---
abstract: |
    Let $K$ be a totally real number field, $\pi$ an irreducible cuspidal representation of $\operatorname{GL}_2(K)\backslash
    \operatorname{GL}_2(\mathbb{A}_K)$ with unitary central character, and $\chi$ a Hecke character of conductor ${\mathfrak{q}}$. Then $L(1/2,\pi\otimes \chi) \ll
    ({\mathcal{N}}{\mathfrak{q}})^{\frac{1}{2} -\frac{1}{8}(1-2\theta)+{\varepsilon}}$, where $0{\leqslant}\theta {\leqslant}1/2$ is any exponent towards the Ramanujan–Petersson conjecture ($\theta=1/9$ is admissible). The proof is based on a spectral decomposition of shifted convolution sums and a generalized Kuznetsov formula.
address:
- 'Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada, M5S 2E4'
- 'Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, Budapest H-1364, Hungary'
author:
- Valentin Blomer
- Gergely Harcos
title: 'Twisted $L$-functions over number fields and Hilbert’s eleventh problem'
---

[^1]

Introduction
============

In a recent article [@BH] the authors developed a new technique to study shifted convolution sums in Hecke eigenvalues of the type $$\label{shift}
 \sum_{n - m = q} \lambda_{\pi_1}(n)\lambda_{\pi_2}(m) W_1(n/Y) W_2(m/Y)$$ for two irreducible cuspidal representations $\pi_1$, $\pi_2$ of $\operatorname{GL}_2({\mathbb{Q}})\backslash\operatorname{GL}_2(\mathbb{A}_{{\mathbb{Q}}})$ with conductor $1$, reasonably regular weight functions $W_1$, $W_2$, a large number $Y>0$, and $q \neq 0$. Such sums play an important role in the theory of automorphic forms, in particular in the study of automorphic $L$-functions, as they constitute 