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On the behavior of Igusa’s local zeta function in towers of field extension. (English) Zbl 0606.14023
Groupe Étude Anal. Ultramétrique, 12e année 1984/85, No. 2, Exposé No. 25, 10 p. (1985).
[For notations see the preceding review.]
The first section of this paper discusses the non-archimedean work of D. Meuser on the determination of the poles of I(s,\(\phi)\) in terms of the numerical data one obtains from a Hironaka resolution of f, and ends with a partially verified conjecture on the realization of the poles as (logarithms of ) eigenvalues of the monodromy action on the cohomology groups of the vanishing cycle complex associated to f. - The second section describes the work of D. Meuser [”Meromorphic continuation of a zeta function of Weil and Igusa type”, Invent. Math. 85, 493-514 (1986; Zbl 0571.12009)] on the behaviour of \(I_ d(s,\phi)\) where the integral is taken over \(K^ n_ d\) \((K_ d\) is the unique unramified extension of K of degree d), and \(\phi\) is the characteristic function of n copies of the ring of integers in K. The results, which are too complicated to state here, depend upon the analysis of the behaviour of the resolutions as d varies.
Reviewer: Lawrence G.Roberts
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G25 Global ground fields in algebraic geometry
11S40 Zeta functions and \(L\)-functions
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