Beyond the Weyl barrier for \(\mathrm{GL}(2)\) exponential sums

In this paper, we use the Bessel \(\delta\)-method, along with new variants of the van der Corput method in two dimensions, to prove non-trivial bounds for \(\mathrm{GL}(2)\) exponential sums beyond the Weyl barrier. More explicitly, for sums of \(\mathrm{GL}(2)\) Fourier coefficients twisted by \(e(f(n))\), with length \(N\) and phase \(f(n)=N^{\beta} \log n / 2\pi\) or \(a n^{\beta}\), non-trivial bounds are established for \( \beta < 1.63651... \), which is beyond the Weyl barrier at \(\beta = 3/2\).