DOI: 10.1017/s0010437x26103327 ISSN: 0010-437X

Relative trace formula and twisted L -functions: the Burgess bound

Liyang Yang
Show PDF Cite

Abstract

Let F be a number field,

pi π $\pi$
either a unitary cuspidal automorphic representation of
normal upper G normal upper L left parenthesis 2 right parenthesis divided by upper F G L ( 2 ) / F $\mathrm{GL}(2)/F$
or a unitary Eisenstein series, and
chi χ $\chi$
a unitary Hecke character of analytic conductor
upper C left parenthesis chi right parenthesis period C ( χ ) . $C(\chi).$
We develop a regularized relative trace formula to prove a refined hybrid subconvex bound for
upper L left parenthesis 1 divided by 2 comma pi times chi right parenthesis period L ( 1 / 2 , π × χ ) . $L(1/2,\pi\times\chi).$
In particular, we obtain the Burgess subconvex bound
L ( 1 / 2 , π × χ ) ≪ π , F , ε C ( χ ) 1 2 − 1 8 + ε . \begin{align*}L(1/2,\pi\times\chi)\ll_{\pi,F,\varepsilon}C(\chi)^{\frac{1}{2}-\frac{1}{8}+\varepsilon}.\end{align*}

More from our Archive