Modular embeddings for some non-arithmetic Fuchsian groups.

*(English)*Zbl 0717.14014It 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.

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 |