Sub-Weyl bound for $GL(2)$ $L$-functions

We begin by briefly introducing the subconvexity problem for $L$-functions and the delta method, which has proven to be a powerful line of attack in this context. As an application, for a $SL(2,\mathbb{Z})$ form $f$, we obtain the sub-Weyl bound:
$$L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-\delta+\varepsilon}$$ for some explicit $\delta>0$, thereby crossing the Weyl barrier for the first time beyond $GL(1)$. The proof uses a refinement of the 'trivial' delta method.