Koszul complexes, differential operators, and the Weil-Tate reciprocity law
The paper studies the geometry of the Weil-Tate multiplicative reciprocity law in connection with Koszul complexes and with the Krichever map. If \(f\), \(g\) are two meromorphic functions on a projective algebraic curve \(X\) (holomorphic outside some smooth point \(a\in X\)), with dominant terms \(z^{-n}\) and \(z^{-m}\) with respect to some local parameter \(z\) at \(a\), then the reciprocity law reads: \[ \prod _{x\in X\backslash \{ a\} }f(x)^{v_x(g)}=(-1)^{mn} \prod _{x\in X\backslash \{ a\} }g(x)^{v_x(f)}.\tag \(*\) \] A direct algebraic proof of \((*)\) is given based on the fact that the left hand-side is det\((f,A/gA)\) where \(A=H^0(X\backslash \{ a\} ,{\mathcal O}_X)\). This can be described as the determinant of the Koszul double complex for \(f\) and \(g\) acting on \(A\). This Koszul complex approach is related to the proof of \((*)\) given by \textit{E. Previato} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 2, 167-171 (1991; Zbl 0739.30034)] where a construction based on differential operators and due to Krichever is used.
scientific article; zbMATH DE number 1523166
scientific article