Duke Mathematical Journal

The spectral decomposition of shifted convolution sums

Valentin Blomer and Gergely Harcos

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Abstract

Let $\pi_1$, $\pi_2$ be cuspidal automorphic representations of ${\rm PGL}_2(\mathbb{R})$ of conductor $1$ and Hecke eigenvalues $\lambda_{\pi_{1, 2}}(n)$, and let $h>0$ be an integer. For any smooth compactly supported weight functions $W_{1, 2}:\mathbb{R}^\times\to\mathbb{C}$ and any $Y>0$, a spectral decomposition of the shifted convolution sum \[ \sum_{m\pm n=h}\frac{\lambda_{\pi_1}(|m|)\lambda_{\pi_2}(|n|)}{\sqrt{|mn|}} W_1\Big(\frac{m}{Y}\Big)W_2\Big(\frac{n}{Y}\Big) \] is obtained. As an application, a spectral decomposition of the Dirichlet series \[ \sum_{\substack{m,n\geq 1 m-n=h}} \frac{\lambda_{\pi_1}(m)\lambda_{\pi_2}(n)}{(m+n)^{s}} \Big(\frac{\sqrt{mn}}{m+n}\Big)^{100} \] is proved for $\mathfrak{R}s > 1/2$ with polynomial growth on vertical lines in the $s$-aspect and uniformity in the $h$-aspect

Article information

Source
Duke Math. J. Volume 144, Number 2 (2008), 321-339.

Dates
First available: 14 August 2008

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1218716301

Digital Object Identifier
doi:10.1215/00127094-2008-038

Mathematical Reviews number (MathSciNet)
MR2437682

Zentralblatt MATH identifier
05317182

Subjects
Primary: 11F30: Fourier coefficients of automorphic forms 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula
Secondary: 11F12: Automorphic forms, one variable 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

Citation

Blomer, Valentin; Harcos, Gergely. The spectral decomposition of shifted convolution sums. Duke Mathematical Journal 144 (2008), no. 2, 321--339. doi:10.1215/00127094-2008-038. http://projecteuclid.org/euclid.dmj/1218716301.


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