Mathematics Colloquium

An old and fruitful tradition in algebraic geometry goes about proving that a complex-analytic object, which could be constructed transcendentally and studied with topological methods, is at the end of the day an algebro-geometric object. We speak of an arithmetic algebraization when we add a number theory dimension to this picture. In the most elementary form dealing with a formal power series in one or several complex variables, arithmetic algebraization means a criterion on the power series to be rational (or algebraic, or holonomic) under the appropriate combination of integer Taylor coefficients with an analytic continuation to a suitably large domain. We will give an introduction to these criteria, both old and new, and show how they may be used by discussing some recent theorems on the separation of roots of integer polynomials, the structure of the congruence-preserving mappings (Hall and Ruzsa's 'pseudopolynomials'), and the theory of modular forms on noncongruence groups.