English
Zeta functions and monodromy
Post-edited
Auteurs : Veys, Wim (Auteur de la conférence)
CIRM (Editeur )
11S40 - Zeta functions and $L$-functions
11S80 - Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.)
14D05 - Structure of families (Picard-Lefschetz, monodromy, etc.)
14J17 - Singularities
14E18 - Arcs and motivic integration
CIRM (Editeur )
Résumé :
The $p$-adic Igusa zeta function, topological and motivic zeta function are (related) invariants of a polynomial $f$, reflecting the singularities of the hypersurface $f = 0$. The first one has a number theoretical flavor and is related to counting numbers of solutions of $f = 0$ over finite rings; the other two are more geometric in nature. The monodromy conjecture relates in a mysterious way these invariants to another singularity invariant of $f$, its local monodromy. We will discuss in this survey talk rationality issues for these zeta functions and the origins of the conjecture.
Codes MSC : 11S40 - Zeta functions and $L$-functions
11S80 - Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.)
14D05 - Structure of families (Picard-Lefschetz, monodromy, etc.)
14J17 - Singularities
14E18 - Arcs and motivic integration
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de Publication : 17/02/15 Date de Captation : 03/02/15 Sous Collection : Research talks Catégorie arXiv : Algebraic Geometry ; Number Theory Domaine(s) : Géométrie Complexe & géométrie Algébrique ; Théorie des Nombres Format : QuickTime (.mov) Durée : 01:03:24 Audience : Chercheurs Download : https://videos.cirm-math.fr/2015-02-03_Veys.mp4 
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Informations sur la Rencontre
Nom de la Rencontre : Applications of Artin approximation in singularity theory / Applications de l'approximation de Artin en théorie des singularitésOrganisateurs de la Rencontre : Hauser, Herwig ; Rond, Guillaume Dates : 02/02/15 - 06/02/15 Année de la rencontre : 2015 URL de la Rencontre : https://conferences.cirm-math.fr/1474.html
Données de citation
DOI : 10.24350/CIRM.V.18690903Citer cette vidéo: Veys, Wim (2015). Zeta functions and monodromy. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18690903 URI : http://dx.doi.org/10.24350/CIRM.V.18690903 |
Bibliographie
- Denef, J., & Loeser, F. (1998). Motivic Igusa zeta functions. Journal of Algebraic Geometry 7(3), 505-537 - https://www.zbmath.org/?q=an:0943.14010
- Igusa, J. (2000). An introduction to the theory of local zeta functions. Providence, RI: American Mathematical Society. (Studies in Advanced Mathematics, 14) - https://www.zbmath.org/?q=an:0959.11047
- Nicaise, J. (2010). An introduction to $p$-adic and motivic zeta functions and the monodromy conjecture. In G. Bhowmik, K. Matsumoto, & H. Tsumura (Eds.), Algebraic and analytic aspects of zeta functions and $L$-functions (pp. 115-140). Tokyo: Mathematical Society of Japan. (MSJ Memoirs, 21) - https://www.zbmath.org/?q=an:1275.11147
- Veys, W. (1993). Poles of Igusa's local zeta function and monodromy. Bulletin de la Société Mathématique de France, 121(4), 545-598 - http://smf4.emath.fr/Publications/Bulletin/121/pdf/smf_bull_121_545-598.pdf
- Veys, W. (2006). Vanishing of principal value integrals on surfaces. Journal für die reine und angewandte Mathematik, 598, 139-158 - http://www.dx.doi.org/10.1515/CRELLE.2006.072
