An introduction to the theory of local zeta functions

Includes index

Includes bibliography (p. 227-230) and references

1.Preliminaries 1.1.Review of some basic theorems 1.2.Noetherian rings 1.3.Hilbert's theorems 2.Implicit function theorems and K-analytic manifolds 2.1.Implicit function theorem 2.2.Implicit function theorem (non-archimedean case) 2.3.Weierstrass preparation theorem 2.4.K-analytic manifolds and differential forms 2.5.Critical sets and critical values 3.Hironaka's desingularization theorem 3.1.Monoidal transformations 3.2.Hironaka's desingularization theorem (analytic form) 3.3.Desingularization of plane curves 4.Bernstein's theory 4.1.Bernstein's polynomial bf(s) 4.2.Some properties of bf(s) 4.3.Reduction of the proof 4.4.A general theorem on D-modules 4.5.Completion of the proof 5.Archimedean local zeta functions 5.1.The group Ω(K[×]) 5.2.Schwartz space S(Kn) 5.3.Local zeta function Zφ(ω) 5.4.Complex power ω(f) via desingularization Contents note continued: 5.5.An application 6.Prehomogeneous vector spaces 6.1.Sato's b-function b(s) 6.2.The Γ-function (a digression) 6.3.b(s) = bf(s) and the rationality of the zeros 7.Totally disconnected spaces and p-adic manifolds 7.1.Distributions in totally disconnected spaces 7.2.The case of homogeneous spaces 7.3.Structure of eigendistributions 7.4.Integration on p-adic manifolds 7.5.Serre's theorem on compact p-adic manifolds 7.6.Integration over the fibers 8.Local zeta functions (p-adic case) 8.1.Selfduality of K and some lemmas 8.2.p-adic zeta function Zφ(ω) 8.3.Weil's functions Fφ(i) and F*φ(i*) 8.4.Relation of Fφ(i) and Zφ(ω) 8.5.Poles of ω(f) for a group invariant f 9.Some homogeneous polynomials 9.1.Quadratic forms and Witt's theorem 9.2.Quadratic forms over finite fields 9.3.Classical groups over finite fields Contents note continued: 9.4.Composition and Jordan algebras 9.5.Norm forms and Freudenthal quartics 9.6.Gauss' identity and its corollaries 10.Computation of Z(s) 10.1.Z(ω) in some simple cases 10.2.A p-adic stationary phase formula 10.3.A key lemma 10.4.Z(s) for a Freudenthal quartic 10.5.Z(s) for the Gramian det(txhx) 10.6.An integration formula 10.7.Z(s) for det(txhx) in product forms 11.Theorems of Denef and Meuser 11.1.Regular local rings 11.2.Geometric language 11.3.Hironaka's desingularization theorem (algebraic form) 11.4.Weil's zeta functions over finite fields 11.5.Degree of Z(s) 11.6.The field Ke (a digression) 11.7.Functional equation of Z(s).

An introductory presentation to the theory of local zeta functions. As distributions, and mostly in the archimedian case, local zeta functions are called complex powers. The volume contains major results on complex powers by Atiyah, Bernstein, I.M. Gelfand, and S.I. Gelfand.