具有正常曲率和模微分方程的度量

IF 1.8 2区 数学 Q1 MATHEMATICS
Jianbin Guo, Changshou Lin, Yifan Yang
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引用次数: 4

摘要

设H = H∪Q∪{∞},其中H是复上半平面,且Q(z)是权4在SL(2, z)上的亚纯模形式,使得微分方程L:本文研究了L在H上是明显的,即L的任意两个非零解之比在H上是单值亚纯的问题。这样的模微分方程与曲率为1/2且在z 7→γ·z下对所有γ∈SL(2, z)不变的共形度量ds = eu|dz|2在H上的存在性密切相关。设±κ∞为L在∞处的局部指数。在k∞∈1 2z的情况下,我们得到了以下结果:(a)一个Q(Z)的完备刻划,使得L在i =√−1或ρ =(1 +√3i)/2时在H上只有一个奇点(直到SL(2,Z)-等价)是明显的;(b)一个Q(Z)的完备刻划使得L在H上只有i和ρ处有奇点是明显的。我们给出了两个证明,一个是用黎曼的存在性定理,另一个是用埃雷门科关于球上共形度量的存在性定理。在κ∞/∈1 2z的情况下,令r∞∈(0,1 /2)定义为r∞≡±κ∞模1。设r∞/∈{1/ 12,5 /12}。Eremenko和Tarasov先前结果的一个特例,证明了1/12 < r∞< 5/12是不变度量存在的充分必要条件。阈值情况r∞∈{1/ 12,5 /12}更为微妙。我们证明了在阈值情况下,当且仅当L有两个线性无关的解,其平方是权- 2的亚纯模形式,且在SL(2,Z)上有一对共轭字符,则存在不变度量。在不存在的情况下,我们的例子表明L的单态数据与椭圆曲线y = x−1728的周期有关。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Metrics with Positive constant curvature and modular differential equations
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.
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