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For a Riemann surface $\Sigma$ there is the following explicit construction of line bundles (from Donaldson: Riemann Surfaces, section 12.1.2): let $p \in \Sigma$. Let $D$ be a disk around $p$ with complex coordinate $z$. Define $L_1=(\Sigma \setminus \{p\}) \times \mathbb{C}$ and $L_2=D \times \mathbb{C}$ and identify $(x,\lambda) \in L_1$ with $(x, z(x)\lambda) \in L_2$. Denote the resulting line bundle by $L_p$.

Consider the elliptic curve $X = T^2 \cong \mathbb{C}/\langle 1,i\rangle$ and let $p \in X$ be some point. Then $L_p \rightarrow X$ is a degree $1$ line bundle. By Lemma 3.10 in these lecture notes, we have $h^0(L_p)=1$. So there must be a holomorphic section of $L_p$, unique up to scaling.

Question: What are all holomorphic sections of $L_p$?

I tried to write down a section as $s_1:X \setminus \{p\} \rightarrow L_1$, $s_1(x)=(x,1)$, and $s_2:D \rightarrow D \times \mathbb{C}$, $s_2(x)=(x,z(x))$. That defines a map $s: X \rightarrow L_p$.

What I find unsatisfying is that I also know that holomorphic sections of $L_p$ are in one-to-one correspondence with meromorphic functions on $X$ that have at most a simple pole at $p$. The section $s$ of $L_p$ corresponds to a constant function on $X$, so does not make use of that simple pole. It seems there ought to be another section, besides mine, that corresponds to a meromorphic function with a simple pole. But I cannot find this section or meromorphic function.

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    $\begingroup$ Could you explain why you think that sections should be meromorphic functions with a simple pole at $p$? The section $s$ you wrote down has a simple zero at $p$, maybe you messed up poles and zeroes somewhere? $\endgroup$ Jun 14, 2022 at 15:16
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    $\begingroup$ There is no other (linearly independent) section. As you yourself say, the space of global sections is one dimensional and the sections are those which have poles of AT WORST order 1 at p. Since your section doesn't even have a pole, it's all good. The global sections are exactly corresponding to the constant functions. $\endgroup$ Jun 17, 2022 at 21:59
  • $\begingroup$ @red_trumpet I got this from Donaldson: Riemann Surfaces, p.183: "... functions with at worst a simple pole at $p$". And thanks, Cranium Clamp, I'll write an answer to this question. $\endgroup$
    – user505117
    Jun 19, 2022 at 16:18

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It is correct that holomorphic sections of $L_p$ are in one-to-one correspondence with complex-valued functions on $X$ that have a worst a simple pole in $p$. This is stated in Donaldson, p.183. The proof is easy: send a meromorphic function $f$ on $X$ with at worst a simple pole in $p$ to the section of $L_p$ defined by $s_1: X \setminus \{p\} \rightarrow \mathbb{C}$, $s_1(x)=f(x)$, and $s_2:D \rightarrow \mathbb{C}$, $s_2(x)=z(x) \cdot f(x)$.

For elliptic curves, one can identify the meromorphic functions with at worst a fixed number of poles explicitly. Here is how to do this: on an elliptic curve, all meromorphic functions are rational functions in the Weierstrass $P$-function and its derivative $P'$, see e.g. https://en.wikipedia.org/wiki/Elliptic_function#Weierstrass_%E2%84%98-function.

So, we have:

  • Meromorphic functions with at most 1 pole is linear span of: constant function
  • At most 2 poles: constant function, $P$
  • At most 3 poles: constant function, $P$, $P'$
  • At most 4 poles: constant function, $P$, $P'$, $P^2$
  • At most 5 poles: constant function, $P$, $P'$, $P^2$, $PP'$
  • At most 6 poles: constant function, $P$, $P'$, $P^2$, $PP'$, $P^3$

In the last line it's worth to note: $P^3$ and $(P')^2$ are linearly dependent (this is stated on the Wikipedia page linked above). So, the space of meromorphic functions with at most 6 poles (at specified locations) is $6$-dimensional. This confirms a special case of Lemma 3.10 from the lecture notes linked in the question, namely $h^0(L_{d\cdot p})=d$.

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