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On the degree of a local zeta function. (English) Zbl 0627.14020
Let \(K\) be a number field, let \(R\) be its ring of integers, and let \(\{\) \(v\}\) be the set of finite places of \(K\). For a finite place v of K, denote by \(|.|_ v\) the associated absolute value on \(K\), denote by \(K_ v\) the completion of K with respect to \(|.|_ v,\) denote by \(R_ v\) the ring of integers of \(K_ v\), and denote by \(k_ v\) the residue field of \(R_ v\). Then \(k_ v\) is a finite field with \(q_ v\) elements. For a homogeneous polynomial \(f(x)=f(x_ 1,...,x_ n)\in K[x_ 1,...,K_ n]\) of degree \(m\) and a finite \(place\quad v,\) consider \(Z(t)=\int_{R^ n_ v}| f(x)|^ s_ v| dx|_ v\quad\) where \(s\in {\mathbb{C}}\) with \(Re(s)>0\) and \(t=q_ v^{- s}\). Due to Igusa, \(Z(t)\) is a rational function of t. Writing \(Z(t)=P(t)/Q(t)\), we define \(\deg Z(t) = \deg P(t)-\deg Q(t).\)
The author proves that if f is non-degenerate with respect to its Newton polyhedron then the degree \(\deg(Z(t))=-m\) for almost all finite places v of K. This establishes a conjecture of Igusa for “generic” homogeneous polynomials.
Reviewer: W.Lütkebohmert
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11S40 Zeta functions and \(L\)-functions
Full Text: Numdam EuDML
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