Jürgen Wolfart

Werte hypergeometrischer funktionen

SummaryA partial result on a problem of Siegel is given: Hypergeometric functions with rational parameters have transcendental values in almost all algebraic points — up to some natural exceptions; these exceptions are the well-known algebraic functions and an (unexpected) second class of examples related to certain Shimura-curves.

Fuchsian triangle groups and Grothendieck dessins. Variations on a theme of Belyi

According to a theorem of Belyi, a smooth projective algebraic curve is defined over a number field if and only if there exists a non-constant element of its function field ramified only over 0, 1 and ∞. The existence of such a Belyi function is equivalent to that of a representation of the curve as a possibly compactified quotient space of the Poincaré upper half plane by a subgroup of finite index in a Fuchsian triangle group. On the other hand, Fuchsian triangle groups arise in many contexts, such as in the theory of hypergeometric functions and certain triangular billiard problems, which would appear at first sight to have no relation to the Galois problems that motivated the above discovery of Belyi. In this note we review several results related to Belyis theorem and we develop certain aspects giving examples. For preliminary accounts, see the preprint [Wo1], the conference proceedings article [BauItz] and the “Comptes Rendus” note [CoWo2].

Semi-arithmetic Fuchsian groups and modular embeddings

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 Γ be a cofinite Fuchsian group and let Γ 2 be the subgroup generated by the squares of the elements of Γ. Then Γ is semi-arithmetic if Γ is contained in an arithmetic group Δ acting on a product H r of upper halfplanes. Equivalently, Γ is semi-arithmetic if all traces of elements of Γ 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 Γ admits a modular embedding, meaning that there exists an arithmetic group Δ acting on H r , a natural group inclusion formula here and a compatible holomorphic embedding formula here that is with formula here for all γ∈Γ and all z ∈ H .

Der Überlagerungsradius gewisser algebraischer Kurven und die Werte der Betafunktion an rationalen Stellen

ZusammenfassungFür abelsche Varietäten mit komplexer Multiplikation wird die Dimension des berechnet, der von den Perioden erster und zweiter Art erzeugt wird. Angewandt auf die Jacobi-Varietät der Fermat-Kurven, ergibt dieses Resultat zusammen mit den Kriterien von Shimura-Taniyama und Deligne-Koblitz-Ogus einen optimalen Satz über die lineare Unabhängigkeit der WerteB(a1,b1),...,B(an,bn) der Betafunktion an rationalen Stellenaj,bj: Sie sind nur in dem offensichtlichen Fall-linear unabhängig, wenn diese Abhängigkeit bereits aus den klassischen Gauß-Legendre-Identitäten für die Werte der Γ-Funktion folgt.Dieser Satz gibt widerum eine Teilantwort auf eine Transzendenzfrage, die S. Lang für die Uniformisierungtheorie aufgeworfen hat: SeiX eine glatte projektive algebraische Kurve, definiert über und vom Geschlechtg>1, und sei

GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS

\varphi :U_r : = \{ \zeta \in \left. \mathbb{C} \right|\left| \zeta \right|< r\} \to X

Algebraic Values of Schwarz Triangle Functions

Eine normalisierte holomorphe Überlagerungsabbildung, d.h. mitϕ(0)∈X() und mit einer über definierten Tangentialabbildungϕ′(0); ist dann der „Überlagerungsradius”r transzendent? Die Antwort ist „ja”, wennX viele Automorphismen besitzt-z. B. für Fermat-Kurven oder die Kleinsche Quadrikund wenn ϕ(0) Fixpunkt ist, denn in diesem Fall läßt sich der Überlagerungsradius als Quotient von-linear unabhängigen Betawerten schreiben.

Darstellungen von SL(2,ℤ/pλℤ) und Thetafunktionen. II

Historical and introductory background.- Graph embeddings.- Dessins and triangle groups.- Galois actions.- Quasiplatonic surfaces, and automorphisms.- Regular maps.- Regular embeddings of complete graphs.- Wilson operations.- Further examples.- Arithmetic aspects.- Beauville surfaces.

Untergruppen der GL2, welche durch homogene lineare Kongruenzen definiert sind

We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph K n,n , where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Skoviera about regular embeddings of the graphs K n,n [ 7 ] and generalises the analogous results for maps obtained in [ 9 ], partly using different methods.