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Galois actions on Néron models of Jacobians

Lars H. Halle (2010)

Annales de l’institut Fourier

Let X be a smooth curve defined over the fraction field K of a complete discrete valuation ring R . We study a natural filtration of the special fiber of the Néron model of the Jacobian of X by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for X over R , and in particular are independent of the residue characteristic. Furthermore, we obtain information about...

Galois orbits and equidistribution: Manin-Mumford and André-Oort.

Andrei Yafaev (2009)

Journal de Théorie des Nombres de Bordeaux

We overview a unified approach to the André-Oort and Manin-Mumford conjectures based on a combination of Galois-theoretic and ergodic techniques. This paper is based on recent work of Klingler, Ullmo and Yafaev on the André-Oort conjecture, and of Ratazzi and Ullmo on the Manin-Mumford conjecture.

Generalised elliptic functions

Matthew England, Chris Athorne (2012)

Open Mathematics

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstraß ℘-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study. The first approach discussed...

Generic Newton polygons of Ekedahl-Oort strata: Oort’s conjecture

Shushi Harashita (2010)

Annales de l’institut Fourier

We study the moduli space of principally polarized abelian varieties in positive characteristic. In this paper we determine the Newton polygon of any generic point of each Ekedahl-Oort stratum, by proving Oort’s conjecture on intersections of Newton polygon strata and Ekedahl-Oort strata. This result tells us a combinatorial algorithm determining the optimal upper bound of the Newton polygons of principally polarized abelian varieties with a given isomorphism type of p -kernel.

Genres de Todd et valeurs aux entiers des dérivées de fonctions L

Christophe Soulé (2005/2006)

Séminaire Bourbaki

La géométrie d’Arakelov étudie les fibrés vectoriels sur une variété algébrique X définie sur les entiers, munis d’une métrique hermitienne lisse sur le fibré holomorphe associé (sur la variété analytique des points complexes de X ). Un théorème de “Riemann-Roch arithmétique” calcule le covolume du réseau euclidien des sections globales d’un tel fibré. Dans cette formule, le genre de Todd comporte un terme complémentaire, défini par une série formelle dont les coefficients font intervenir les valeurs...

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