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Theta-functions on noncommutative tori

Albert Schwarz
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Abstract

Ordinary theta-functions can be considered as holomorphic sections of line bundles over tori. We show that one can define generalized theta-functions as holomorphic elements of projective modules over noncommutative tori (theta-vectors). The theory of these new objects is not only more general, but also much simpler than the theory of ordinary theta-functions. It seems that the theory of theta-vectors should be closely related to Manin's theory of quantized theta-functions, but we don't analyze this relation.

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Albert Schwarz. 2001-07-25. Theta-functions on noncommutative tori. https://arxiv.org/abs/math/0107186

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