This volume contains the proceedings of the 2019 Lluis A. Santalo Summer School on $p$-Adic Analysis, Arithmetic and Singularities, which was held from June 24-28, 2019, at the Universidad Internacional Menendez Pelayo, Santander, Spain.
The main purpose of the book is to present and analyze different incarnations of the local zeta functions and their multiple connections in mathematics and theoretical physics. Local zeta functions are ubiquitous objects in mathematics and theoretical physics. At the mathematical level, local zeta functions contain geometry and arithmetic information about the set of zeros defined by a finite number of polynomials. In terms of applications in theoretical physics, these functions play a central role in the regularization of Feynman amplitudes and Koba-Nielsen-type string amplitudes, among other applications.
This volume provides a gentle introduction to a very active area of research that lies at the intersection of number theory, $p$-adic analysis, algebraic geometry, singularity theory, and theoretical physics. Specifically, the book introduces $p$-adic analysis, the theory of zeta functions, Archimedean, $p$-adic, motivic, singularities of plane curves and their Poincare series, among other similar topics. It also contains original contributions in the aforementioned areas written by renowned specialists.
This book is an important reference for students and experts who want to delve quickly into the area of local zeta functions and their many connections in mathematics and theoretical physics.
This book is published in cooperation with Real Sociedad Matematica Espanola.
Surveys: E. Leon-Cardenal, Archimedean zeta functions and oscillatory
integrals
J. J. Moyano-Fernandez, Generalized Poincare series for plane curve
singularities
N. Potemans and W. Veys, Introduction to $p$-adic Igusa zeta functions
J. Viu-Sos, An introduction to $p$-adic and motivic integration, zeta
functions and invariants of singularities
W. A. Zuniga-Galindo, $p$-adic analysis: A quick introduction
Articles: E. Artal Bartolo and M. Gonzalez Villa, On maximal order poles of
generalized topological zeta functions
J. I. Cogolludo-Agustin, T. Laszlo, J. Martin-Morales, and A. Nemethi, Local
invariants of minimal generic curves on rational surfaces
J. Nagy and A. Nemethi, Motivic Poincare series of cusp surface
singularities
C. D. Sinclair, Non-Archimedean electrostatics
Carlos Galindo, Universidad Jaume I, Castello de la Plana, Spain.
Alejandro Melle Hernandez, Universidad Complutense de Madrid, Spain.
Julio Jose Moyano-Fernandez, Universitat Jaume I, Castello de la Plana, Spain.
Wilson A. Zuniga-Galindo, University of Texas, Brownsville, TX, and Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nactional, Unidad Quer, Santiago de Queretaro, Qro., Mexico.
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