Explicit subconvexity for ${\mathrm{GL}}_2$

Abstract

We make the subconvex exponent for ${\mathrm{GL}}_2$ cuspidal representation in the work of Michel \& Venkatesh explicit. The result depends on an effective dependence on the `fixed' ${\mathrm{GL}}_2$ representation in our former work on the subconvex bounds for twists by Hecke characters, which in turn depends on the ${\mathrm{L}}^4$-norm of the test function.

Accepted version in Mathematika