Séminaires : Séminaire Théorie des Nombres

Equipe(s) : tn,
Responsables :Ziyang Gao, Marc Hindry, Bruno Kahn, João Pedro P. dos Santos
Email des responsables : ziyang.gao@imj-prg.fr
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Orateur(s) Robin de Jong - ,
Titre Logarithmic equidistribution of division points on superelliptic curves
Date22/03/2010
Horaire14:00 à 16:00
RésumeA 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.
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