Free algebras of automorphic forms on the upper half-plane
From the text: ``Let \(\Gamma\) be a discrete automorphism group of the upper half-plane \(H\) such that the quotient space \(X=H/\Gamma\) is compact. A function \(a(\gamma,z)\) of two variables \(\gamma\in\Gamma\) and \(z \in H\) is called a cocycle (or an automorphy factor) of \(\Gamma\) if it (a) is holomorphic in \(z\) for each \(\gamma\) and nowhere vanishes on \(H\) and (b) satisfies the condition \[ a(\gamma_1,\gamma_2,z) =a( \gamma_1,\gamma_2z)a(\gamma_2,z). \] A function \(f(z)\) holomorphic on \(H\) is called an \(a-\Gamma\)-automorphic form (or simply a form) of integral weight \(l\) if it satisfies the functional equation \[ f(\gamma z)=a^l(\gamma,z)f(z). \] Let \(A_l\) be the \(\mathbb{C}\)-linear space of these forms. Consider the graded algebra \[ A=A(\Gamma,a)=\bigotimes_{l\in \mathbb{Z}}A_l. \] The objective of the paper is to indicate all pairs \((\Gamma, a)\) for which the algebra \(A(\Gamma,a)\) is free, i.e., isomorphic to the graded algebra \(\mathbb{C}[X,Y]\) of polynomials in two variables.'' The amount of space required to define the notation makes a precise statement of the results (i.e., the enumeration of the free algebras) inappropriate. That said, once the notation is set up, the result is elegant. Last, the abstract, though not the title, mentions the genus 0 restriction.
scientific article; zbMATH DE number 2065923