Koszul complexes, differential operators, and the Weil-Tate reciprocity law. (English) Zbl 0989.14004
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 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.
Reviewer: V.P.Kostov (Nice)
MSC:
| 14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |
| 30F30 | Differentials on Riemann surfaces |
| 30D30 | Meromorphic functions of one complex variable (general theory) |
Keywords:
Weil-Tate reciprocity law; Koszul double complex; Krichever map; Nakayashiki-Mukai Fourier transform; differential operatorsCitations:
Zbl 0739.30034References:
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