Connecting interpolation and multiplicity estimates in commutative algebraic groups

Abstract

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 used in transcendental number theory and algebraic independence. Let $\gamma _1,\ldots ,\gamma _l$ denote a family of generators of $\Gamma$ and, for any $S>1$, let $\Gamma (S)$ be the set of elements $n_1\gamma _1+\cdots +n_l\gamma _l$ 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 $P \in H^0(\overline{G} ,\mathcal{O}(D))$ 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.