Pseudodifferential Operators on Q_p and L-Series

2020/02/28 by Parikshit Dutta, Dutta, Parikshit, Debashis Ghoshal +1
Mathematics · #11M06 #47G30 #65T60 #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2003.00901

openalex publication_date 2020/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define a family of pseudodifferential operators on the Hilbert space L²(Q_p) of complex valued square-integrable functions on the p-adic number field 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 L²(Q_p). We also extend this to the L-functions associated with modular (cusp) forms. Wavelets on L²(Q_p) are common sets of eigenfunctions of these operators.

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