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Questions about half-integral weight modular forms, and more generally automorphic representations associated to metaplectic groups.

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Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4?

This is nothing like a complete answer (EDIT-now it seems to be--see below), but it may suggest a fruitful attack, based on the comment (see below for a version incorporated into this answer) that I m …
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