Short Second Moment of $\mathrm{GL}(2)$ L-Functions via Trivial Delta

Abstract

Let $f$ be a holomorphic cusp form for $SL(2,\mathbb{Z})$. In this paper, we prove for $T^{\frac{1}{3}} \ll M \ll T^{\frac{1}{2}}$, the following short interval second moment \begin{equation*} \int_T^{T+M}\left|L\left(\frac12+it,f\right) \right|^2 dt \ll_{f,\epsilon} T^\epsilon \left( M + \frac{T}{M}+\sqrt{TM}\right). \end{equation*} The proof uses the trivial delta method along with conductor lowering technique.