Metrics with Positive constant curvature and modular differential equations

Abstract

Let H = H∪Q∪{∞}, where H is the complex upper half-plane, and Q(z) be a meromorphic modular form of weight 4 on SL(2,Z) such that the differential equation L : y(z) = Q(z)y(z) is Fuchsian on H. In this paper, we consider the problem when L is apparent on H, i.e., the ratio of any two nonzero solutions of L is single-valued and meromorphic on H. Such a modular differential equation is closely related to the existence of a conformal metric ds = eu|dz|2 on H with curvature 1/2 that is invariant under z 7→ γ · z for all γ ∈ SL(2,Z). Let ±κ∞ be the local exponents of L at ∞. In the case κ∞ ∈ 1 2 Z, we obtain the following results: (a) a complete characterization of Q(z) such that L is apparent on H with only one singularity (up to SL(2,Z)-equivalence) at i = √ −1 or ρ = (1 + √ 3i)/2, and (b) a complete characterization of Q(z) such that L is apparent on H with singularities only at i and ρ. We provide two proofs of the results, one using Riemann’s existence theorem and the other using Eremenko’s theorem on the existence of conformal metric on the sphere. In the case κ∞ / ∈ 1 2 Z, we let r∞ ∈ (0, 1/2) be defined by r∞ ≡ ±κ∞ mod 1. Assume that r∞ / ∈ {1/12, 5/12}. A special case of an earlier result of Eremenko and Tarasov says that 1/12 < r∞ < 5/12 is the necessary and sufficient condition for the existence of the invariant metric. The threshold case r∞ ∈ {1/12, 5/12} is more delicate. We show that in the threshold case, an invariant metric exists if and only if L has two linearly independent solutions whose squares are meromorphic modular forms of weight −2 with a pair of conjugate characters on SL(2,Z). In the non-existence case, our example shows that the monodromy data of L are related to periods of the elliptic curve y = x − 1728.