The spectral decomposition of  |θ|2

Paul D. Nelson

doi:10.1007/s00209-020-02665-8

The spectral decomposition of |θ|²

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

4 Citations (Scopus)

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.

Original language	English
Journal	Mathematische Zeitschrift
Volume	298
Issue	3-4
Pages (from-to)	1425-1447
Number of pages	23
ISSN	0025-5874
DOIs	https://doi.org/10.1007/s00209-020-02665-8
Publication status	Published - Aug 2021
Externally published	Yes

Keywords

Half-integral weight modular forms
Spectral decomposition
Theta functions

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10.1007/s00209-020-02665-8

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Cite this

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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.",

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N2 - 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.

AB - 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.

KW - Half-integral weight modular forms

KW - Spectral decomposition

KW - Theta functions

UR - https://www.scopus.com/pages/publications/85097666539

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DO - 10.1007/s00209-020-02665-8

M3 - Journal article

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JO - Mathematische Zeitschrift

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