Eine arithmetische Eigenschaft automorpher Formen zu gewissen nicht-arithmetischen Gruppen

Summary

LetΔ⊂SL₂

denote a Fuchsian triangle group andA an elliptic fixed point of δ. For any weightk and any multiplier systemv whose values are roots of unity, there is a basis of automorphic forms with expansions

$$f\left( z \right) = \left( {z - \bar A} \right)^{ - k} \sum\limits_{n = 0}^\infty {r_n b^n } \left( {\frac{{z - A}}{{z - A}}} \right)^n ,$$

Where allr _n are rational andb is a constant depending only on Δ andA. The computation ofb yields a product of Γ-values at rational arguments up to an algebraic factor. If ∞ is a cusp of Δ, the same basis has — up to constant factors —Fourier expansions of type

$$\sum\limits_{n = 1}^\infty {r_n a^n e^{\frac{{2\pi i}}{t}\left( {n - s} \right)z} } ,$$

Where allr _n are rational,s∈ℝ depends onv, anda is a constant depending only on Δ. For reasonably normalized Δ, the computation ofa leads to a rational or algebraic number for arithmetic groups Δ and a transcendental number$\alpha _1^{\beta _1 } ...\alpha _m^{\beta _m }$ with algebraic α _j , β _j in all other cases.

These results hold also for normal subgroups ϕ of Δ with finite abelian factor group Δ/ϕ.