Parshin, A. N. A holomorphic version of the Tate-Iwasawa method for unramified \(L\)-functions. I. (English. Russian original) Zbl 1318.11149 Sb. Math. 205, No. 10, 1473-1491 (2014); translation from Mat. Sb. 205, No. 10, 107-124 (2014). This paper does not contain new results, but rather a new methodology in studying the problem of meromorphic continuation and the existence of a functional equation for zeta and \(L\)-functions of one-dimensional arithmetical schemes.Let \(C\) be a smooth, projective, geometrically irreducible curve over a finite field \(k=\mathbb F_q\) and let \(K = k(C)\) be the field of rational functions on \(C\). Denote by \(K_x\) the completion of \(K\) at \(x\in C\) and define the adele ring \(A\) on \(C\) by \(A:= \prod_{x \in C}'K_x\), where the product is the restricted (adelic) product of the \(K_x\)’s, with respect to the discrete valuation subrings \(\mathcal O_x\) in \(K_x\). Also, let \(\mathcal O := \prod_{x \in C}'\mathcal O_x\). Let \(\mathcal D(A)\) be the vector space of all complex-valued locally constant functions on \(A\) with compact support and let \(f\) be a function from \(\mathcal D(A)^{\mathcal O^*}\).Let \(Div^0(C)\) denote the group of divisors of order 0 of \(C\) and \(Div_l(C)\) denote the group of divisors linearly equivalent to \(0\). The finite group \(Div^0(C) / Div_l(C)\) is denoted by \(\Phi\) and its dual by \(\tilde{\Phi}\).Let \(X\) be a variety which is a union of projective lines \(P^1_{\chi}\), \(\chi \in \pi_0(X)\) over \(C\) and let \(T_0:=Hom(\Gamma_{(0)}, \mathbb C^*)\), where \(\Gamma_{(0)}\) is the valuation group at \(0\).In author’s language, the result can be stated as follows:Theorem. There exists a unique rational function \(F\) on \(X\) such that \(F=\tilde{f}_{(0_\chi)}\) in \(K_{(0_\chi)}\) and \(F = \tilde{f}_{(\infty_\chi)}\) in \(K_{(\infty_\chi)}\) for all \(\chi \in \tilde{\Phi}\). Moreover, \(F\) is regular outside the points \(0_\chi\), \(\infty_\chi\), \(\chi \in \tilde{\Phi}\), \(z=1\), \(z=q^{-1}\) on \(T_0\) and may have points of order \(\leq 1\) at \(z=1\), \(z=q^{-1}\) on \(T_0\).The analytic continuation and the formula for residues come from the above theorem while the functional equation comes from the symmetry with respect to \(i^*\), where \(i\) is the involution \(i : T_0 \to T_0\) defined by \(i(z) = q^{-1}z^{-1}\). Reviewer: Stelian Mihalas (Timişoara) Cited in 3 Documents MSC: 11R56 Adèle rings and groups 11R54 Other algebras and orders, and their zeta and \(L\)-functions 11M41 Other Dirichlet series and zeta functions Keywords:zeta function; analytic continuation; Poisson formula; sum of residues; cousin problem; unramified \(L\)-functions; adeles PDFBibTeX XMLCite \textit{A. N. Parshin}, Sb. Math. 205, No. 10, 1473--1491 (2014; Zbl 1318.11149); translation from Mat. Sb. 205, No. 10, 107--124 (2014) Full Text: DOI