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I'm interested in nowhere zero holomorphic functions that are not the exponential of a holomorphic function, i.e. the space $\mathcal{O}^*(X)/\exp(\mathcal{O}(X))$, where $X$ is an affine curve. In particular are there any explicit examples of these kinds of functions when $X = \mathbb{C}^*$ or a punctured torus?

I know that such functions must exist - from the short exact sequence $$0\to 2\pi i\mathbb{Z}\to\mathcal{O} \to \mathcal{O}^*\to 0$$ and the fact that $H^1(X,\mathcal{O}) = 0$ the connecting homomorphism $\mathcal{O}^*(X)\to H^1(X,\mathbb{Z})$ must be surjective - but I am not aware of an explicit construction.

Edit: As pointed out in the comments this question is completely obvious when $X=\mathbb{C}^*$ as you can take $z\mapsto z^n$ to represent any class in $H^1(X,\mathbb{Z})$ (woops!). I'm still interested in the case of the punctured torus though.

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    $\begingroup$ For $\Bbb C^\times$, I'd try the identity ... $\endgroup$ – Hagen von Eitzen May 28 '18 at 5:56
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While I appreciate that this isn't an exhaustive answer to the question, I think it will be useful to discuss the obstructions that make it impossible to find logarithms of nowhere-vanishing functions on punctured tori (or on any other punctured Riemann surface, for that matter).

So suppose $X = \mathbb C / \Lambda - \{ z_1, \dots, z_m \}$ is a your punctured torus. Notice that if $f$ is a meromorphic function $f$ on the whole torus $\mathbb C / \Lambda$, whose zeros and poles are contained inside $\{ z_1, \dots, z_m \}$, then $f$ can be viewed as a nowhere-vanishing holomorphic function on $X$.

For such an $f$ to have a well-defined logarithm $\log f$ on $X$, it must be the case that $ \oint_\gamma \frac{f'}{f} = 0$ for all closed contours $\gamma \subset X$. This is because, if $\log f$ exists, then $\frac{f'}{f} = (\log f)'$, and $\oint_\gamma (\log f)' = 0$ for any closed contour $\gamma$ by the fundamental theorem of calculus.

So what does this mean for the zeroes and poles at the puncture points? Well, let's pick a puncture point $z_j$, and let's take $\gamma$ to a loop around $z_j$, small enough such that $\gamma$ wraps no other puncture point except $z_j$. Then $\oint_{\gamma} \frac{f'}{f}$ is equal to the order of the zero or pole of $f$ at $z_j$.

The conclusion, therefore, is that $f$ cannot admit a logarithm on $X=\mathbb C / \Lambda - \{z_1, \dots, z_m \}$ unless the order of its zero or pole at every puncture point is trivial.

For example:

  • If $f$ is the Weierstrass elliptic function $\wp(z, \Lambda)$, then $f$ has a double pole at $z = 0$ and a pair of simple zeroes at $z = \pm z_0$ for some $z_0$. We can then view $f$ as a nowhere-vanishing holomorphic function on $X = \mathbb C / \Lambda - \{0, z_0, - z_0 \}$. By our discussion above, $f$ has no logarithm on $X$.

  • If $f$ is the Weierstrass derivative $\wp'(z, \Lambda)$, then $f$ has a triple pole at $z = 0$, and three simple zeroes at $z = \frac 1 2, \frac \tau 2, \frac {\tau + 1}{2}$ (where the lattice $\Lambda = \mathbb Z \oplus \tau \mathbb Z$). If we view $f$ as a nowhere-vanishing holomorphic function on $X = \mathbb C / \Lambda - \{ 0, \frac 1 2, \frac \tau 2, \frac {\tau + 1}{2} \}$, then $f$ has no logarithm on $X$.

I'm highlighting these examples because they are very natural examples to consider when thinking about the punctured torus as an affine curve (which was your mindset when you posed the question). Indeed, viewing the torus as a cubic in $\mathbb {CP}^2$, the functions $\wp(z, \Lambda)$ and $\wp'(z, \Lambda)$ are the projections to coordinate axes.

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