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Local and Global Zeta-Functions of Singular Algebraic Curves

✍ Scribed by Karl-Otto Stöhr

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
420 KB
Volume
71
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let X be a complete singular algebraic curve defined over a finite field of q elements. To each local ring O of X there is associated a zeta-function O(s) that encodes the numbers of ideals of given norms. It splits into a finite sum of partial zeta-functions, which are rational functions in q &s . We provide explicit formulae for the partial zeta-functions and prove that the quotient of the zeta-functions of O and its normalization O is a polynomial in q &s of degree not larger than the conductor degree of O. The global zeta-function OX (s), defined by encoding the numbers of coherent ideal sheaves of given degrees, satisfies the global functional equation if and only if X is a Gorenstein curve. We introduce a modified zeta-function, which always satisfies the functional equation and which in the Gorenstein case coincides with `OX (s). We prove that the two global zeta-functions have the same residue at s=0, and that this residue determines the number of the rational points of the compactified Jacobian of X.

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