Research and Innovation Articles
Published 2019-02-19
Keywords
- Local zeta functions,
- p-adic analysis,
- local fields,
- stationary phase formula
How to Cite
León-Cardenal, E., & Zúñiga-Galindo, W. A. (2019). An Introduction to the Theory of Local Zeta Functions from Scratch. Revista Integración, Temas De matemáticas, 37(1), 45–76. https://doi.org/10.18273/revint.v37n1-2019004
Abstract
This survey article aims to provide an introduction to the theory of local zeta functions in the p-adic framework for beginners. We also give an extensive guide to the current literature on local zeta functions and its connections with other fields in mathematics and physics.
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References
Albeverio S., Khrennikov A. and Shelkovich V.M., “Theory of p-adic distributions: linear and nonlinear models”, in London Mathematical Society, Lecture Note Series, 370, Cambridge University Press, Cambridge (2010), xvi+351 pp.
Aref’eva I., Dragović B.G. and Volovich I.V., “On the adelic string amplitudes”, Phys. Lett. B 209 (1988), No. 4, 445–450.
Arnold V.I., Gusein-Zade S.M. and Varchenko A.N., Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals. Translated from the Russian by Hugh Porteous. Translation revised by the authors and James Montaldi. Monographs in Mathematics, 83. Birkhauser Boston, Inc., Boston, MA, 1988.
Atiyah M.F., “Resolution of Singularities and Division of Distributions”, Comm. pure Appl. Math. 23 (1970), 145–150.
Belkale P. and Brosnan P., “Periods and Igusa local zeta functions”, Int. Math. Res. Not. 49 (2003), 2655–2670.
Bernstein I.N., “Modules over the ring of differential operators; the study of fundamental solutions of equations with constant coefficients”, Funct. Anal. Appl. 5 (1972), No. 2, 1–16.
Bleher P.M., “Analytic continuation of massless Feynman amplitudes in the Schwartz space S′”, Rep. Math. Phys. 19 (1984), No. 1, 117–142.
Bocardo-Gaspar M., García-Compeán H. and Zúñiga-Galindo W.A., “Regularization of p-adic String Amplitudes, and Multivariate Local Zeta Functions”, Lett. Math. Phys.(2018), 1–38. https://doi.org/10.1007/s11005-018-1137-1.
Bocardo-Gaspar M., García-Compeán H. and Zúñiga-Galindo W.A., “ On p-Adic string amplitudes in the limit p approaches to one”, J. High Energ. Phys. (2018), No. 8, 043. https://doi.org/10.1007/JHEP08(2018)043.
Bogner C. and Weinzierl S., “Periods and Feynman integrals”, J. Math. Phys. 50 (2009), No. 4, 042302, 16 pp.
[11] Bollini C.G., Giambiagi J.J. and González Domínguez A., “Analytic regularization and the divergences of quantum field theories”, Il Nuovo Cimiento 31 (1964), No. 3, 550–561.
Brekke L., Freund P.G.O., Olson M. and Witten E., “Non-Archimedean string dynamics”, Nuclear Phys. B 302 (1988), No. 3, 365–402.
Brekke L. and Freund P.G.O., “p-adic numbers in physics”, Phys. Rep. 233 (1993), No. 1, 1–66.
Denef J., “Report on Igusa’s Local Zeta Function”, Séminaire Bourbaki 43 (1990–1991), exp. 741; Astérisque 201–202–203, 359–386 (1991).
Denef J. and Loeser F., “Caractéristiques D’Euler-Poincaré, Fonctions Zeta locales et modifications analytiques”, J. Amer. Math. Soc. 5 (1992), No. 4, 705–720.
Denef J. and Loeser F., “Motivic Igusa zeta functions”, J. Algebraic Geom. 7 (1998), No. 3, 505–537.
Denef J. and Loeser F., “On some rational generating series occuring in arithmetic geometry”, in Geometric aspects of Dwork theory Vol I, II, Walter de Gruyter, Berlin, (2004), 509–526.
Frampton P.H. and Okada Y., “p-adic string N-point function”, Phys. Rev. Lett. 60 (1988), No. 6, 484–486.
Freund P.G.O. and Olson M., “Non-Archimedean strings”, Phys. Lett. B 199 (1987), No. 2, 186–190.
Freund P.G.O. and Witten E., “Adelic string amplitudes”, Phys. Lett. B 199 (1987), No. 2, 191–194.
Gel’fand I.M. and Shilov G.E., Generalized Functions, Academic Press, New York and London, 1977.
Ghoshal D., “Noncommutative p-tachyon”, Proc. Steklov Inst. Math. 245 (2004), No. 2, 83–90.
Ghoshal D., “p-adic string theories provide lattice discretization to the ordinary string worldsheet”, Phys. Rev. Lett. 97 (2006), 151601.
Ghoshal D. and Sen A., “Tachyon condensation and brane descent relations in p-adic string theory”, Nucl. Phys. B 584 (2000), No. 1-2, 300–312.
Gouvêa F., p-adic numbers: An introduction, Universitext, Springer-Verlag, Berlin, 1997.
Halmos P., Measure Theory, D. Van Nostrand Company Inc., New York, 1950.
Hloušek Z. and Spector D., “p-adic string theories”, Ann. Physics 189 (1989), No. 2, 370–431.
Igusa J.-I., Lectures on forms of higher degree, Tata Institute of Fundamental Research, Lectures on Mathematics and Physics vol. 59, The Narosa Publishing House, New Delhi, 1978.
Igusa J.-I., “A stationary phase formula for p-adic integrals and its applications”, in Algebraic geometry and its applications (West Lafayette, IN, 1990), , Springer, New York (1994), 175–194.
Igusa J.-I., “On local zeta functions” [translation of S¯ugaku 46 (1994), no. 1, 2338], in Selected papers on number theory and algebraic geometry, 1–20, Amer. Math. Soc. Transl. Ser. 2, 172, Amer. Math. Soc., Providence, RI, 1996.
Igusa J.-I., “An introduction to the theory of local zeta functions”, in AMS/IP Studies in Advanced Mathematics, 14, American Mathematical Society, Providence, RI; International Press, Cambridge, MA (2000), xii+232 pp.
Katok S., “p-adic analysis compared with real”, in Student Mathematical Library, 37, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA (2007), xiv+152 pp.
Khrennikov A., Kozyrev S. and Zúñiga-Galindo W.A., “Ultrametric Pseudodifferential Equations and Applications”, in Encyclopedia of Mathematics and its Applications vol. 168, Cambridge University Press (2018), xv+237 pp.
Kimura T., Introduction to prehomogeneous vector spaces, Translated from the 1998 Japanese original by Makoto Nagura and Tsuyoshi Niitani and revised by the author, Translations of Mathematical Monographs, 215. American Mathematical Society, Providence, RI, 2003.
Koblitz N., “p-adic Numbers, p-adic Analysis and Zeta functions”, Second edition, in Graduate Texts in Mathematics, 58, Springer-Verlag, New York (1984), xii+150 pp.
Lang S., “Real and functional analysis”, Third edition, in Graduate Texts in Mathematics, 142, Springer-Verlag, New York (1993), xiv+580 pp.
Lerner È.Y., “Feynman integrals of a p-adic argument in a momentum space. I. Convergence”, Theoret. and Math. Phys. 102 (1995), No. 3, 267–274.
Lerner È. Y., “Feynman integrals of a p-adic argument in a momentum space. II. Explicit formulas”, Theoret. and Math. Phys. 104 (1995), No. 3, 1061–1077.
Lerner È. Y. and Missarov M. D., “p-adic Feynman and string amplitudes”, Comm. Math. Phys. 121 (1989), No. 1, 35–48.
Lin S., “Ideal-theoretic strategies for asymptotic approximation of marginal likelihood integrals”, J. Algebr. Stat. 8 (2017), No. 1, 22–55.
Loeser F., “Fonctions zêta locales d’Igusa à plusieurs variables, intégration dans les fibres, et discriminants”, Ann. Sci. École Norm. Sup. 22 (1989), No. 3, 435–471.
Marcolli M., Feynman motives, World Scientific Publishing, NJ, 2010.
Mendoza-Martínez M.L., Vallejo J.A. and Zúñiga-Galindo W.A., “Acausal quantum theory for non-Archimedean scalar fields”, arXiv:1805.08613.
Meuser D., “A survey of Igusa’s local zeta function”, Amer. J. Math. 138 (2016), No. 1, 149–179.
Nicaise J., “An introduction to p-adic and motivic zeta functions and the monodromy conjecture”, in Algebraic and analytic aspects of zeta functions and L-functions, MSJ Mem., 21, Math. Soc. Japan, Tokyo (2010), 141–166.
Robert A., A course in p-adic Analysis, Springer-Verlag, New York, 2000.
Schikhof W.H., “Ultrametric calculus: An introduction to p-adic analysis”, in Cambridge Studies in Advanced Mathematics, 4, Cambridge University Press, Cambridge (2006), xii+306 pp.
Smirnov V.A., “Renormalization in p-adic quantum field theory”, Modern Phys. Lett. A 6 (1991), No. 15, 1421–1427.
Smirnov V.A., “Calculation of general p-adic Feynman amplitude”, Comm. Math. Phys. 149 (1992), No. 3, 623–636.
Speer E.R., “Generalized Feynman amplitudes”, in Annals of Mathematics Studies vol. 62, Princeton University Press (1969), vi+120 pp.
Speer E.R., “Ultraviolet and infrared singularity structure of generic Feynman amplitudes”, Ann. Inst. H. Poincaré Sect. A (N.S.) 23 (1975), No. 1, 1–21.
Spokoiny B.L., “Quantum Geometry of Non-archimedean Particles and Strings”, Phys. Lett. B 208 (1988), No. 3-4, 401–406.
Taibleson M.H., Fourier analysis on local fields, Princeton University Press, Tokyo, 1975.
Varadarajan V.S., Reflections on quanta, symmetries, and supersymmetries, Springer, New
York, 2011.
VeysW., “Arc spaces, motivic integration and stringy invariants”, in Singularity theory and its applications, Adv. Stud. Pure Math., 43, Math. Soc. Japan, Tokyo (2006), 529–572.
Vladimirov V.S., Volovich I.V., and Zelenov E.I., “p-adic analysis and mathematical physics”, in Series on Soviet and East European Mathematics 1, World Scientific, River Edge, NJ (1994), xx+319 pp.
Volovich I.V., “p-adic string”, Classical Quantum Gravity 4 (1987), No. 4, L83–L87.
Watanabe S., “Algebraic geometry and statistical learning theory”, in Cambridge Monographs on Applied and Computational Mathematics, 25, Cambridge University Press, Cambridge (2009), viii+286 pp.
Weil A., “Sur la formule de Siegel dans la théorie des groupes classiques”, Acta Math. 113 (1965), 1–87.
Zúñiga-Galindo W.A., “Igusa’s local zeta functions of semiquasihomogeneous polynomials”, Trans. Amer. Math. Soc. 353 (2001), No. 8, 3193–3207.
Zúñiga-Galindo W.A., “Local zeta functions and Newton polyhedra”, Nagoya Math. J. 172 (2003), 31–58.
Zúñiga-Galindo W.A., “Pseudodifferential equations over non-Archimedean spaces”, in Lectures Notes in Mathematics vol. 2174, Springer, Cham. (2016), xvi+175 pp.
Aref’eva I., Dragović B.G. and Volovich I.V., “On the adelic string amplitudes”, Phys. Lett. B 209 (1988), No. 4, 445–450.
Arnold V.I., Gusein-Zade S.M. and Varchenko A.N., Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals. Translated from the Russian by Hugh Porteous. Translation revised by the authors and James Montaldi. Monographs in Mathematics, 83. Birkhauser Boston, Inc., Boston, MA, 1988.
Atiyah M.F., “Resolution of Singularities and Division of Distributions”, Comm. pure Appl. Math. 23 (1970), 145–150.
Belkale P. and Brosnan P., “Periods and Igusa local zeta functions”, Int. Math. Res. Not. 49 (2003), 2655–2670.
Bernstein I.N., “Modules over the ring of differential operators; the study of fundamental solutions of equations with constant coefficients”, Funct. Anal. Appl. 5 (1972), No. 2, 1–16.
Bleher P.M., “Analytic continuation of massless Feynman amplitudes in the Schwartz space S′”, Rep. Math. Phys. 19 (1984), No. 1, 117–142.
Bocardo-Gaspar M., García-Compeán H. and Zúñiga-Galindo W.A., “Regularization of p-adic String Amplitudes, and Multivariate Local Zeta Functions”, Lett. Math. Phys.(2018), 1–38. https://doi.org/10.1007/s11005-018-1137-1.
Bocardo-Gaspar M., García-Compeán H. and Zúñiga-Galindo W.A., “ On p-Adic string amplitudes in the limit p approaches to one”, J. High Energ. Phys. (2018), No. 8, 043. https://doi.org/10.1007/JHEP08(2018)043.
Bogner C. and Weinzierl S., “Periods and Feynman integrals”, J. Math. Phys. 50 (2009), No. 4, 042302, 16 pp.
[11] Bollini C.G., Giambiagi J.J. and González Domínguez A., “Analytic regularization and the divergences of quantum field theories”, Il Nuovo Cimiento 31 (1964), No. 3, 550–561.
Brekke L., Freund P.G.O., Olson M. and Witten E., “Non-Archimedean string dynamics”, Nuclear Phys. B 302 (1988), No. 3, 365–402.
Brekke L. and Freund P.G.O., “p-adic numbers in physics”, Phys. Rep. 233 (1993), No. 1, 1–66.
Denef J., “Report on Igusa’s Local Zeta Function”, Séminaire Bourbaki 43 (1990–1991), exp. 741; Astérisque 201–202–203, 359–386 (1991).
Denef J. and Loeser F., “Caractéristiques D’Euler-Poincaré, Fonctions Zeta locales et modifications analytiques”, J. Amer. Math. Soc. 5 (1992), No. 4, 705–720.
Denef J. and Loeser F., “Motivic Igusa zeta functions”, J. Algebraic Geom. 7 (1998), No. 3, 505–537.
Denef J. and Loeser F., “On some rational generating series occuring in arithmetic geometry”, in Geometric aspects of Dwork theory Vol I, II, Walter de Gruyter, Berlin, (2004), 509–526.
Frampton P.H. and Okada Y., “p-adic string N-point function”, Phys. Rev. Lett. 60 (1988), No. 6, 484–486.
Freund P.G.O. and Olson M., “Non-Archimedean strings”, Phys. Lett. B 199 (1987), No. 2, 186–190.
Freund P.G.O. and Witten E., “Adelic string amplitudes”, Phys. Lett. B 199 (1987), No. 2, 191–194.
Gel’fand I.M. and Shilov G.E., Generalized Functions, Academic Press, New York and London, 1977.
Ghoshal D., “Noncommutative p-tachyon”, Proc. Steklov Inst. Math. 245 (2004), No. 2, 83–90.
Ghoshal D., “p-adic string theories provide lattice discretization to the ordinary string worldsheet”, Phys. Rev. Lett. 97 (2006), 151601.
Ghoshal D. and Sen A., “Tachyon condensation and brane descent relations in p-adic string theory”, Nucl. Phys. B 584 (2000), No. 1-2, 300–312.
Gouvêa F., p-adic numbers: An introduction, Universitext, Springer-Verlag, Berlin, 1997.
Halmos P., Measure Theory, D. Van Nostrand Company Inc., New York, 1950.
Hloušek Z. and Spector D., “p-adic string theories”, Ann. Physics 189 (1989), No. 2, 370–431.
Igusa J.-I., Lectures on forms of higher degree, Tata Institute of Fundamental Research, Lectures on Mathematics and Physics vol. 59, The Narosa Publishing House, New Delhi, 1978.
Igusa J.-I., “A stationary phase formula for p-adic integrals and its applications”, in Algebraic geometry and its applications (West Lafayette, IN, 1990), , Springer, New York (1994), 175–194.
Igusa J.-I., “On local zeta functions” [translation of S¯ugaku 46 (1994), no. 1, 2338], in Selected papers on number theory and algebraic geometry, 1–20, Amer. Math. Soc. Transl. Ser. 2, 172, Amer. Math. Soc., Providence, RI, 1996.
Igusa J.-I., “An introduction to the theory of local zeta functions”, in AMS/IP Studies in Advanced Mathematics, 14, American Mathematical Society, Providence, RI; International Press, Cambridge, MA (2000), xii+232 pp.
Katok S., “p-adic analysis compared with real”, in Student Mathematical Library, 37, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA (2007), xiv+152 pp.
Khrennikov A., Kozyrev S. and Zúñiga-Galindo W.A., “Ultrametric Pseudodifferential Equations and Applications”, in Encyclopedia of Mathematics and its Applications vol. 168, Cambridge University Press (2018), xv+237 pp.
Kimura T., Introduction to prehomogeneous vector spaces, Translated from the 1998 Japanese original by Makoto Nagura and Tsuyoshi Niitani and revised by the author, Translations of Mathematical Monographs, 215. American Mathematical Society, Providence, RI, 2003.
Koblitz N., “p-adic Numbers, p-adic Analysis and Zeta functions”, Second edition, in Graduate Texts in Mathematics, 58, Springer-Verlag, New York (1984), xii+150 pp.
Lang S., “Real and functional analysis”, Third edition, in Graduate Texts in Mathematics, 142, Springer-Verlag, New York (1993), xiv+580 pp.
Lerner È.Y., “Feynman integrals of a p-adic argument in a momentum space. I. Convergence”, Theoret. and Math. Phys. 102 (1995), No. 3, 267–274.
Lerner È. Y., “Feynman integrals of a p-adic argument in a momentum space. II. Explicit formulas”, Theoret. and Math. Phys. 104 (1995), No. 3, 1061–1077.
Lerner È. Y. and Missarov M. D., “p-adic Feynman and string amplitudes”, Comm. Math. Phys. 121 (1989), No. 1, 35–48.
Lin S., “Ideal-theoretic strategies for asymptotic approximation of marginal likelihood integrals”, J. Algebr. Stat. 8 (2017), No. 1, 22–55.
Loeser F., “Fonctions zêta locales d’Igusa à plusieurs variables, intégration dans les fibres, et discriminants”, Ann. Sci. École Norm. Sup. 22 (1989), No. 3, 435–471.
Marcolli M., Feynman motives, World Scientific Publishing, NJ, 2010.
Mendoza-Martínez M.L., Vallejo J.A. and Zúñiga-Galindo W.A., “Acausal quantum theory for non-Archimedean scalar fields”, arXiv:1805.08613.
Meuser D., “A survey of Igusa’s local zeta function”, Amer. J. Math. 138 (2016), No. 1, 149–179.
Nicaise J., “An introduction to p-adic and motivic zeta functions and the monodromy conjecture”, in Algebraic and analytic aspects of zeta functions and L-functions, MSJ Mem., 21, Math. Soc. Japan, Tokyo (2010), 141–166.
Robert A., A course in p-adic Analysis, Springer-Verlag, New York, 2000.
Schikhof W.H., “Ultrametric calculus: An introduction to p-adic analysis”, in Cambridge Studies in Advanced Mathematics, 4, Cambridge University Press, Cambridge (2006), xii+306 pp.
Smirnov V.A., “Renormalization in p-adic quantum field theory”, Modern Phys. Lett. A 6 (1991), No. 15, 1421–1427.
Smirnov V.A., “Calculation of general p-adic Feynman amplitude”, Comm. Math. Phys. 149 (1992), No. 3, 623–636.
Speer E.R., “Generalized Feynman amplitudes”, in Annals of Mathematics Studies vol. 62, Princeton University Press (1969), vi+120 pp.
Speer E.R., “Ultraviolet and infrared singularity structure of generic Feynman amplitudes”, Ann. Inst. H. Poincaré Sect. A (N.S.) 23 (1975), No. 1, 1–21.
Spokoiny B.L., “Quantum Geometry of Non-archimedean Particles and Strings”, Phys. Lett. B 208 (1988), No. 3-4, 401–406.
Taibleson M.H., Fourier analysis on local fields, Princeton University Press, Tokyo, 1975.
Varadarajan V.S., Reflections on quanta, symmetries, and supersymmetries, Springer, New
York, 2011.
VeysW., “Arc spaces, motivic integration and stringy invariants”, in Singularity theory and its applications, Adv. Stud. Pure Math., 43, Math. Soc. Japan, Tokyo (2006), 529–572.
Vladimirov V.S., Volovich I.V., and Zelenov E.I., “p-adic analysis and mathematical physics”, in Series on Soviet and East European Mathematics 1, World Scientific, River Edge, NJ (1994), xx+319 pp.
Volovich I.V., “p-adic string”, Classical Quantum Gravity 4 (1987), No. 4, L83–L87.
Watanabe S., “Algebraic geometry and statistical learning theory”, in Cambridge Monographs on Applied and Computational Mathematics, 25, Cambridge University Press, Cambridge (2009), viii+286 pp.
Weil A., “Sur la formule de Siegel dans la théorie des groupes classiques”, Acta Math. 113 (1965), 1–87.
Zúñiga-Galindo W.A., “Igusa’s local zeta functions of semiquasihomogeneous polynomials”, Trans. Amer. Math. Soc. 353 (2001), No. 8, 3193–3207.
Zúñiga-Galindo W.A., “Local zeta functions and Newton polyhedra”, Nagoya Math. J. 172 (2003), 31–58.
Zúñiga-Galindo W.A., “Pseudodifferential equations over non-Archimedean spaces”, in Lectures Notes in Mathematics vol. 2174, Springer, Cham. (2016), xvi+175 pp.