Given an elliptic curve $E$ (over $\mathbb{C}$) and line bundle $L$, one can identify $H^0(E,L)$ with a particular space of theta functions. 


Serre duality gives a perfect pairing between $H^0(E,L)$ and $H^1(E,L^{-1})$, does this pairing have a nice expression in terms of theta functions?


I'm either looking for some pairing on the space of theta functions themselves, or a particular way to interpret $H^1(E,L^{-1})$ as objects somehow dual to theta functions. 



Theta functions on an elliptic curve and Serre duality
Let $X$ be an elliptic curve $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice of real rank $2$ in $\mathbb{C}$. A theta function is a holomorphic section of a line bundle $L$ on $X$ whose transition from $U$ to $U + \ell$ is given by$$f(z + \ell) = e^{a\_\ell z + b\_\ell} f(z).$$ Here we have $\ell \in \Lambda$, where $a\_\ell, b\_\ell \in \mathbb{C}$ and $f$ is an entire function on $\mathbb{C}$. This theta function can be regarded as an element of $\text{H}^0(X, L)$ when $\text{H}^0(X, L)$ is the kernel and $\text{H}^1(X, L)$ is the cokernel of the map$$C\_{0, 0}^\infty(X, L) \overset{\overline{\partial}}{\to} C\_{0, 1}^\infty(X, L),$$where $C\_{0, q}^\infty(X, L)$ is the space of all $L$-valued $C^\infty(0, q)$-forms on $X$ for $q = 0, 1$. Thus $\text{H}^1(X, L^{-1})$ is the cokernel of$$C\_{0,0}^\infty(X, L^{-1}) \overset{\overline{\partial}}{\to} C\_{0, 1}^\infty(X, L^{-1}).$$We introduce a smooth metric $H(z) > 0$ for $L$. The compatibility condition for $H(z)$ is$$H(z + \ell) \left|e^{a\_\ell z + b\_\ell}\right|^2 = H(z)$$for $\ell \in \Lambda$ so that the pointwise norm square$$H(z + \ell)|f(z + \ell)|^2 = H(z)|f(z)|^2$$is a well-defined function on $X$. One way to explicitly write down such a metric $H(z)$ is to choose $H(z) = e^{-\gamma|z|^2}$ so that the curvature$${{\sqrt{-1}}\over{2\pi}}\partial\overline{\partial}(-\log H(z)) = \gamma {{\sqrt{-1}}\over{2\pi}} \text{d}z \wedge \text{d}\overline{z}$$of the metric $H(z)$ of $L$ is positive when $\gamma$ is a positive constant. The compatibility condition$$H(z + \ell) \left|e^{a\_\ell z + b\_\ell}\right|^2 = H(z)$$corresponds to the conditions $a\overline{\ell} = a\_\ell$ and ${1\over2}|\ell|^2 = b\_\ell$. With the use of the metric $H(z)^{-1}$ for $L^{-1}$, the cokernel $\text{H}^1(X, L^{-1})$ of$$C\_{0,0}^\infty(X, L^{-1}) \overset{\overline{\partial}}{\to} C\_{0,1}^\infty(X, L^{-1})$$can be identified as the kernel of the $\overline{\partial}^\*$ of the map $\overline{\partial}$ (with respect to the metric $H(z)^{-1}$ of $L^{-1}$). An element $\text{H}^1(X, L^{-1})$ can now be described by a $C^\infty$ $(0, 1)$-form $g(z)\,\text{d}\overline{z}$ on $\mathbb{C}$ such that:


(i) $g(z + \ell) = e^{-a\_\ell z - b\_\ell}g(z)$ for $z \in \mathbb{C}$ and $\ell \in L$ and


(ii) $e^{\gamma z \overline{z}} g(z)$ is antiholomorphic, which means that $\overline{\partial}^\* g(z) = -H(z)\partial\_z(H(z)^{-1}g(z))$ (with $H(z) = e^{-\gamma z\overline{z}}$) is identically zero.


In other words, $g(z)\,\text{d}\overline{z}$ defines a $L^{-1}$-valued $(0, 1)$-form on $X$ so that (i) states the fact that $g(z)\,\text{d}\overline{z}$ is $L^{-1}$-valued and (ii) states the fact that $g(z)\,\overline{z}$ is harmonic with respect to the metric $H(z)^{-1}$ of $L^{-1}$. Note that harmonic means both $\overline{\partial}$-closed and $\overline{\partial}^\*$-closed, but $\overline{\partial}$-closedness is automatic for a $(0, 1)$-form on the compact Riemann surface $X$.


The Serre duality comes from the pairing$$\text{H}^0(X, L \otimes K\_X) \times \text{H}^1(X, L^{-1}) \ni (f, g\,\text{d}\overline{z}) \mapsto \int\_X f(z)\,\text{d}z \wedge g(z)\,\text{d}\overline{z}$$because $K\_X$ is trivial so that we can identify $f \in \text{H}^0(X, L)$ with $f\,dz \in \text{H}^0(X, L \otimes K\_X)$.


