Poles of a two-variable 𝑃-adic complex power

Abstract

Abstract. For almost all P-adic completions of an algebraic number field, if s G C is a pole of f = ff\f(x, y) \s \dx\K \ dy \K , where / is a polynomial whose only singular point is the origin,/(0,0) = 0, and/is irreducible in K[ [x, y]], then Re(i) is -1 or one of an explicitly given set of rational numbers, whose cardinality is the number of characteristic exponents of / = 0. 0. Introduction. Let F be an algebraic number field, Kp a F-adic completion of F with ring of integers F, maximal ideal F, group of units Fx , and residue class field R/P of cardinality q. The Haar measure on Kp such that F has measure 1 is called the usual Haar measure, and its product measure is the usual Haar measure on Kp. The absolute value IL on Kn is defined as 101^ = 0 and \d(tx)\Kp=\t\Kp\dx\Kp for every t in Kp -{0}.Let f E K[x, y] have a singularity only at (0,0),/(0,0) = 0, and suppose that/is irreducible in K[ [x, y]], where K is the algebraic closure of K.Our purpose is to investigate the poles of the meromorphic continuation of the complex-valued function fs=SJ)^y^M^My\x, where j is a complex variable.Igusa has given [4, p. 310], in a more general setting, a set of candidates which contains the poles of fs and, in the situation described above, has determined the pole of/1 closest to the origin [3, p. 367].Here we show that for almost all F-adic completions of K, if j is a pole of fs, Re(j) is -1 or one of an explicitly given set of quotients of integers called "numerical data" of desingularization. Only one such quotient is associated with each characteristic exponent of / = 0.Every exceptional curve in the desingularization of / = 0, not only the relatively few we associate below with the characteristic exponents, has a pair of numerical data whose quotient appears, at first glance, to give a negative real pole of fs. We eliminate false candidates by using a relationship between the numerical data, and also an argument involving the Newton polygon. In the process, the behavior of a previously studied function defined on the set of exceptional curves is clarified.

Computing motivic zeta functions on log smooth models

“…Analogous results have previously appeared in the literature for the p-adic zeta function [Lo88,St83], the topological zeta function [Ve97] and the so-called "naïve" motivic zeta function [Ro04]. 8.3.…”

A new generating function for calculating the Igusa local zeta function

“…The sharpest possible lower bound on p-divisibility is given in [22], which is extended to general algebraic sets in [16]. Numerous other works study the poles of the local zeta function [25,34,20,35,3,36,4,7,5,44,45,30,47,38,39,23,24]. Many authors have labored on the calculation of local zeta functions in various situations [28,37,1,5,42,43,21,29,30,9], and many works either use local zeta functions or else apply the methods developed for obtaining them [8,46,48,31,17,18,49,40,33,50].…”

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