Lower bound for the poles of Igusa's p-adic zeta functions
| dc.creator | Segers, Dirk | |
| dc.date | 2005-09-02 | |
| dc.date.accessioned | 2026-07-07T06:34:10Z | |
| dc.date.available | 2026-07-07T06:34:10Z | |
| dc.description | 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 χbe a character of R^{\times}. Let M_i(u) be the number of solutions of f=u in (R/P^i)^n for i \in \mathbb{Z}_{\geq 0} and u \in R/P^i. These numbers are related with Igusa's p-adic zeta function Z_{f,χ}(s) of f. We explain the connection between the M_i(u) and the smallest real part of a pole of Z_{f,χ}(s). We also prove that M_i(u) is divisible by q^{\ulcorner(n/2)(i-1)\urcorner}, where the corners indicate that we have to round up. This will imply our main result: Z_{f,χ}(s) has no poles with real part less than -n/2. We will also consider arbitrary K-analytic functions f. | |
| dc.description | 10 pages | |
| dc.identifier | https://arxiv.org/abs/math/0509043 | |
| dc.identifier | http://arxiv.org/abs/math/0509043 | |
| dc.identifier | Math. Ann. 336, 659-669 (2006) | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/99436 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Primary 11D79; 11S80; Secondary 14B05 | |
| dc.title | Lower bound for the poles of Igusa's p-adic zeta functions | |
| dc.type | text |