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Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there is an orthonormal basis whose sections lift to theta functions on the complex plane.

My question is, what if we try to compare the Hilbert spaces obtained for different $\tau$? One method would be to look at the Hitchin connection. However, if we instead tried to use the Blattner-Kostant-Souriau (BKS) pairing between polarizations, would this just amount to doing a Fourier transform of sorts? In particular will this just give us something proportional to the Hermite-Jacobi action?

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Why is this downvoted? (This is not meant to directly disagree, I am pretty clueless on this subject; so there might well be a good reason I miss.) – user9072 Aug 29 '12 at 15:28
I voted down because you have no real question here. An expert may be aware of 'what hilbert space obtained for different $\tau$'s' you are referring to, but i don't. I think the statement of your question should (i) clarify your 2nd sentence, what explicitly is the pairing (who is orthonormal with respect to what)? (ii) give explanation for why sections lift to theta functions, ie. explicitly identify the theta function, and (iii) recall the defition--for us--of what is the BKS pairing. But this only if you want others to benefit or think about your question. If for-experts-only, leave as is. – J. Martel Aug 29 '12 at 15:32
I think every statement concerning theta functions needs to be explicit as possible. The correspondence between theta functions and sections of line bundles, as described in Mumfords Tata Lectures on Theta III', is not just that sections` are theta functions. This question drops some big words, but no mathematics. Not yet, anyway. Anybody who wants to understand one iota $(\iota)$ of theta functions has to start bottom-up everytime, my opinion. Plus what this pairing between polarizations? Why does the OP not generously recall this? Either they don't know, or they're too lazy. – J. Martel Aug 29 '12 at 15:38
@J. Martel: thank you for the eleabortation. As said I just could not judge for myself, and sometimes people just misclick and this can create a dynamic (e.g., I just looked at the question because it had a negative score, to see what is going on). Thanks again for taking the time to explain. – user9072 Aug 29 '12 at 16:14
@Blake: You have explicit series expansions, giving you bases for the theta functions of order $k$ for a given $\tau$. I.e., you have the dependence on $\tau$ explicitly! – Peter Dalakov Aug 29 '12 at 22:54

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