The arithmetic of power series

Abstract: A holomorphic function in a neighborhood of z=0 is algebraic if it satisfies a polynomial equation with coefficients in Q[z]. An example of such a function is the square root of 1-4z. The power series expansion of this function turns out to have coefficients which are all integers, and a theorem of Eisenstein says that (up to scaling and suitably interpreted!) this is true of all algebraic functions. But under what conditions can we deduce that a power series with integral coefficients is an algebraic function? We shall explain how this problem is related not only to complex analysis and number theory, but also to Klein’s famous observation that not all finite index subgroups of SL_2(Z) are determined by congruence conditions.