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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, doi: 10.5802/aif.2454
Article PDF | Reviews MR 2543659 | Zbl 1192.14020 | 1 citation in Cedram
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 $0$.

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