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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
MSC:
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11S40 Zeta functions and \(L\)-functions
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References:
[1] Danilov, V.I. : The geometry of toric varieties . Russian Math. Surveys 33: 2 (1978) 97-154. · Zbl 0425.14013 · doi:10.1070/RM1978v033n02ABEH002305
[2] Denef, J. : Poles of complex powers and Newton polyhedron . To appear in Groupe d’Etude d’Analyse Ultrametrique, Paris ’84-’85. · Zbl 0591.14016 · numdam:GAU_1984-1985__12_1_A10_0 · eudml:91932
[3] Igusa, J.-I. : Some observations on higher degree characters . Amer. J. Math. 99 (1977) 393-417. · Zbl 0373.12008 · doi:10.2307/2373827
[4] Igusa, J.-I. : Some results on p-adic complex powers . Amer. J. Math. 106 (1984) 1013-1032. · Zbl 0589.14023 · doi:10.2307/2374271
[5] Igusa, J.-I. : Some aspects of the arithmetic theory of polynomials . Preprint (1986).
[6] Kouchnirenko, A.G. : Polyèdres de Newton et nombres de Milnor . Invent. Math. 32 (1976) 1-32. · Zbl 0328.32007 · doi:10.1007/BF01389769 · eudml:142365
[7] Lichtin, B. : Estimation of Lojasiewicz exponents and Newton polygons . Invent. Math. 64 (1981) 417-429. · Zbl 0556.32003 · doi:10.1007/BF01389274 · eudml:142829
[8] Lichtin, B. and Meuser, D. : Poles of a local zeta function and Newton polygons . Compositio Math. 55 (1985) 313-332. · Zbl 0606.14022 · numdam:CM_1985__55_3_313_0 · eudml:89724
[9] Shimura, G. : Reduction of algebraic varieties with respect to a discrete valuation of the basic field . Amer. J. Math. 77 (1955) 134-176. · Zbl 0065.36701 · doi:10.2307/2372425
[10] Varchenko, A.N. : Zeta function of monodromy and Newton’s diagram . Invent. Math. 37 (1977) 253-267. · Zbl 0333.14007 · doi:10.1007/BF01390323 · eudml:142438
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