The finiteness of derivation-invariant prime ideals and the algebraic independence of the Eisenstein series

Abstract

We prove that for a graded algebra A with a derivation D-A satisfying certain conditions, and a bi-graded algebra A[q] with an extended derivation D of D-A, there are only finitely many D-A- and D-invariant (or differential with respect to D-A and D) principal prime ideals of A and of A[q], respectively. As its application, we prove the algebraic independence of the values of certain Eisenstein series for the arithmetic Hecke groups H(m) for m = 4, 6, infinity, which is an extension of Nesterenko's result for the Eisenstein series for SL2(Z).