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Journal of the London Mathematical Society 2000 61(1):13-24; doi:10.1112/S0024610799008315
© 2000 by London Mathematical Society
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© The London Mathematical Society

Semi-Arithmetic Fuchsian Groups and Modular Embeddings

Paul Schmutz Schaller and Jürgen Wolfart

Section de Mathématiques, Université de Genève Case Postale 240, CH-1211 Genève 24, Switzerland, Paul.Schmutz{at}math.unige.ch
Mathematisches Seminar, Goethe Universität Robert Mayer-Straße 6–10, D-60054 Frankfurt-am-Main, Germany, wolfart{at}math.uni-frankfurt.de

Received 26 January 1998. Revision received 4 June 1998.

Arithmetic Fuchsian groups are the most interesting and most important Fuchsian groups owing to their significance for number theory and owing to their geometric properties. However, for a fixed signature there exist only finitely many non-conjugate arithmetic Fuchsian groups; it is therefore desirable to extend this class of Fuchsian groups. This is the motivation of our definition of semi-arithmetic Fuchsian groups. Such a group may be defined as follows (for the precise formulation see Section 2). Let {Gamma} be a cofinite Fuchsian group and let {Gamma}2 be the subgroup generated by the squares of the elements of {Gamma}. Then {Gamma} is semi-arithmetic if {Gamma} is contained in an arithmetic group {Delta} acting on a product Hr of upper halfplanes. Equivalently, {Gamma} is semi-arithmetic if all traces of elements of {Gamma}2 are algebraic integers of a totally real field. Well-known examples of semi-arithmetic Fuchsian groups are the triangle groups (and their subgroups of finite index) which are almost all non-arithmetic with the exception of 85 triangle groups listed by Takeuchi [16].

While it is still an open question as to what extent the non-arithmetic Fuchsian triangle groups share the geometric properties of arithmetic groups, it is a fact that their automorphic forms share certain arithmetic properties with modular forms for arithmetic groups. This has been clarified by Cohen and Wolfart [5] who proved that every Fuchsian triangle group {Gamma} admits a modular embedding, meaning that there exists an arithmetic group {Delta} acting on Hr, a natural group inclusion

f:{Gamma}->{Delta}

and a compatible holomorphic embedding

F:H->Hr

that is with

F(yZ)=f(y)F(z)

for all {gamma}isin{Gamma} and all zisinH.


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