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.