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

Abstract

We prove that for a graded algebra $$\mathcal {A}$$ with a derivation $$D_{\mathcal {A}}$$ satisfying certain conditions, and a bi-graded algebra $$\mathcal {A}[q]$$ with an extended derivation D of $$D_{\mathcal {A}}$$ , there are only finitely many $$D_{\mathcal {A}}$$ - and D-invariant (or differential with respect to $$D_{\mathcal {A}}$$ and D) principal prime ideals of $$\mathcal {A}$$ and of $$\mathcal {A}[q]$$ , respectively. As its application, we prove the algebraic independence of the values of certain Eisenstein series for the arithmetic Hecke groups H(m) for $$m=4,6, \infty $$ , which is an extension of Nesterenko’s result for the Eisenstein series for SL $$_2(\mathbb {Z})$$ .