Uniformization of Drinfeld Modular Curves

These are notes from a talk in the Arithmetic Geometry Learning Seminar on Drinfeld modules during the fall of 2017. In this talk, we discuss the uniformization of the Drinfeld modular curves (of fixed rank, with level structure), following [Dri74, §6]. In addition, we introduce the relevant rigid-analytic background. The exposition draws heavily on [DH87, Gek86]. 1. Uniformization of Complex Modular Curves If N ∈ Z>0, consider the functor FN : (Sch /Z[ 1 N ]) op → (Sets) given by S 7→ { elliptic curves E over S, equipped with a full level-N structure }/ ', where a full level-N structure on an elliptic curve E over S is the choice of an isomorphism (Z/NZ) ' −→ E[N ] of S-group schemes. Theorem 1.1. If N ≥ 3, the functor FN is representable by a smooth affine Z[ 1 N ]-scheme Y (N) of finite type. Taking the C-points of the moduli scheme Y (N) gives a complex manifold Y(N) := Y (N)(C) called the modular curve of level N over C, whose points parametrize elliptic curves over C together with an isomorphism (of abelian groups) between the N -torsion of the elliptic curve and (Z/NZ). Let H := {τ ∈ C : im(τ) > 0} be the complex upper half-plane. Recall that SL(2,Z) acts on H (on the left) by fractional linear transformations. We will be particularly interested in the action of the principal congruence subgroup Γ(N) := { γ = ( a b c d ) ∈ SL(2,Z) : γ ≡ ( 1 0 0 1 ) mod N } of level N ; equivalently, Γ(N) is the kernel of the “reduction mod N” map SL(2,Z) → SL(2,Z/NZ). The action of Γ(N) on H is discrete (equivalently, properly discontinuous), so the quotient Γ(N)\H exists as a complex-analytic space. For any τ ∈ H, the lattice Λτ := Z + Zτ gives rise to an elliptic curve Eτ := C/Λτ over C, along with a full level-N structure (Z/nZ) ' −→ Eτ [N ], sending the two generators of (Z/NZ) to 1 N , τ N ∈ Eτ [N ]. This construction gives rise a surjective, Γ(N)-invariant, holomorphic map H −→ Y(N) that induces an isomorphism Γ(N)\H ' −→ Y(N) of complex-analytic spaces. The realization of Y(N) as the quotient of the complex manifold H by the discrete action of the group Γ(N) is often referred to as the (complex-analytic) uniformization of the modular curve of level N . The goal of today’s lecture is to explain the analogue of the above uniformization theorem for the moduli spaces of Drinfeld modules of fixed rank and with some level structure. To do so, we must first discuss the correct analytic framework. Date: October 29, 2017.