Jacobian Varieties

Abstract

The main objective of this chapter is the uniformization of the Jacobian of a smooth projective curve $X_{K}$ over a non-Archimedean field $K$ and its relationship to a semi-stable reduction $\widetilde{X}$ of $X_{K}$.

We assume that the reader is familiar with the notion of the Jacobian variety of a smooth projective curve over a field; see for instance the article (Milne in Arithmetic Geometry, Springer, Berlin, 1986) or (Bosch et al. in Néron Models, vol. 21, Springer, Berlin/Heidelberg/New York, 1990; Chap. 9). For our purpose it is necessary to have analyzed the generalized Jacobian of a semi-stable curve $\widetilde{X}$, especially its representation as a torus extension of the Jacobian of its normalization $\widetilde{X}'$. In Sects. 5.1 and 5.2 we reassemble the main results we need in the sequel.

In Sect. 5.3 it is shown that the generalized Jacobian $\widetilde{J}:=\operatorname{Jac}{\widetilde{X}}$ has a lifting $\overline{J}_{K}$ as an open rigid analytic subgroup of $J_{K}:=\operatorname{Jac}{X_{K}}$ and that $\overline{J}_{K}$ has a smooth formal $R$-model $\overline{J}$ with semi-abelian reduction. $\overline{J}$ is a formal torus extension of a formal abelian $R$-scheme $B$ with reduction $\widetilde{B}=\operatorname{Jac}{\widetilde{X}'}$.

The generic fiber $\overline{J}_{K}$ of $\overline{J}$ is the largest connected open subgroup of $J_{K}$ which admits a smooth formal $R$-model; this is discussed in Sect. 5.4 in a more general context. The relationship between the maximal formal torus $\overline {T}$ of $\overline{J}$ and the group $H^{1}(X_{K},\mathbb{Z})$ shows that the inclusion map $\overline{T}_{K}\hookrightarrow\overline{J}_{K}$ from the generic fiber $\overline{T}_{K}$ of the formal torus $\overline{T}$ to $\overline{J}_{K}$ extends to a rigid analytic group homomorphism $T_{K}\to J_{K}$, where $T_{K}$ is the affine torus which contains $\overline {T}_{K}$ as the torus of units.

The push-out $\widehat{J}_{K}:=T_{K}\amalg_{\overline{T}}\overline {J}_{K}$ is a rigid analytic group which contains $\overline{J}_{K}$ as an open rigid analytic subgroup and the inclusion $\overline {J}_{K}\hookrightarrow J_{K}$ extends to a surjective homomorphism $\widehat{J}_{K}\to J_{K}$ of rigid analytic groups. The kernel of the latter map is a lattice $M$ in $\widehat{J}_{K}$ and makes $J_{K}=\widehat {J}_{K}/M$ into a quotient of the “universal covering” $\widehat{J}_{K}$. The representation $J_{K}=\widehat{J}_{K}/M$ is called the Raynaud representation of $J_{K}$.

Since every abelian variety is isogenous to a subvariety of a product of Jacobians, one can transfer the results to abelian varieties. This implies Grothendieck’s semi-abelian reduction theorem for abelian varieties; cf. (Grothendieck et al. in Groupes de Monodromie en Géométrie Algébrique, vols. 288, 340, Springer, Berlin/Heidelberg/New York, 1972).

We want to mention that there are also contributions by Fresnel, Reversat and van der Put (in Rigid Analytic Geometry and Its Applications, vol. 218, Birkhäuser Boston, Inc., Boston 2004) and (in Bull. Soc. Math. Fr. 117(4):415–444, 1989).