Shifted convolution sums and Burgess type subconvexity over number fields

Abstract

Let $F$ be a number field and $\pi$ an irreducible cuspidal representation of $\mathrm{GL}_{2}(F)\backslash\mathrm{GL}_{2}(\mathbf{A})$ with unitary central character. Then the bound $$L(1/2,\pi\otimes\chi)\ll_{F,\pi,\chi_{\infty},\varepsilon} \mathcal{N}(\frak{q})^{3/8+\theta/4+\varepsilon}$$ holds for any Hecke character $\chi$ of conductor $\frak{q}$, where $\theta$ is any constant towards the Ramanujan-Petersson conjecture ($\theta=7/64$ is admissible). The proof is based on a spectral decomposition of shifted convolution sums.