This volume contains the proceedings of the 2019 Llu´ıs A. Santal´o Summer School on p-Adic Analysis, Arithmetic and Singularities, which was held from June 24–28, 2019, at the Universidad Internacional Men´endez 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 Archimedean, p-adic, and motivic zeta functions, singularities of plane curves and their Poincar´e 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.
Author(s): Carlos Galindo, Alejandro Melle Hernandez, Julio Jose Moyano-fernandez, Wilson A. Zuniga-Galindo
Series: Contemporary Mathematics, 778
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 331
City: Providence
Cover
Title page
Contents
Preface
Speakers and members of the organizing committee
Part I: Surveys
Archimedean zeta functions and oscillatory integrals
1. Introduction
2. Archimedean local zeta functions
3. The Bernstein-Sato polynomial
4. Oscillatory integrals
5. Some generalizations
Acknowledgements
References
Generalized Poincaré series for plane curve singularities
1. Introduction
2. Technical tools
3. Poincaré series of a local branch
4. Integration with respect to the Euler characteristic
5. Poincaré series as an integral with respect to the Euler characteristic
6. Generalized Poincaré series
7. Generalized Poincaré series in terms of an embedded resolution
References
Introduction to ?-adic Igusa zeta functions
1. Polynomial congruences
2. ?-adic Igusa (local) zeta functions
3. ?-adic manifolds and rationality of the zeta function
4. Denef’s formula
5. Back to polynomial congruences
6. Igusa zeta function for plane curves
7. Topological and motivic zeta function
8. Miscellaneous
Acknowledgment
References
An introduction to ?-adic and motivic integration, zeta functions and invariants of singularities
Introduction
1. Prehistory: Counting \mathds{?}_{?}-points, ?-adic integration and the Igusa zeta function
2. Motivic integration
3. Applications to singularities and the motivic zeta function
Acknowledgments
References
?-Adic analysis: A quick introduction
1. Introduction
2. ?-Adic numbers: essential facts
3. Integration in ℚ_{?}ⁿ
4. Change of variables formula
5. Additive characters
6. Fourier Analysis on ℚ_{?}ⁿ
7. The ?²-theory
8. ? as a topological vector space
9. The space of distributions on ℚ_{?}ⁿ
10. The Fourier transform on ?′
References
Part II: Articles
On maximal order poles of generalized topological zeta functions
1. Introduction
2. Generalized local topological zeta functions
3. Examples with several poles of order two
4. Poles distinct from -1/?
References
Local invariants of minimal generic curves on rational surfaces
1. Introduction
2. Preliminaries
3. Old and new results for cyclic quotient singularities
4. Delta invariant of minimal generic curves on rational singularities
5. Minimal generic curves on quotient singularities
6. Proof of Theorem 5.1
References
Motivic Poincaré series of cusp surface singularities
1. Introduction
2. Preliminaries regarding normal surface singularities
3. The extension of the series to cusp singularities
References
Non-Archimedean electrostatics
1. Introduction
2. Non-Archimedean fields
3. Electrostatics
4. The canonical ensemble
5. The grand canonical ensemble
6. Multi-component ensembles
Acknowledgements
References
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