In this thesis we study moduli stacks \calM_\Phi^n, indexed by an integer n>0 and a CM-type (K,\Phi), which parametrize abelian schemes equipped with action by \OK and an \OK-linear principal polarization, such that the representation of \OK on the relative Lie algebra of the abelian scheme consists of n copies of each character in \Phi. We do this by systematically applying Serre's tensor construction, and for that we first establish a general correspondence between polarizations on abelian schemes M\otimes_R A arising from this construction and polarizations on the abelian scheme A, along with positive definite hermitian forms on the module M. Next we describe a tensor product of categories and apply it to the category \Herm_n(\OK) of finite non-degenerate positive-definite \OK-hermitian modules of rank n and the category fibred in groupoids \calM_\Phi^1 of principally polarized CM abelian schemes. Assuming n is prime to the class number of K, we show that Serre's tensor construction provides an identification of this tensor product with a substack of the moduli space \calM_\Phi^n, and that in some cases, such as when the base is finite type over \CC or an algebraically closed field of characteristic zero, this substack is the entire space. We then use this characterization to describe the Galois action on \calM_\Phi^n(\overline{\QQ}), by using the description of the action on \calM_\Phi^1(\overline{\QQ}) supplied by the main theorem of complex multiplication.

In this thesis we study moduli stacks \calM_\Phi^n, indexed by an integer n>0 and a CM-type (K,\Phi), which parametrize abelian schemes equipped with action by \OK and an \OK-linear principal polarization, such that the representation of \OK on the relative Lie algebra of the abelian scheme consists of n copies of each character in \Phi. We do this by systematically applying Serre's tensor construction, and for that we first establish a general correspondence between polarizations on abelian schemes M\otimes_R A arising from this construction and polarizations on the abelian scheme A, along with positive definite hermitian forms on the module M. Next we describe a tensor product of categories and apply it to the category \Herm_n(\OK) of finite non-degenerate positive-definite \OK-hermitian modules of rank n and the category fibred in groupoids \calM_\Phi^1 of principally polarized CM abelian schemes. Assuming n is prime to the class number of K, we show that Serre's tensor construction provides an identification of this tensor product with a substack of the moduli space \calM_\Phi^n, and that in some cases, such as when the base is finite type over \CC or an algebraically closed field of characteristic zero, this substack is the entire space. We then use this characterization to describe the Galois action on \calM_\Phi^n(\overline{\QQ}), by using the description of the action on \calM_\Phi^1(\overline{\QQ}) supplied by the main theorem of complex multiplication.

Cette thèse est divisée en deux parties. Dans la première partie, nous introduisons une nouvelle condition, appelée additivité torique, sur une famille de variétés abéliennes qui dégénèrent en un schéma semi-abelien au-dessus d’un diviseur à croisements normaux. La condition ne dépend que du module de Tate T l A(K sep ) de la fibre générique. Nous montrons que l’additivité torique est une condition suffisante pour l’existence d’un modèle de Néron, si la base est un schéma de caractéristique nulle. Dans le cas de la jacobienne d’une courbe lisse à réduction semi-stable, on obtient le même résultat sans aucune hypothèse sur la caractéristique de base; et nous montrons que l’additivité torique est aussi nécessaire pour l’existence d’un modèle de Néron, si la base est un schéma de caractéristique nulle. Dans la deuxième partie, on considère le cas d’une famille de courbes nodales sur un anneau de valuation discrète. On donne une condition combinatoire sur le graphe dual de la fibre spéciale, appelée semi-factorialité, qui équivaut au fait que tous les faisceaux inversibles sur la fibre générique s’étendent en des faisceaux inversibles sur l’espace total de la courbe. Il est démontré par la suite que cette condition est automatiquement satisfaite après un éclatement centré aux points fermés non-réguliers de la famille de courbes. On applique le résultat ci-dessus pour généraliser un théorème de Raynaud sur le modèle de Néron des jacobiennes de courbes lisses, au cas des courbes nodales. / This thesis is subdivided in two parts. In the first part, we introduce a new condition, called toric-additivity, on a family of abelian varieties degenerating to a semi-abelian scheme over a normal crossing divisor. The condition depends only on the Tate module TlA(Ksep) of the generic fibre, for a prime l invertible on the base. We show that toric-additivity is a sufficient condition for the existence of a Néron model if the base is a Q-scheme. In the case of the jacobian of a smooth curve with semi-stable reduction, we obtain the same result without assumptions on the base characteristic; and we show that toric-additivity is also necessary for the existence of a Néron model, when the base is a Q-scheme. In the second part, we consider the case of a family of nodal curves over a discrete valuation ring, having split singularities. We say that such a family is semi factorial if every line bundle on the generic fibre extends to a line bundle on the total space. We give a necessary and sufficient condition for semi- factoriality, in terms of combinatorics of the dual graph of the special fibre. In particular, we show that performing one blow-up with center the non regular closed points yields a semi-factorial model of the generic fibre. As an application, we extend the result of Raynaud relating Néron models of smooth curves and Picard functors of their regular models to the case of nodal curves having a semi-factorial model.