On the motivic oscillation index and bound of exponential sums modulo p via analytic isomorphisms (original) (raw)

On the Poles of Maximal Order of the Topological Zeta Function

Bulletin of the London Mathematical Society, 1999

The global and local topological zeta functions are singularity invariants associated to a polynomial f and its germ at 0, respectively. By de nition these zeta functions are rational functions in one variable and their poles are negative rational numbers. In this paper we study their poles of maximal possible order. When f is non degenerate with respect to its Newton polyhedron we prove that its local topological zeta function has at most one such pole, in which case it is also the largest pole; concerning the global zeta function we give a similar result. Moreover for any f we show that poles of maximal possible order are always of the form ?1=N with N a positive integer.

The motivic zeta function and its smallest poles

Journal of Algebra, 2007

Let f be a regular function on a nonsingular complex algebraic variety of dimension d. We prove a formula for the motivic zeta function of f in terms of an embedded resolution. This formula is over the Grothendieck ring itself, and specializes to the formula of Denef and Loeser over a certain localization. We also show that the space of n-jets satisfying f = 0 can be partitioned into locally closed subsets which are isomorphic to a cartesian product of some variety with an affine space of dimension dn/2 . Finally, we look at the consequences for the poles of the motivic zeta function.

The Igusa Local Zeta Function Associated with the Singular Cases of the Determinant and the Pfaffian

Journal of Number Theory, 1996

This paper describes the theory of the Igusa local zeta function associated with a polynomial f (x) with coefficients in a p-adic local field K. Results are given in two cases where f (x) is the determinant of a Hermitian matrix of degree m with coefficients in: (1) a ramified quadratic extension of K; and (2) the unique quaternion division algebra over K. | f | s (8)= | K n | f (x)| s K 8(x) dx, in which | } | K is an absolute value in K, 8 is a Schwartz Bruhat function, and dx is a Haar measure on K n. The complex parameter s above is restricted to the right half plane and a fundamental theorem states that | f | s has a meromorphic continuation to the whole s-plane. Furthermore, if K is a p-adic field with q as the cardinality of its residue field, then | f | s (8) is a rational function of t=q &s. This theorem was proved by Atiyah, Bernstein, S. I. Gel'fand, and Igusa in several papers published between 1969 and 1975 [1, 3, 9]. In the p-adic case, these complex powers are called Igusa local zeta functions. Any discussion of developments in this field should also mention the earlier works of Gel'fand and Shilov [7] in which this theorem was proved for a quite general f (x) and the works of Sato and others on prehomogeneous vector spaces [18, 19].

Iwasawa-Tate on ζ-functions and L-functions

2011

After a too-brief introduction to adeles and ideles, we sketch proof of analytic continuation and functional equation of Riemann’s zeta, in the modern form due independently to Iwasawa and Tate about 1950. The sketch is repeated for Dedekind zeta functions of number fields, noting some additional complications. The sketch is repeated again for Hecke’s (größencharakter) L-functions, noting further complications.

Quasi-ordinary power series and their zeta functions

2005

The main objective of this paper is to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, we compute the local Denef-Loeser motivic zeta function Z DL (h, T) of a quasi-ordinary power series h of arbitrary dimension over an algebraically closed field of characteristic zero from its characteristic exponents without using embedded resolution of singularities. This allows us to effectively represent Z DL (h, T) = P (T)/Q(T) such that almost all the candidate poles given by Q(T) are poles. Anyway, these candidate poles give eigenvalues of the monodromy action of the complex of nearby cycles on h −1 (0). In particular we prove in this case the monodromy conjecture made by Denef-Loeser for the local motivic zeta function and the local topological zeta function. As a consequence, if h is a quasi-ordinary polynomial defined over a number field we prove the Igusa monodromy conjecture for its local Igusa zeta function.

A non-vanishing theorem for zeta functions ofGL n

Inventiones Mathematicae, 1976

A and lI its ring of adeles and group of ideles. We regard the group G,= GL(n) as an F-group and, for n > 2, denote by Z, its center. Assume n => 2. A function ~ on the quotient *-G. (V) \ G. (A) G nis said to be cuspidal if the integral (1.1) q~v(g)= S (o(ug)du, U*=U(F)\U(A), U* vanishes for all g 6 G, (A) and all horicycles U of G,. (A horicycle is the unipotent radical of a proper F-parabolic subgroup of G,.) If co is a character (of modulus one) oflI/F • we denote by L2 (~, n) the space of all functions f on G,(A) such that f(Tag)=~o(a)f(g) for 7~G,(F), a~Z,(A)~II, ]f(g)]2 dg< +~. Z.(A) Gn(F)-. Gn(A) We denote by L2o (co, n) the closed subspace of cuspidal elements of L 2 (co, n). The representation of G,(A) on L2(co, n) (by right translations) decomposes into a discrete sum of irreducible representations, each occurring with multiplicity one [15, 2, 13]. To an irreducible component 7r of L2o (o0, n), one can attach as in [5] an L-function L(s, ~z) which satisfies a functional equation (1.2) L(s,n)=~(s, rr)L(1-s,~), where ~ is the representation contragredient to ft. (For other references see [1]). Of course ~ is a component of L 2 (co-~, n). Our main result is as follows:

On Igusa zeta functions of monomial ideals

Proceedings of the American Mathematical Society, 2007

We show that the real parts of the poles of the Igusa zeta function of a monomial ideal can be computed from the torus-invariant divisors on the normalized blow-up of the affine space along the ideal. Moreover, we show that every such number is a root of the Bernstein-Sato polynomial associated to the monomial ideal.