The spectral decomposition of |θ|2

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Abstract

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

OriginalsprogEngelsk
TidsskriftMathematische Zeitschrift
Vol/bind298
Nummer3-4
Sider (fra-til)1425-1447
Antal sider23
ISSN0025-5874
DOI
StatusUdgivet - aug. 2021
Udgivet eksterntJa

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