Relative Trace Formula and Twisted $L$-functions: the Burgess Bound

Let $F$ be a number field, $\pi$ either a unitary cuspidal automorphic representation of $\mathrm{G}\mathrm{L}(2)\mathrm{/}F$ or a unitary Eisenstein series, and $\chi$ a unitary Hecke character of analytic conductor $C(\chi )\mathrm{.}$ We develop a regularized relative trace formula to prove a refined hybrid subconvex bound for $L(1\mathrm{/}2,\pi \times \chi )\mathrm{.}$ In particular, we obtain the Burgess subconvex bound \begin{align*} L(1/2,π\timesχ)\ll_{π,F,\varepsilon}C(χ)^{\frac{1}{2}-\frac{1}{8}+\varepsilon}, \end{align*} where the implied constant depends on $\pi ,$ $F$ and $\epsilon \mathrm{.}$