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

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