Metrics with Positive constant curvature and modular differential equations

Let $\mathbb{H}^\ast = \mathbb{H} \cup \mathbb{Q} \cup \{ \infty \}$, where $\mathbb{H}$ is the complex upper half-plane, and $Q(z)$ be a meromorphic modular form of weight $4$ on $\mathrm{SL}(2, \mathbb{Z})$ such that the differential equation $\mathcal{L}:y''(z) = Q(z)y(z)$ is Fuchsian on $\mathbb{H}^\ast$. In this paper, we consider the problem when $\mathcal{L}$ is apparent on $\mathbb{H}$, i.e., the ratio of any two nonzero solutions of $\mathcal{L}$ is single-valued and meromorphic on $\mathbb{H}$. Such a modular differential equation is closely related to the existence of a conformal metric $ds^2 = e^u|dz|^2$ on $\mathbb{H}$ with curvature $1/2$ that is invariant under $z \mapsto \gamma \cdot z$ for all $\gamma \in \mathrm{SL}(2, \mathbb{Z})$.

Let $\pm \kappa_\infty$ be the local exponents of $\mathcal{L}$ at $\infty$. In the case $\kappa_\infty \in \frac12 \mathbb{Z}$, we obtain the following results:

(a) a complete characterization of $Q(z)$ such that $\mathcal{L}$ is apparent on $\mathbb{H}$ with only one singularity (up to $\mathrm{SL}(2, \mathbb{Z})$-equivalence) at $i = \sqrt{-1}$ or $\rho = (1+ \sqrt{3i})/2$, and

(b) a complete characterization of $Q(z)$ such that $\mathcal{L}$ is apparent on $\mathbb{H}^\ast$ with singularities only at $i$ and $\rho$.

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 $\kappa_\infty \notin \frac12 \mathbb{Z}$, we let $r_\infty \in(0,1/2)$ be defined by $r_\infty \equiv \pm \kappa_\infty \operatorname{mod} 1$. Assume that $r_\infty \notin \{1/12,5/12 \}$. A special case of an earlier result of Eremenko and Tarasov says that $1/12<r_\infty<5/12$ is the necessary and sufficient condition for the existence of the invariant metric. The threshold case $r_\infty \in \{1/12,5/12 \}$ is more delicate. We show that in the threshold case, an invariant metric exists if and only if $\mathcal{L}$ has two linearly independent solutions whose squares are meromorphic modular forms of weight $-2$ with a pair of conjugate characters on $\mathrm{SL}(2, \mathbb{Z})$. In the non-existence case, our example shows that the monodromy data of $\mathcal{L}$ are related to periods of the elliptic curve $y^2 = x^3-1728$.

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Jia-Wei Guo is supported by Grant 109-2115-M-002-017-MY2 of Ministry of Science and Technology, Taiwan

Yifan Yang is supported by Grant 109-2115-M-002-010-MY3 of Ministry of Science and Technology, Taiwan.

Received 26 February 2021

Published 22 March 2022