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Modular embeddings for some non-arithmetic Fuchsian groups. (English) Zbl 0717.14014
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

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
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