Abstract
In this articlewe introduce a new type of local zeta functions and study some connections with pseudodifferential operators in the framework of non-Archimedean fields. The new local zeta functions are defined by integrating complex powers of norms of polynomials multiplied by infinitely pseudo-differentiable functions. In characteristic zero, the new local zeta functions admit meromorphic continuations to the whole complex plane, but they are not rational functions. The real parts of the possible poles have a description similar to the poles of Archimedean zeta functions. But they can be irrational real numbers while in the classical case are rational numbers. We also study, in arbitrary characteristic, certain connections between local zeta functions and the existence of fundamental solutions for pseudodifferential equations.
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References
S. Albeverio, A. Yu. Khrennikov and V. M. Shelkovich, Theory of p-Adic Distributions: Linear and Nonlinear Models (Cambridge University Press, 2010).
V. I. Arnold, S. M. Gussein-Zade and A. N. Varchenko, Singularités des applications différentiables, Vol II. (Éditions Mir,Moscou, 1986).
M. F. Atiyah, “Resolution of singularities and division of distributions,” Comm. Pure Appl. Math. 23, 145–150 (1970).
P. Belkale and P. Brosnan, “Periods and Igusa local zeta functions,” Int. Math. Res. Not. 49, 2655–2670 (2003).
P. Berthelot, “Introduction à la théorie arithmétique des D-modules,” Cohomologies p-adiques et applications arithmétiques, II, Astérisque 279, 1–80 (2002).
I. N. Bernstein, “Modules over the ring of differential operators; the study of fundamental solutions of equations with constant coefficients,” Func. Anal. Appl. 5 (2), 1–16 (1972).
C. G. Bollini, J. J. Giambiagi and A. González Domínguez, “Analytic regularization and the divergencies of quantum field theories,” Il Nuovo Cim. XXXI (3), 550–561 (1964).
M. Bocardo-Gaspar, H. García-Compeán and W. A. Zúñiga-Galindo, “Regularization of p-adic string amplitudes, and multivariate local zeta functions,” arXiv:1611.03807, (2016).
J. Cassaigne and V. Maillot, “Hauteur des hypersurfaces et fonctions zêta d’Igusa,” J. Number Th. 83 (2), 226–255 (2000).
J. Denef, “Report on Igusa’s Local zeta function,” Séminaire Bourbaki 43 (1990-1991), exp. 741; Astérisque 201-202-203, 359–386 (1991).
J. Denef and F. Loeser, “Motivic Igusa zeta functions,” J. Alg. Geom. 7, 505–537 (1998).
I. M. Gel’fand and G. E. Shilov, Generalized Functions. Vol. 1. Properties and Operations (AMS Chelsea Publishing, 2010).
I. M. Gel’fand and G. E. Shilov, Generalized Functions. Vol. 2. Spaces of Fundamental and Generalized Functions (AMS Chelsea Publishing, 2010).
I. M. Gel’fand and N. Ya. Vilenkin, Generalized Functions. Vol. 4. Applications of Harmonic Analysis (AMS Chelsea Publishing, 2010).
T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, White Noise. An Infinite-Dimensional Calculus (Kluwer Academic Publishers Group, 1993).
J.-I. Igusa, An Introduction to the Theory of Local Zeta Functions, AMS/IP Studies in Advanced Mathematics, (2000).
J.-I. Igusa, “A stationary phase formula for p-adic integrals and its applications,” Algebraic Geometry and its Applications, pp. 175–194 (Springer-Verlag, 1994).
J.-I. Igusa, Forms of Higher Degree (The Narosa Publishing House, 1978).
H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero,” Ann.Math. 79, 109–326 (1964).
A. N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields (Marcel Dekker, 2001).
F. Loeser, “Fonctions zêta locales d’Igusa à plusiers variables, intégration dans les fibres, et discriminants,” Ann. Sc. Ec.Norm. Sup. 22 (3), 435–471 (1989).
M. Mustaţâ, “Bernstein-Sato polynomials in positive characteristic,” J. Algebra 321 (1), 128–151 (2009).
N. Obata, White Noise Calculus and Fock Space, Lecture Notes in Mathematics 1577 (Springer-Verlag, 1994).
B. Rubin, “Riesz potentials and integral geometry in the space of rectangular matrices,” Adv. Math. 205 (2), 549–598 (2006).
E. R. Speer, Generalized Feynman Amplitudes, Annals ofMathematics Studies 62 (Princeton University Press, 1969).
M. H. Taibleson, Fourier Analysis on Local Fields (Princeton University Press, 1975).
W. Veys and W. A. Zúniga-Galindo, “Zeta functions for analytic mappings, log-principalization of ideals and Newton polyhedra,” Trans. Amer.Math. Soc. 360, 2205–2227 (2008).
W. Veys and W. A. Zúniga-Galindo, “Zeta functions and oscillatory integrals for meromorphic functions,” Adv. Math. 311, 295–337 (2017).
V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, 1994).
A. Weil, Basic Number Theory (Springer-Verlag, 1967).
K. Yosida, Functional analysis (Springer-Verlag, 1965).
W. A. Zúñiga-Galindo, “Non-Archimedean white noise, pseudodifferential stochastic equations, andmassive Euclidean fields,” J. Fourier Anal. Appl. 23 (2), 288–323 (2017).
W. A. Zúñiga-Galindo, Pseudodifferential Equations over Non-Archimedean Spaces, Lectures Notes in Mathematics 2174 (Springer, 2016).
W. A. Zuniga-Galindo, “Local zeta functions and Newton polyhedra,” Nagoya Math. J. 172, 31–58 (2003).
W. A. Zuniga-Galindo, “Fundamental solutions of pseudo-differential operators over p-adic fields,” Rend. Sem. Mat. Univ. Padova 109, 241–245 (2003).
W. A. Zúñiga-Galindo, “Igusa’s local zeta functions of semiquasihomogeneous polynomials,” Trans. Amer. Math. Soc. 353 (8), 3193–3207 (2001).
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Zúñiga-Galindo, W.A. Local zeta functions, pseudodifferential operators and Sobolev-type spaces over non-Archimedean local fields. P-Adic Num Ultrametr Anal Appl 9, 314–335 (2017). https://doi.org/10.1134/S2070046617040069
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DOI: https://doi.org/10.1134/S2070046617040069