The shifted convolution problem for Fourier coefficients of Siegel modular forms of degree $2$

We provide a power-saving bound for certain smoothed shifted convolution sums for Fourier coefficients of Siegel cusp forms. This result is the first nontrivial estimate for a shifted convolution sum with two cusp forms on a group of higher rank than $\GL_2$. Our approach is based on a novel automorphic reinterpretation of the delta method of Duke, Friedlander, and Iwaniec. The method reduces the problem to the estimation of Fourier coefficients of Siegel Poincare series, which is ultimately based on the Weil bound.