Number Theory Seminar

This talk focuses on the application of a relative trace formula to establish an enhanced hybrid subconvex bound for $L(1/2,\pi\times\chi)$, where $\pi$ denotes a unitary automorphic representation of $\mathrm{GL}(2)$ over a number field $F$ and $\chi$ represents a Hecke character. Our approach leads to the derivation of the Burgess subconvex bound, which can be succinctly stated as:

\begin{align*}

L(1/2,\pi\times\chi)\ll_{\pi,F,\varepsilon}C(\chi)^{\frac{1}{2}-\frac{1}{8}+\varepsilon},

\end{align*}

where $C(\chi)$ refers to the analytic conductor of $\chi$.