# Séminaires : Séminaire Théorie des Nombres

 Equipe(s) : fa, tn, tga, Responsables : Marc Hindry, Bruno Kahn, Wieslawa Niziol, Cathy Swaenepoel Email des responsables : cathy.swaenepoel@imj-prg.fr Salle : Adresse : Description http://www.imj-prg.fr/tn/STN/stnj.html

 Orateur(s) Robin de Jong - , Titre Logarithmic equidistribution of division points on superelliptic curves Date 22/03/2010 Horaire 14:00 à 16:00 Diffusion Résume A superelliptic curve is a curve over a number field $K$ given by anequation $y^N=f(x)$, with suitable conditions on $f$ and $N$. On such curves onehas the notion of $n$-division points, generalizing the notion of $n$-torsionpoints on elliptic curves. We prove that the N\'eron-Tate height,restricted to the canonical image of $X$ in its jacobian, satisfies a Mahlertype formula, i.e. can be written as a sum, over all places of $K$, ofcertain local logarithmic integrals over $X$. Also we prove that for almostall algebraic points on $X$ these local integrals can be computed byaveraging over the n-division points of $X$, and letting $n$ tend to infinity.For elliptic curves these results were shown by Everest-ni Fhlathuin andEverest-Ward. Our proofs involve, among other things, an application ofpotential theory on (Berkovich) analytic curves, and an application ofFaltings's diophantine approximation on abelian varieties. Salle Adresse