A vanishing result for Igusa’s p-adic zeta functions with character
نویسنده
چکیده
Let K be a p-adic field and let f be a K-analytic function on an open and compact subset of K3. Let R be the valuation ring of K and let χ be an arbitrary character of R×. Let Zf,χ(s) be Igusa’s p-adic zeta function. In this paper, we prove a vanishing result for candidate poles of Zf,χ(s). This result implies that Zf,χ(s) has no pole with real part less than −1 if f has no point of multiplicity 2.
منابع مشابه
Lower bound for the poles of Igusa’s p-adic zeta functions
Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P . Let f be a polynomial over R in n > 1 variables and let χ be a character of R×. Let Mi(u) be the number of solutions of f = u in (R/P i)n for i ∈ Z≥0 and u ∈ R/P i. These numbers are related with Igusa’s p-adic zeta function Zf,χ(s) of f . We explain the connection betwe...
متن کاملOn the smallest poles of Igusa’s p-adic zeta functions
Let K be a p-adic field. We explore Igusa’s p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of Kn. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than −1/2 for n = 2 and less than −1 for n = 3 which occur as ...
متن کاملLocal Zeta Functions Supported on Analytic Submanifolds and Newton Polyhedra
The local zeta functions (also called Igusa’s zeta functions) over p-adic fields are connected with the number of solutions of congruences and exponential sums mod pm. These zeta functions are defined as integrals over open and compact subsets with respect to the Haar measure. In this paper, we introduce new integrals defined over submanifolds, or more generally, over non-degenerate complete in...
متن کاملIgusa’s Local Zeta Functions of Semiquasihomogeneous Polynomials
In this paper, we prove the rationality of Igusa’s local zeta functions of semiquasihomogeneous polynomials with coefficients in a non-archimedean local field K. The proof of this result is based on Igusa’s stationary phase formula and some ideas on Néron π-desingularization.
متن کاملVanishing of Principal Value Integrals on Surfaces
Principal value integrals are associated to multi–valued rational differential forms with normal crossings support on a non–singular algebraic variety. We prove their vanishing on rational surfaces in the context of a conjecture of Denef–Jacobs. As an application we obtain a strong vanishing result for candidate poles of p–adic and motivic Igusa zeta functions.
متن کامل