Schwarzian Equations and Equivariant Functions

In this review article we show how the theory of Schwarzian differential equations leads to an interesting class of meromorphic functions on the upper-half plane ℍ $${\mathbb H}$$ named equivariant functions. These functions have the property that their Schwarz derivatives are weight 4 automorphic forms for a discrete subgroup Γ of PSL 2 ( ℝ ) $${\mbox{PSL}_2({\mathbb R})}$$ . It turns out that these functions must satisfy the relation f ( γ τ ) = ρ ( γ ) f ( τ ) , τ ∈ ℍ , γ ∈ Γ , $$\displaystyle f(\gamma \tau )\,=\, \rho (\gamma )f(\tau )\, ,\ \, \tau \in {\mathbb H}\;,\ \gamma \in {\varGamma }, $$ where ρ is a 2-dimensional complex representation of Γ and the matrix action on both sides is by linear fractional transformation. When ρ is the identity representation ρ(γ) = γ, the equivariant functions are parameterized by scalar automorphic forms, while if ρ is an arbitrary representation they are parameterized by vector-valued automorphic forms with multiplier ρ. If Γ is a modular subgroup we obtain important applications to modular forms for Γ as well as a description in terms of elliptic functions theory. We also prove the existence of equivariant functions for the most general case by constructing a vector bundle attached to the data (Γ, ρ) and applying the Kodaira vanishing theorem.