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

Let F be a number field, pi π π 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 GL(2)/F or a unitary Eisenstein series, and chi χ χ a unitary Hecke character of analytic conductor upper C left parenthesis chi right parenthesis period C ( χ ) . C(χ). 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,π×χ). In particular, we obtain the Burgess subconvex bound L ( 1 / 2 , π × χ ) ≪ π , F , ε C ( χ ) 1 2 − 1 8 + ε