Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I want to see more concrete description of sections. Let me start with simple example - line bundle $\mathcal{O}(3x)$ over $E$, then $H^0(E, \mathcal{O}(3x))$ is a $3$-dimensional space, and basis is $1, \sigma, \sigma'$, where $\sigma$ is a Weierstrass function.

My first question is there any description of this kind for sections of arbitrary line bundle $\mathcal{L}$ of degree, say, $3$?

Second question. Let me choose a vector bundle of rank two, say as kernel of some map, or some extension of a line bundle by a line bundle. Is there a way to understand its sections as $2$-component vector functions, with special properties, known as some interesting analytic functions?

I want to understand the following example. Let $\mathcal{L}$ be a line bundle of degree $3$ and $\mathcal{L} \ncong \mathcal{O}(3x)$, vector bundle $K$ of rank $2$ is defined by short exact sequence $$ 0 \to K \to \operatorname{Hom}(\mathcal{O}, \mathcal{L})\otimes \mathcal{O} \to \mathcal{L} \to 0 $$ then $K(1)$ (I choose $\mathcal{O}(1) \cong \mathcal{O}(3x))$ has three global sections. What are they as periodic meromorphic vector functions?



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