Table of contents for this issue | Previous article | Next article
Antoine Chambert-Loir; Amaury Thuillier
Mesures de Mahler et équidistribution logarithmique
(Mahler measures and logarithmic equidistribution)
Annales de l'institut Fourier, 59 no. 3 (2009), p. 977-1014
Article: subscription required | Reviews MR 2543659
Class. Math.: 14G40, 14G22, 32U05
Keywords: Mahler measures, equidistribution, Arakelov geometry, points of small height, dynamical systems, Berkovich analytic spaces
[1] Ahmed Abbes & Thierry Bouche, “Théorème de Hilbert–Samuel « arithmétique »”, Ann. Inst. Fourier (Grenoble) 45 (1995), p. 375-401
Cedram | MR 1343555 | Zbl 0818.14011
[2] Pascal Autissier, “Points entiers sur les surfaces arithmétiques”, J. reine angew. Math. 531 (2001), p. 201-235 MR 1810122 | Zbl 1007.11041
[3] Pascal Autissier, “Équidistribution des sous-variétés de petite hauteur”, J. Théor. Nombres Bordeaux 18 (2006), p. 1-12, arXiv:math.NT/0404355
Cedram | MR 2245872 | Zbl pre05070444
[4] Pascal Autissier, “Sur une question d’équirépartition de nombres algébriques”, C. R. Acad. Sci. Paris Sér. I Math. 342 (2006), p. 639-641 MR 2225867 | Zbl 1129.11046
[5] Matthew Baker & Robert Rumely, “Equidistribution of small points, rational dynamics, and potential theory”, Ann. Inst. Fourier (Grenoble) 56 (2006), p. 625-688, arXiv:math.NT/0407426
Cedram | MR 2244226 | Zbl pre05176555
[6] Eric Bedford & B.A. Taylor, “A new capacity for plurisubharmonic functions”, Acta Math. 149 (1982), p. 1-40 MR 674165 | Zbl 0547.32012
[7] Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs 33, American Mathematical Society, 1990 MR 1070709 | Zbl 0715.14013
[8] Spencer Bloch, Henri Gillet & Christophe Soulé, “Non-archimedean Arakelov theory”, J. Algebraic Geometry 4 (1995), p. 427-485 MR 1325788 | Zbl 0866.14011
[9] Enrico Bombieri & Jeffrey Vaaler, “On Siegel’s lemma”, Invent. Math. 73 (1983), p. 11-32 MR 707346 | Zbl 0533.10030
[10] S. Bosch, U. Güntzer & R. Remmert, Non-Archimedean analysis. A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften 261, Springer-Verlag, 1984 MR 746961 | Zbl 0539.14017
[11] Jean-Benoît Bost, Henri Gillet & Christophe Soulé, “Heights of projective varieties and positive Green forms”, J. Amer. Math. Soc. 7 (1994), p. 903-1027 Zbl 0973.14013
[12] N. Bourbaki, Intégration, chapitres 1 à 4, Springer-Verlag, 2007, réimpression de l’édition Hermann de 1965
[13] Antoine Chambert-Loir, “Mesures et équidistribution sur des espaces de Berkovich”, J. reine angew. Math. 595 (2006), p. 215-235, arXiv:math.NT/0304023 MR 2244803 | Zbl 1112.14022
[14] Sinnou David & Patrice Philippon, “Minorations des hauteurs normalisées des sous-variétés des tores”, Ann. Scuola Norm. Sup. Pisa 28 (1999), p. 489-543
Numdam | MR 1736526 | Zbl 1002.11055
[15] Jean-Pierre Demailly, “Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines”, Mém. Soc. Math. France (1985)
Numdam | MR 813252 | Zbl 0579.32012
[16] Jean-Pierre Demailly, Monge-Ampère operators, Lelong numbers and Intersection theory, Complex analysis and geometry, Plenum, 1993, p. 115–193 MR 1211880 | Zbl 0792.32006
[17] Graham Everest & Thomas Ward, Heights of polynomials and entropy in algebraic dynamics, Springer-Verlag London Ltd., 1999 MR 1700272 | Zbl 0919.11064
[18] Gerd Faltings, “Diophantine approximation on abelian varieties”, Ann. of Math. 133 (1991), p. 549-576 MR 1109353 | Zbl 0734.14007
[19] Charles Favre & Juan Rivera-Letelier, “Equidistribution quantitative des points de petite hauteur sur la droite projective”, Math. Ann. 335 (2006), p. 311-361, arXiv:math.NT/0407471 MR 2221116 | Zbl pre05035986
[20] Takao Fujita, “Approximating Zariski decomposition of big line bundles”, Kodai Math. J. 17 (1994), p. 1-3
Article | MR 1262949 | Zbl 0814.14006
[21] William Fulton, Intersection theory, Springer-Verlag, 1998 MR 1644323 | Zbl 0885.14002
[22] Henri Gillet & Christophe Soulé, “Amplitude arithmétique”, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), p. 887-890 MR 974432 | Zbl 0676.14007
[23] Henri Gillet & Christophe Soulé, “Arithmetic intersection theory”, Publ. Math. Inst. Hautes Études Sci. 72 (1990), p. 94-174
Numdam | MR 1087394 | Zbl 0741.14012
[24] Walter Gubler, “Local heights of subvarieties over non-archimedean fields”, J. reine angew. Math. 498 (1998), p. 61-113 MR 1629925 | Zbl 0906.14013
[25] Wayne M. Lawton, “A generalization of a theorem of Kronecker”, J. Sci. Fac. Chiang Mai Univ. 4 (1977), p. 15-23 Zbl 0447.12004
[26] Robert Lazarsfeld, Positivity in algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete 48–49, Springer-Verlag, 2004 Zbl 1093.14500
[27] Vincent Maillot, “Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables”, Mém. Soc. Math. France (2000)
Numdam | Zbl 0963.14009
[28] Atsushi Moriwaki, “Continuity of volumes on arithmetic varieties”, posted on May 13, 2008, PII S 1056-3911(08)00500-6 (à paraître) ; arXiv:math.NT/0612269, 2008
arXiv
[29] Patrice Philippon, “Lemmes de zéros dans les groupes algébriques commutatifs”, Bull. Soc. Math. France 114 (1986), p. 355-383
Numdam | MR 878242 | Zbl 0617.14001
[30] Patrice Philippon, “Sur des hauteurs alternatives, I”, Math. Ann. 289 (1991), p. 255-283
Article | MR 1092175 | Zbl 0726.14017
[31] Jorge Pineiro, Lucien Szpiro & Thomas J. Tucker, Mahler measure for dynamical systems on ${\mathbb{P}}^1$ and intersection theory on a singular arithmetic surface, Geometric methods in algebra and number theory, Birkhäuser Boston, 2005, p. 219–250 MR 2166086 | Zbl 1101.11020
[32] Robert Rumely, On Bilu’s Equidistribution theorem, in Spectral problems in geometry and arithmetic, Contemp. Math., 1999, p. 159-166 MR 1710794 | Zbl 1029.11030
[33] Robert Rumely, Chi Fong Lau & Robert Varley, “Existence of the sectional capacity”, Mem. Amer. Math. Soc. 145 (2000) MR 1677934 | Zbl 0987.14018
[34] Lucien Szpiro & Tom Tucker, “Equidistribution and generalized Mahler measures”, preprint, 2005
arXiv
[35] Lucien Szpiro, Emmanuel Ullmo & Shou-Wu Zhang, “Équidistribution des petits points”, Invent. Math. 127 (1997), p. 337-348 MR 1427622 | Zbl 0991.11035
[36] Amaury Thuillier, Théorie du potentiel sur les courbes en géométrie non archimédienne. Applications à la théorie d’Arakelov, thesis, Université de Rennes 1, 2005
Article
[37] Emmanuel Ullmo, “Positivité et discrétion des points algébriques des courbes”, Ann. of Math. 147 (1998), p. 167-179 MR 1609514 | Zbl 0934.14013
[38] Xinyi Yuan, “Big line bundles on arithmetic varieties”, Invent. Math. 173 (2008), p. 603-649, arXiv:math.NT/0612424 MR 2425137 | Zbl 1146.14016
[39] Shou-Wu Zhang, “Positive line bundles on arithmetic varieties”, J. Amer. Math. Soc. 8 (1995), p. 187-221 MR 1254133 | Zbl 0861.14018
[40] Shou-Wu Zhang, “Small points and adelic metrics”, J. Algebraic Geometry 4 (1995), p. 281-300 MR 1311351 | Zbl 0861.14019
[41] Shou-Wu Zhang, “Equidistribution of small points on abelian varieties”, Ann. of Math. 147 (1998), p. 159-165 MR 1609518 | Zbl 0991.11034
Antoine Chambert-Loir; Amaury Thuillier
Mesures de Mahler et équidistribution logarithmique
(Mahler measures and logarithmic equidistribution)
Annales de l'institut Fourier, 59 no. 3 (2009), p. 977-1014
Article: subscription required | Reviews MR 2543659
Class. Math.: 14G40, 14G22, 32U05
Keywords: Mahler measures, equidistribution, Arakelov geometry, points of small height, dynamical systems, Berkovich analytic spaces
Résumé - Abstract
Let $X$ be a projective integral scheme over a number field $F$; let $L$ be a ample line bundle on $X$ together with a semi-positive adelic metric in the sense of Zhang. The main results of this article are
(1)A formula which allows to compute the local heights (relative to $L$) of a Cartier divisor $D$ on $X$ as generalized “Mahler measures”, i.e., integrals of Green functions for $D$ against measures attached to $L$;
(2)A theorem of equidistribution of points of “small” height valid for functions with logarithmic singularities along a divisor $D$, provided the height of $D$ is “minimal”. In the context of algebraic dynamics, “small” means of height converging to $0$, and “minimal” means height $0$.
Bibliography
Cedram | MR 1343555 | Zbl 0818.14011
[2] Pascal Autissier, “Points entiers sur les surfaces arithmétiques”, J. reine angew. Math. 531 (2001), p. 201-235 MR 1810122 | Zbl 1007.11041
[3] Pascal Autissier, “Équidistribution des sous-variétés de petite hauteur”, J. Théor. Nombres Bordeaux 18 (2006), p. 1-12, arXiv:math.NT/0404355
Cedram | MR 2245872 | Zbl pre05070444
[4] Pascal Autissier, “Sur une question d’équirépartition de nombres algébriques”, C. R. Acad. Sci. Paris Sér. I Math. 342 (2006), p. 639-641 MR 2225867 | Zbl 1129.11046
[5] Matthew Baker & Robert Rumely, “Equidistribution of small points, rational dynamics, and potential theory”, Ann. Inst. Fourier (Grenoble) 56 (2006), p. 625-688, arXiv:math.NT/0407426
Cedram | MR 2244226 | Zbl pre05176555
[6] Eric Bedford & B.A. Taylor, “A new capacity for plurisubharmonic functions”, Acta Math. 149 (1982), p. 1-40 MR 674165 | Zbl 0547.32012
[7] Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs 33, American Mathematical Society, 1990 MR 1070709 | Zbl 0715.14013
[8] Spencer Bloch, Henri Gillet & Christophe Soulé, “Non-archimedean Arakelov theory”, J. Algebraic Geometry 4 (1995), p. 427-485 MR 1325788 | Zbl 0866.14011
[9] Enrico Bombieri & Jeffrey Vaaler, “On Siegel’s lemma”, Invent. Math. 73 (1983), p. 11-32 MR 707346 | Zbl 0533.10030
[10] S. Bosch, U. Güntzer & R. Remmert, Non-Archimedean analysis. A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften 261, Springer-Verlag, 1984 MR 746961 | Zbl 0539.14017
[11] Jean-Benoît Bost, Henri Gillet & Christophe Soulé, “Heights of projective varieties and positive Green forms”, J. Amer. Math. Soc. 7 (1994), p. 903-1027 Zbl 0973.14013
[12] N. Bourbaki, Intégration, chapitres 1 à 4, Springer-Verlag, 2007, réimpression de l’édition Hermann de 1965
[13] Antoine Chambert-Loir, “Mesures et équidistribution sur des espaces de Berkovich”, J. reine angew. Math. 595 (2006), p. 215-235, arXiv:math.NT/0304023 MR 2244803 | Zbl 1112.14022
[14] Sinnou David & Patrice Philippon, “Minorations des hauteurs normalisées des sous-variétés des tores”, Ann. Scuola Norm. Sup. Pisa 28 (1999), p. 489-543
Numdam | MR 1736526 | Zbl 1002.11055
[15] Jean-Pierre Demailly, “Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines”, Mém. Soc. Math. France (1985)
Numdam | MR 813252 | Zbl 0579.32012
[16] Jean-Pierre Demailly, Monge-Ampère operators, Lelong numbers and Intersection theory, Complex analysis and geometry, Plenum, 1993, p. 115–193 MR 1211880 | Zbl 0792.32006
[17] Graham Everest & Thomas Ward, Heights of polynomials and entropy in algebraic dynamics, Springer-Verlag London Ltd., 1999 MR 1700272 | Zbl 0919.11064
[18] Gerd Faltings, “Diophantine approximation on abelian varieties”, Ann. of Math. 133 (1991), p. 549-576 MR 1109353 | Zbl 0734.14007
[19] Charles Favre & Juan Rivera-Letelier, “Equidistribution quantitative des points de petite hauteur sur la droite projective”, Math. Ann. 335 (2006), p. 311-361, arXiv:math.NT/0407471 MR 2221116 | Zbl pre05035986
[20] Takao Fujita, “Approximating Zariski decomposition of big line bundles”, Kodai Math. J. 17 (1994), p. 1-3
Article | MR 1262949 | Zbl 0814.14006
[21] William Fulton, Intersection theory, Springer-Verlag, 1998 MR 1644323 | Zbl 0885.14002
[22] Henri Gillet & Christophe Soulé, “Amplitude arithmétique”, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), p. 887-890 MR 974432 | Zbl 0676.14007
[23] Henri Gillet & Christophe Soulé, “Arithmetic intersection theory”, Publ. Math. Inst. Hautes Études Sci. 72 (1990), p. 94-174
Numdam | MR 1087394 | Zbl 0741.14012
[24] Walter Gubler, “Local heights of subvarieties over non-archimedean fields”, J. reine angew. Math. 498 (1998), p. 61-113 MR 1629925 | Zbl 0906.14013
[25] Wayne M. Lawton, “A generalization of a theorem of Kronecker”, J. Sci. Fac. Chiang Mai Univ. 4 (1977), p. 15-23 Zbl 0447.12004
[26] Robert Lazarsfeld, Positivity in algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete 48–49, Springer-Verlag, 2004 Zbl 1093.14500
[27] Vincent Maillot, “Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables”, Mém. Soc. Math. France (2000)
Numdam | Zbl 0963.14009
[28] Atsushi Moriwaki, “Continuity of volumes on arithmetic varieties”, posted on May 13, 2008, PII S 1056-3911(08)00500-6 (à paraître) ; arXiv:math.NT/0612269, 2008
arXiv
[29] Patrice Philippon, “Lemmes de zéros dans les groupes algébriques commutatifs”, Bull. Soc. Math. France 114 (1986), p. 355-383
Numdam | MR 878242 | Zbl 0617.14001
[30] Patrice Philippon, “Sur des hauteurs alternatives, I”, Math. Ann. 289 (1991), p. 255-283
Article | MR 1092175 | Zbl 0726.14017
[31] Jorge Pineiro, Lucien Szpiro & Thomas J. Tucker, Mahler measure for dynamical systems on ${\mathbb{P}}^1$ and intersection theory on a singular arithmetic surface, Geometric methods in algebra and number theory, Birkhäuser Boston, 2005, p. 219–250 MR 2166086 | Zbl 1101.11020
[32] Robert Rumely, On Bilu’s Equidistribution theorem, in Spectral problems in geometry and arithmetic, Contemp. Math., 1999, p. 159-166 MR 1710794 | Zbl 1029.11030
[33] Robert Rumely, Chi Fong Lau & Robert Varley, “Existence of the sectional capacity”, Mem. Amer. Math. Soc. 145 (2000) MR 1677934 | Zbl 0987.14018
[34] Lucien Szpiro & Tom Tucker, “Equidistribution and generalized Mahler measures”, preprint, 2005
arXiv
[35] Lucien Szpiro, Emmanuel Ullmo & Shou-Wu Zhang, “Équidistribution des petits points”, Invent. Math. 127 (1997), p. 337-348 MR 1427622 | Zbl 0991.11035
[36] Amaury Thuillier, Théorie du potentiel sur les courbes en géométrie non archimédienne. Applications à la théorie d’Arakelov, thesis, Université de Rennes 1, 2005
Article
[37] Emmanuel Ullmo, “Positivité et discrétion des points algébriques des courbes”, Ann. of Math. 147 (1998), p. 167-179 MR 1609514 | Zbl 0934.14013
[38] Xinyi Yuan, “Big line bundles on arithmetic varieties”, Invent. Math. 173 (2008), p. 603-649, arXiv:math.NT/0612424 MR 2425137 | Zbl 1146.14016
[39] Shou-Wu Zhang, “Positive line bundles on arithmetic varieties”, J. Amer. Math. Soc. 8 (1995), p. 187-221 MR 1254133 | Zbl 0861.14018
[40] Shou-Wu Zhang, “Small points and adelic metrics”, J. Algebraic Geometry 4 (1995), p. 281-300 MR 1311351 | Zbl 0861.14019
[41] Shou-Wu Zhang, “Equidistribution of small points on abelian varieties”, Ann. of Math. 147 (1998), p. 159-165 MR 1609518 | Zbl 0991.11034
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310