J. 20, No. 6, 929--936 (2009); translation from Algebra Anal. 20, No. 6, 108--118 (2008; Zbl 1206.11137)] it was shown that the local norm residue symbol has the form \((\frac{\alpha,\beta}{\pi}) = \zeta^{[\alpha,\beta]}\) for \([\alpha,\beta] =\) res\((\Phi(\underline{\alpha}(X),\underline{\beta}(X)) /(\underline{\zeta}(X)^{p^n}-1)\), where \(\Phi\) is an explicitly given function of \(\alpha\) and \(\beta\) and where \(\underline{\zeta}\) is a polynomial with \(\underline{\zeta}(\pi) = \zeta\). The main result of the present paper is that \([\alpha,\beta]\) can be written as a Shnirelman integral: \([\alpha,\beta] = \int_{0,\pi} (\Phi(\underline{\alpha}(X),\underline{\beta}(X)) /(\underline{\zeta}(X)^{p^n}-1)\). It is also shown that this integral can be computed explicitly in the special case of Eisenstein's reciprocity law.