The spectral decomposition of shifted convolution sums

By V. Blomer, Gergely Harcos. 2007

58 citations recorded by the source corpus.

Let pi(1), pi(2)) be cuspidal automorphic representations of PGL(2)(R) Qf 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,W-2 : R-x --> C and any Y > 0, aspectral decomposition of the shifted convolution sum Sigma(m+/-n=h) lambda(pi 1) (vertical bar m vertical bar)lambda(pi 2) (vertical bar n vertical bar)/root vertical bar mn vertical bar W-1(m/Y)W-2(n/Y) is obtained. As an application, aspectral decomposition of the Dirichlet series [GRAPHICS] is proved for Rs > 1/2 with polynomial growth on vertical lines in the S-aspect and uniformity in the h-aspect.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/79433bbdb3e195bb68cdc656baf6d953aef44dd5

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In Algebra and Number Theory

1. V. Blomer
2. Gergely Harcos
3. A. Ubis
4. Peter Sarnak
5. P. Michel
6. R. Munshi
7. M. Young
8. Akshay Venkatesh

In Discrete Mathematics and Combinatorics

1. Peter Sarnak
2. A. Ubis
3. V. Blomer
4. Gergely Harcos
5. R. Munshi
6. Farrell Brumley
7. Nicolas Templier
8. Frank Calegari

In Mathematical Physics

1. V. Blomer
2. Gergely Harcos
3. A. Ubis
4. Peter Sarnak
5. P. Michel
6. R. Munshi
7. M. Young
8. Akshay Venkatesh

Citations in the corpus

• Hybrid bounds for twisted L-functions (2008, background): “3) Very recently the authors found an alternate spectral decomposition that avoids these di‰culties, see [ BH ].”
• The horocycle flow at prime times (2011, methodology): “1 we can substitute y−θ by m (see [3]).”