The Weyl bound for Dirichlet $L$-functions of cube-free conductor

Abstract

We prove a Weyl-exponent subconvex bound for any Dirichlet $L$-function of cube-free conductor. We also show a bound of the same strength for certain $L$-functions of self-dual ${\mathrm{GL}}_2$ automorphic forms that arise as twists of forms of smaller conductor.

Final version, to appear in Annals of Mathematics. We fixed a gap in the proof of the character sum estimate in section 9 using Deligne's semicontinuity theorem