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is the twice punctured complex plane parabolic or hyperbolic? In this sense: does $\mathbb{C}-\{0,1\}$ admit a nonconstant, negative, subharmonic function?


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A subharmonic function which is bounded from above near a polar set automatically extends through it. Therefore, any negative subharmonic function on $\mathbb C \setminus\{0,1\}$ induces a negative subharmonic function on $\mathbb C$, and by Liouville theorem, it must be constant.

So the answer to your question is no.

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The $j$-invariant in the theory of elliptic curves is a covering map from the upper half plane to the complex plane punctured at $0$ and $1$. See Koblitz, Introduction to Elliptic Curves and Modular Forms.

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I don't think you mean the $j$-invariant (which does take both values $0$ and $1$) but rather a modular function for $\Gamma_0(4)$. – Dan Petersen Dec 19 '13 at 16:28

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