Explicit section of explicit line bundle on elliptic curve

complex-geometryelliptic-curvesline-bundlesvector-bundles

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.

Best Answer

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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