Pseudodifferential Operators on ${\mathbf{Q}}_p$ and $L$-Series

Abstract

We define a family of pseudodifferential operators on the Hilbert space $L^2({\mathbf{Q}}_p)$ of complex valued square-integrable functions on the $p$-adic number field ${\mathbf{Q}}_p$. The Riemann zeta-function and the related Dirichlet $L$-functions can be expressed as a trace of these operators on a subspace of $L^2({\mathbf{Q}}_p)$. We also extend this to the $L$-functions associated with modular (cusp) forms. Wavelets on $L^2({\mathbf{Q}}_p)$ are common sets of eigenfunctions of these operators.

1+13 pages, 1 figure