An additive problem in the Fourier coefficients of cusp forms

Abstract

We establish an estimate on sums of shifted products of Fourier coefficients coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the binary additive divisor sum which has been studied extensively. As an application we derive, extending work of Duke, Friedlander and Iwaniec, a subconvex estimate on the critical line for L-functions associated to character twists of these cusp forms.

16 pages, LaTeX2e; v2: lots of changes, Theorem 2 is new, notation changed to standard one, abstract and further references added; v3: minor changes, some restriction imposed in Theorem 2, additional references; v4: introduction revised, references added, typos corrected; v5: final, revised version incorporating suggestions by the referee (e.g. Section 5 was added)

References in corpus (1)

• Uniform approximate functional equation for principal L-functions

Cited by in corpus (9)

• The spectral decomposition of shifted convolution sums
• Shifted convolution sums for $GL(3)\times GL(2)$
• New bounds for automorphic L-functions
• Theta functions, fourth moments of eigenforms, and the sup-norm problem II
• On binary and quadratic divisor problems
• Local integral transforms and global spectral decomposition
• On fundamental Fourier coefficients of Siegel cusp forms of degree 2
• The sixth moment of the Riemann zeta function and ternary additive divisor sums
• Shifted Convolution Sums for $GL(3)\times GL(2)$ Averaged over weighted sets