Analytic P-adic Cell Decomposition and P-adic Integrals

Roughly speaking, the semialgebraic cell decomposition theorem for p-adic numbers describes piecewise the p-adic valuation of p-adic polyno-mials (and more generally of semialgebraic p-adic functions), the pieces being geometrically simple sets, called cells. In this paper we prove a similar cell decomposition theorem to describe piecewise the valuation of analytic functions (and more generally of subanalytic functions) using subanalytic cells. We use this cell decomposition theorem to solve a conjecture of Jan Denef on p-adic subanalytic integration; the conjecture describes the dependence of integrals on p-adic parameters. We also obtain an explicit description of parameter dependence of (sub)-analytic local zeta functions. Further we classify subanalytic sets up to subanalytic bijection.