Cohen, Paula; Wolfart, Jürgen Modular embeddings for some non-arithmetic Fuchsian groups. (English) Zbl 0717.14014 Acta Arith. 56, No. 2, 93-110 (1990). It is shown for modular curves as well as for Shimura curves X of genus \(>1\) that the covering radius \(\rho\) (which is unique up to an algebraic factor) of the normalized universal holomorphic covering \(\phi\) : \({\mathbb{E}}_{\rho}\to X\), \({\mathbb{E}}_{\rho}:=\{z\in {\mathbb{C}}| | z| <\rho \}\) always turns out to be a transcendental number thus answering a question raised by Lang. Moreover this result is still true for curves with covering group \(\Delta\) of finite index in a Fuchsian triangle group having \(\phi\) (0) as nontrivial fixed-point. More precisely in all the above cases \(\rho\) may be expressed as a quotient of a period of the first kind by a period of the second kind on a certain abelian variety with complex multiplication which forces \(\rho\) to be transcendental due to J. Wolfart and G. Wüstholz [Math. Ann. 273, 1-15 (1985; Zbl 0559.14023)]. As the authors remark, the entering of abelian varieties for triangle groups is somewhat strange. The answer to this question is given in the main result: any such curve may be \({\bar {\mathbb{Q}}}\)-rationally mapped into a suitable Shimura variety such that the fixed-point becomes a complex multiplication point. This result is proven in three completely different ways extending by the way the notion of modular embedding introduced by W. F. Hammond [Am. J. Math. 88, 497-516 (1966; Zbl 0144.341)] some years ago. Thoroughly studied examples complete the paper. Reviewer: F.W.Knoeller Cited in 3 ReviewsCited in 30 Documents MSC: 14G35 Modular and Shimura varieties 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 11J91 Transcendence theory of other special functions Keywords:transcendence theory; modular curves; Shimura curves; covering radius; Fuchsian triangle group; period; abelian variety with complex multiplication; modular embedding Citations:Zbl 0575.14026; Zbl 0559.14023; Zbl 0144.341 PDFBibTeX XMLCite \textit{P. Cohen} and \textit{J. Wolfart}, Acta Arith. 56, No. 2, 93--110 (1990; Zbl 0717.14014) Full Text: DOI EuDML