# pairing theta functions for different complex structures

I apologize for my previous attempt to ask this, which was very badly written.

Let us start with $\mathbb{C}\times\mathbb{C}$. To form an Hermitian line bundle over a complex torus with complex structure given by $\tau$ we may quotient $\mathbb{C}\times\mathbb{C}$ by the lattice generated by 1 and $\tau$, using an appropriate cocycle to define the action on the line bundle. Holomorphic sections of this line bundle will lift to linear combinations of theta functions on $\mathbb{C}$ (in an appropriate trivialization$. We can also define an inner product on the space of holomorphic sections in the usual way, by integrating the sesquilinear product of the two sections. However, what if we want to integrate the sesquilinear product of two theta functions which come from different$\tau$? So, for example, if$\theta_1$and$\theta_2$are lifts of holomorphic sections in the$\tau_1$and$\tau_2$complex structures, is there a standard formula for the integral of$\theta_1 \overline{\theta_2}$over the torus? What complicates this seems to be that the bundles (and the tori) are constructed by taking two different quotients, thus one must first find a unitary bundle diffeomorphism between them. In particular, this integral defines a pairing between the holomorphic sections from the$\tau_1$structure and the holomorphic sections coming from$\tau_2\$. Does this pairing induce a linear map between the spaces of functions which is unitary, or projectively unitary? And, is this related to what I have seen called the 'Hermite-Jacobi action?'

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