Cancellation in GL(3)×GL(3) shifted convolution sums with a fixed shift

Establish non-trivial cancellation for the GL(3)×GL(3) shifted convolution sum S_h(N) = ∑_{n≤N} a_{π1}(n) a_{π2}(n+h) with a fixed integer shift h, where a_{πi}(n) are the normalized Fourier coefficients of two Hecke–Maass cusp forms for GL(3); specifically, determine an upper bound that improves upon the trivial bound without averaging over h.

Background

Shifted convolution sums of automorphic coefficients are central in analytic number theory due to connections with equidistribution and subconvexity. For GL(3)×GL(3), obtaining cancellation in sums with a single fixed shift h is notably difficult.

To circumvent this difficulty, existing work considers an additional average over the shift variable h and seeks non-trivial bounds in ranges where H is as small as possible relative to N. This paper advances that approach by proving power-saving bounds for the averaged GL(3) case down to H ≥ N{1/6+ε}, but the fixed-shift problem itself remains unresolved.

References

For $\GL(3)\times \GL(3)$ shifted convolution sums it is a hard open problem to obtain cancellation only in the shifted convolution sum with a fixed shift $h$.

— On shifted convolution sums of $\mathrm{GL}(3)$-Fourier coefficients with an average over shifts (2510.15799 - Pal et al., 17 Oct 2025) in Section 1 (Introduction)