Connecting Interpolation and Multiplicity Estimates in Commutative Algebraic Groups

Let $G$ be a commutative algebraic group embedded in projective space and $\Gamma$ a finitely generated subgroup of $G$. From these data we construct a chain of algebraic subgroups of $G$ which is intimately related to obstructions to multiplicity or interpolation estimates. Let $\gamma_1,...,\gamma_l$ denote a family of generators of $\Gamma$ and, for any $S>1$, let $\Gamma(S)$ be the set of elements $n_1\gamma_1+..+n_l\gamma_l$ with integers $n_j$ such that $|n_j| < S$. Then this chain of subgroups controls, for large values of $S$, the distribution of $\Gamma(S)$ with respect to algebraic subgroups of $G$. As an application we essentially determine (up to multiplicative constants) the locus of common zeros of all $P \in H^0(\barG ,{\cal O}(D))$ which vanish to at least some given order at all points of $\Gamma(S)$. When $D$ is very small this result reduces to a multiplicity estimate; when $D$ is very large it is a kind of interpolation estimate.