Lower bound for the poles of Igusa’s p-adic zeta functions
نویسنده
چکیده
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 between the Mi(u) and the smallest real part of a pole of Zf,χ(s). We also prove that Mi(u) is divisible by qp(n/2)(i−1)q, where the corners indicate that we have to round up. This will imply our main result: Zf,χ(s) has no poles with real part less than −n/2. We will also consider arbitrary K-analytic functions f .
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