Abstract
Let theta be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for langle |theta |^2, varphi rangle as varphi traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically |theta |^2 notin L^2. Applications to the subconvexity, quantum variance and 4-norm problems are indicated.
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