Nombres d'intersection arithmétiques et opérateurs de Hecke sur les courbes modulaires

Abstract : The theme of this thesis is the action of the Hecke operators as correspondances on the modular curves X_0(N). On the one hand,
we study the relation between the Hecke algebra and Arakelov theory; on the other hand, we make an attemp to study the dynamics of the action of the Hecke operators on the set of supersingular elliptic curves.

We consider the modular curve X_0(N) endowed with the Poincaré metric (hyperbolic metric). This metric is singular at the elliptic points and at the cusps. We suppose N squarefree. Let XN be the model of this curve over Spec(Z) given by the modular interpretation obtained by Deligne and Rapoport. We define a generalized arithmetic Chow group CH(N) such that its elements are classes of pairs (D,g) where D is a Weil divisor on XN and g an admissible Green's current with respect to the Poincaré metric. J.-B. Bost and U. Kühn have generalized, independently, Arakelov's arithmetic intersection theory in such a way that a real valued bilinear form is defined on CH(N) x CH(N) in this framework where the metric is singular. We also study a version of CH(N) with real coefficients and up to numeric equivalence which is denoted CH(N)*.

We show that Hecke correspondences act on CH(N) and that this action is self-adjoint with respect to the Bost-Kühn bilinear form. This allows to diagonalize the action on CH(N)* and to define its eigenspaces. As an application we study the relative dualizing sheaf, seen as an element in CH(N)*, and its decomposition in eigencomponents. We compute the self-intersection of the eigencomponent associated to the cusp at infinity using a computation by Ulf Kühn.

The action of Hecke operators on the special fibers of XN defines a dynamic, and this dynamic preserves supersingular points. We study this action on the supersingular points inside the fibers of good reduction and we compute, using results by Deuring and Eichler, the asympotic frequence with which a given supersingular point visits another supersingular point.
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Ricardo Menares. Nombres d'intersection arithmétiques et opérateurs de Hecke sur les courbes modulaires. Mathématiques [math]. Université Paris Sud - Paris XI, 2008. Français. ⟨tel-00360171⟩

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