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Moment Zeta Functions in Arithmetic Geometry
 Department of Mathematics, University of California, Irvine
, 2002
"... Introduction In this expository paper, we discuss several fundamental properties on variation of the HasseWeil zeta function attached to an algebraic variety. The zeta function is a suitable generating function which counts the (closed) points of the algebraic variety. Among the most important pro ..."
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Introduction In this expository paper, we discuss several fundamental properties on variation of the HasseWeil zeta function attached to an algebraic variety. The zeta function is a suitable generating function which counts the (closed) points of the algebraic variety. Among the most important properties are the conjectural meromorphic continuation and the generalized Riemann hypothesis. In characteristic zero case, both conjectures are wide open in general. In positive characteristic case, both conjectures are well understood by the results of Dwork and Deligne on the Weil conjectures. Our main concern here is how the zeta function varies when the algebraic variety moves in an algebraic family. This leads to the introduction of a higher moment zeta function. Again, among the most important properties are the conjectural meromorphic continuation and the generalized Riemann hypothesis. In characteristic zero case, everything remains largely conjectural even for an elliptic curve. In
Lectures on zeta functions over finite fields, Proceedings of 2007 Göttingen summer school on higher dimensional geometry over finite fields, to appear. arXiv:0711.3651
"... 2. Zeta functions over Finite fields 1 3. General properties of Z(f, T). 4 4. Moment Zeta Functions 5 ..."
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2. Zeta functions over Finite fields 1 3. General properties of Z(f, T). 4 4. Moment Zeta Functions 5