Duke Mathematical Journal

The spectral decomposition of shifted convolution sums

Valentin Blomer and Gergely Harcos

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Let π1, π2 be cuspidal automorphic representations of PGL2(R) of conductor 1 and Hecke eigenvalues λπ1,2(n), and let h>0 be an integer. For any smooth compactly supported weight functions W1,2:R×C and any Y>0, a spectral decomposition of the shifted convolution sum m±n=hλπ1(|m|)λπ2(|n|)|mn|W1(mY)W2(nY) is obtained. As an application, a spectral decomposition of the Dirichlet series m,n1mn=hλπ1(m)λπ2(n)(m+n)s(mnm+n)100 is proved for Rs>1/2 with polynomial growth on vertical lines in the s-aspect and uniformity in the h-aspect

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

First available in Project Euclid: 14 August 2008

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Zentralblatt MATH identifier

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}


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

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