Subconvexity bound for ${\rm GL}(2)$ $L$-functions: $t$-aspect
Volume 194 / 2020
Acta Arithmetica 194 (2020), 111-133
MSC: Primary 11F66, 11M41; Secondary 11F55.
DOI: 10.4064/aa180711-9-5
Published online: 2 March 2020
Abstract
Let $F$ be a holomorphic Hecke eigenform or a Hecke–Maass cusp form for the full modular group ${\rm SL}(2, \mathbb {Z})$. We use the circle method to prove the Weyl exponent for ${\rm GL}(2)$ $L$-functions. We show that \[ L ( {1}/{2} + it, F ) \ll _{F, \epsilon } ( 2 + |t| )^{1/3 + \epsilon } \] for any $\epsilon \gt 0.$
