Transcendence of the values of automorphic functions over \({\mathfrak H}\times{\mathfrak H}\)
It is shown that suitably normalized automorphic functions on the product of two upper half planes take transcendental values at all arguments with one coordinate algebraic, the other transcendental. The proof is based on Wüstholz analytic subgroup theorem and a careful analysis of families of abelian varieties parametrized by \({\mathfrak H}^2\). For an alternative approach, see Propositions 6 and 4 in [\textit{H. Shiga} and \textit{J. Wolfart}, J. Reine Angew. Math. 463, 1--25 (1995; Zbl 0827.11043)].
scientific article; zbMATH DE number 2184619
scientific article