Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- 9 Heights and measures on analytic spaces. A survey of recent results, and some remarks
- 10 C-minimal structures without density assumption
- 11 Trees of definable sets in ℤp
- 12 Triangulated motives over noetherian separated schemes
- 13 A survey of algebraic exponential sums and some applications
- 14 A motivic version of p-adic integration
- 15 Absolute desingularization in characteristic zero
- References
9 - Heights and measures on analytic spaces. A survey of recent results, and some remarks
Published online by Cambridge University Press: 07 October 2011
Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- 9 Heights and measures on analytic spaces. A survey of recent results, and some remarks
- 10 C-minimal structures without density assumption
- 11 Trees of definable sets in ℤp
- 12 Triangulated motives over noetherian separated schemes
- 13 A survey of algebraic exponential sums and some applications
- 14 A motivic version of p-adic integration
- 15 Absolute desingularization in characteristic zero
- References
Summary
The first goal of this paper was to survey my definition in [19] of measures on non-archimedean analytic spaces in the sense of Berkovich and to explain its applications in Arakelov geometry. These measures are analogous the measures on complex analytic spaces given by products of first Chern forms of hermitian line bundles. In both contexts, archimedean and non-archimedean, they are related with Arakelov geometry and the local height pairings of cycles. However, while the archimedean measures lie at the ground of the definition of the archimedean local heights in Arakelov geometry, the situation is reversed in the ultrametric case: we begin with the definition of local heights given by arithmetic intersection theory and define measures in such a way that the archimedean formulae make sense and are valid. The construction is outlined in Section 1, with references concerning metrized line bundles and the archimedean setting. More applications to Arakelov geometry and equidistribution theorems are discussed in Section 3.
The relevance of Berkovich spaces in Diophantine geometry has now made been clear by many papers; besides [19] and [20] and the general equidistribution theorem of Yuan [59], I would like to mention the works [38, 39, 40, 30] who discuss the function field case of the equidistribution theorem, as well as the potential theory on non-archimedean curves developed simultaneously by Favre, Jonsson & Rivera-Letelier [32, 33] and Baker & Rumely for the projective line [8], and in general by A. Thuillier's PhD thesis [55].
- Type
- Chapter
- Information
- Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry , pp. 1 - 50Publisher: Cambridge University PressPrint publication year: 2011
References
[1] , and 2000. Line bundles, rational points and ideal classes. Math. Res. Letters, 7, 709–717.CrossRefGoogle Scholar
[2] . 1974. Intersection theory of divisors on an arithmetic surface. Izv. Akad. Nauk SSSR Ser. Mat., 38(6), 1167–1180.Google Scholar
[3] . 2001. Points entiers sur les surfaces arithmétiques. J. reine angew. Math., 531, 201–235.Google Scholar
[4] . 2009. A Finiteness Theorem for Canonical Heights Attached to Rational Maps over Function Fields. J. reine angew. Math., 626, 205–233. arXiv:math.NT/0601046.Google Scholar
[5] , and . 2009. Preperiodic points and unlikely intersections. To appear in Duke math. J; arXiv:0911.0918.
[6] , and . 2009. Metric properties of the tropical Abel-Jacobi map. To appear in Journal of Algebraic Combinatorics. 33 (3), 349–381.CrossRefGoogle Scholar
[7] , and . 2007. Harmonic Analysis on Metrized Graphs. Canad. J. Math., 59(2), 225–275.CrossRefGoogle Scholar
[8] , and . 2010. Potential Theory and Dynamics on the Berkovich Projective Line. Surveys and Mathematical Monographs, vol. 159. Amer. Math. Soc.Google Scholar
[9] . 1993. Survey of pluri-potential theory. In: Several complex variables. Math. Notes, no. 38. Stockholm, 1987/1988: Princeton Univ. Press.Google Scholar
[10] , and 1982. A new capacity for plurisubharmonic functions. Acta Math., 149, 1–40.CrossRefGoogle Scholar
[11] 1998. Fatou components in p-adic dynamics. PhD Thesis., 1–98.
[12] 1990. Spectral theory and analytic geometry over non-Archimedean fields. Mathematical Surveys and Monographs, vol. 33. Providence, RI: American Mathematical Society.Google Scholar
[13] 1999. Smooth p-adic analytic spaces are locally contractible. Invent. Math., 137(1), 1–84.CrossRefGoogle Scholar
[14] , , and . 1995. Nonarchimedean Arakelov theory. J. Algebraic Geometry, 4, 427–485.Google Scholar
[15] . 1980. Points of finte order on abelian varieties. Izv. Akad. Nauk. SSSR Ser. Mat., 44(4), 782–804, 973.Google Scholar
[16] . 1999. Potential theory and Lefschetz theorems for arithmetic surfaces. Ann. Sci. École Norm. Sup., 32(2), 241–312.CrossRefGoogle Scholar
[17] , and . 1993. Canonical heights on varieties with morphisms. Compositio Math., 89, 163–205.Google Scholar
[18] . 2000. Points de petite hauteur sur les variétés semi-abéliennes. Ann. Sci. École Norm. Sup., 33(6), 789–821.CrossRefGoogle Scholar
[19] . 2006. Mesures et équidistribution sur des espaces de Berkovich. J. reine angew. Math., 595, 215–235. math.NT/0304023.Google Scholar
[20] , and . 2009. Mesures de Mahler et équidistribution logarithmique. Ann. Inst. Fourier (Grenoble), 59(3), 977–1014.CrossRefGoogle Scholar
[21] , and . 2008. Difference fields and descent in algebraic dynamics. I. J. Inst. Math. Jussieu, 7(4), 653–686.CrossRefGoogle Scholar
[23] . 2009. Zhang's Conjecture and the Effective Bogomolov Conjecture over function fields. arXiv:0901.3945.
[24] . 2009. Canonical height and logarithmic equidistribution on superelliptic curves. arXiv:0911.1271.
[25] 1987. Le déterminant de la cohomologie. Pages 93–177 of: Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985). Contemp. Math., vol. 67. Providence, RI: Amer. Math. Soc.CrossRefGoogle Scholar
[26] . 1985. Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines. Mém. Soc. Math. France, 124.Google Scholar
[27] . 1993. Monge-Ampère operators, Lelong numbers and Intersection theory. Pages 115–193 of: Complex analysis and geometry. Univ. Ser. Math. New York: Plenum.CrossRefGoogle Scholar
[28] 1999. On the elliptic analogue of Jensen's formula. J. London Math. Soc. (2), 59(1), 21–36.CrossRefGoogle Scholar
[29] , and . 1996. The elliptic Mahler measure. Math. Proc. Cambridge Philos. Soc., 120(1), 13–25. Heights and measures on analytic spaces 49CrossRefGoogle Scholar
[30] 2009. Equidistribution of dynamically small subvarieties over the function field of a curve. Acta Arith., 137(4), 345–389.CrossRefGoogle Scholar
[31] . 1984. Calculus on arithmetic surfaces. Ann. of Math. (2), 119(2), 387–424.CrossRefGoogle Scholar
[32] , and . 2004. The valuative tree. Lecture Notes in Mathematics, vol. 1853. Berlin: Springer-Verlag.Google Scholar
[33] , and . 2007. Théorie ergodique des fractions rationnelles sur un corps ultramétrique. arXiv:0709.0092.
[34] , and . 1990. Arithmetic intersection theory. Publ. Math. Inst. Hautes Études Sci., 72, 94–174.CrossRefGoogle Scholar
[35] . 1997. Heights of subvarieties over M-fields. Pages 190–227 of: (ed), Arithmetic geometry. Symp. Math., vol. 37.Google Scholar
[36] . 1998. Local heights of subvarieties over non-archimedean fields. J. reine angew. Math., 498, 61–113.Google Scholar
[37] . 2003. Local and canonical heights of subvarieties. Ann. Scuola Norm. Sup. Pisa, 2(4), 711–760.Google Scholar
[38] . 2007a. The Bogomolov conjecture for totally degenerate abelian varieties. Invent. Math., 169(2), 377–400.CrossRefGoogle Scholar
[39] . 2007b. Tropical varieties for non-Archimedean analytic spaces. Invent. Math., 169(2), 321–376.CrossRefGoogle Scholar
[40] . 2008. Equidistribution over function fields. Manuscripta Math., 127(4), 485–510.CrossRefGoogle Scholar
[41] . 2010. Non-archimedean canonical measures on abelian varieties. Compositio Math. 146 (3), 683–730.CrossRefGoogle Scholar
[42] . 2004. Admissible metrics for line bundles on curves and abelian varieties over non-Archimedean local fields. Arch. Math. (Basel), 82(2), 128–139.CrossRefGoogle Scholar
[43] , and . 1994. Superattractive fixed points in Cn. Indiana Univ. Math. J., 43(1), 321–365.CrossRefGoogle Scholar
[44] , and 2009. Nonarchimedean Green functions and dynamics on projective space. Math. Z., 262(1), 173–197.CrossRefGoogle Scholar
[45] , and . 2001. Homological mirror symmetry and torus fibrations. Pages 203–263 of: Symplectic geometry and mirror symmetry (Seoul, 2000). World Sci. Publ., River Edge, NJ.CrossRefGoogle Scholar
[46] , and . 2006. Affine structures and non-Archimedean analytic spaces. Pages 321–385 of: The unity of mathematics. Progr. Math., vol. 244. Boston, MA: Birkhäuser Boston.Google Scholar
[47] . 2000. Géométrie d'Arakelov des variétés toriques et fibrés en droites intégrables. Mém. Soc. Math. France, 129.Google Scholar
[48] . 1997. On the preperiodic points of an endomorphism ofP1 × P1. PhD Thesis, Columbia University.Google Scholar
[49] , , and 2009. Adynamical pairing between two rational maps. arXiv:0911.1875.
[50] . 1983a. Courbes sur une variété abélienne et points de torsion. Invent. Math., 71(1), 207–233.CrossRefGoogle Scholar
[51] . 1983b. Sous-variétés d'une variété abélienne et points de torsion. Pages 327–352 of: , and (eds), Arithmetic and Geometry. Papers dedicated to I.R. Shafarevich. Progr. Math., no. 35. Birkhäuser.Google Scholar
[52] . 2003. Dynamique des fonctions rationnelles sur des corps locaux. Astérisque, xv, 147–230. Geometric methods in dynamics. II.Google Scholar
[53] . 1993. An effective Matsusaka big theorem. Ann. Inst. Fourier (Grenoble), 43(5), 1387–1405.CrossRefGoogle Scholar
[54] , , and . 1997. Équidistribution des petits points. Invent. Math., 127, 337–348.CrossRefGoogle Scholar
[55] . 2005. Théorie du potentiel sur les courbes en géométrie non archimédienne. Applications à la théorie d'Arakelov. Ph.D. thesis, Université de Rennes 1.Google Scholar
[56] . 1994. Fatou sets in complex dynamics on projective spaces. J. Math. Soc. Japan, 46(3), 545–555.CrossRefGoogle Scholar
[57] . 1998. Positivité et discrétion des points algébriques des courbes. Ann. of Math., 147(1), 167–179.CrossRefGoogle Scholar
[58] , and . 2009. Calabi theorem and algebraic dynamics. http://www.math.columbia.edu/~szhang/papers/calabi%20and%20semigroup.pdf.
[59] . 2008. Big line bundles on arithmetic varieties. Invent. Math., 173, 603–649. arXiv:math.NT/0612424.CrossRefGoogle Scholar
[62] . 1995a. Positive line bundles on arithmetic varieties. J. Amer. Math. Soc., 8, 187–221.CrossRefGoogle Scholar
[64] . 1998. Equidistribution of small points on abelian varieties. Ann. of Math., 147(1), 159–165.CrossRefGoogle Scholar
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