Reduction of points in the group of components, by Dino J. Lorenzini

Let $K$ be a complete discrete valuation field with ring of integers $\co_K$. Let $X/K$ be a proper smooth curve and let $A/K$ denote its jacobian. Let $P$ and $Q$ belong to $X(K)$. The divisor $P - Q$ defines a $K$-rational point of $A/K$. In this article, we study the reduction of $P - Q$ in the N\'eron model ${\cal A}_K/{\cal O}_K$ of $A/K$ in terms of the reductions of the points $P$ and $Q$ in a regular model $\cx/\co_K$ of $X/K$. The author introduced earlier two functorial filtrations of the prime-to-$p$ part of the group of component $\Phi_K$ of ${\cal A}_K/{\cal O}_K$. Filtrations for the full group $\Phi_K$ were later introduced by Bosch and Xarles. An example of a functorial subgroup of $\Phi_K$ occuring in one of the filtrations is the group $\Psi_{K,L}$, the kernel of the natural map $\Phi_K \longrightarrow \Phi_L $, where $L/K$ denotes the minimal extension of $K$ such that $A_L/L$ has semistable reduction. Given two points $P$ and $Q$ in $X(K)$, it is natural to wonder whether it is possible to predict when the reduction of $P-Q$ in $\Phi_K$ belongs to one of the functorial subgroups mentioned above. This question is not easy since even deciding whether the reduction of $P-Q$ is trivial is in general not obvious. We give in this paper a sufficient condition on the special fiber of a model $\cx$ for the image of $P-Q$ in $\Phi_K$ to belong to the subgroup $\Psi_{K,L}$. When this condition is satisfied, we are able to provide a formula for the order of this image. We conjecture that the sufficient condition alluded to above is also necessary and we provide evidence in support of this conjecture. We also discuss cases where the image of $P-Q$ belongs to a functorial subgroup of $\Psi_{K,L}$, using a pairing associated to $\Phi_K$.

Dino J. Lorenzini <lorenzini@math.uga.edu>