On double shifted convolution sum of $SL(2, \mathbb{Z})$ Hecke eigen forms

TL;DR

This paper establishes a nontrivial upper bound for a double shifted convolution sum involving Fourier coefficients of automorphic forms, under specific growth conditions on the shift parameter.

Contribution

It provides the first nontrivial bound for a complex triple shifted convolution sum of automorphic form coefficients with averaging over shifts.

Findings

Derived a nontrivial upper bound for the sum under H ≥ N^{1/2+ε}

Extended understanding of shifted convolution sums in automorphic forms

Applied advanced analytic techniques to bound complex sums

Abstract

Let $\lambda_i (n)$ $i= 1, 2, 3$ denote the normalised Fourier coefficients of holomorphic eigenform or Maass cusp form. In this paper we shall consider the sum: \[ S:= \frac{1}{H}\sum_{h\leq H} V\left( \frac{h}{H}\right)\sum_{n\leq N} \lambda_1 (n) \lambda_2 (n+h) \lambda_3 (n+ 2h)W\left( \frac{n}{N} \right), \] \noindent where $V$ and $W$ are smooth bump functions, supported on $[1, 2]$. We shall prove a nontrivial upper bound, under the assumption that $H\geq N^{1/2+ \epsilon}$.