Short Second Moment Bound for GL(2) $L$-functions in $q$-Aspect
Agniva Dasgupta
TL;DR
The paper proves a Lindelöf-on-average bound for the second moment of $L$-functions attached to a level $1$ cusp form in the $q$-aspect, with twists by characters modulo $q^{2/3}$ and $q=p^3$, showing $$ onsuperscript{} \\sum_{\psi\pmod{q^{2/3}}} \left|L(f\otimes(\alpha\cdot\psi),\tfrac12)\right|^2 \ll_{f,\varepsilon} q^{\frac{2}{3}+\varepsilon}. $$This near-Lindelöf bound follows a strategy combining an approximate functional equation with a shifted-convolution reduction, a delta-symbol method to separate variables, Voronoi summation (including a modified version), a Bruggeman–Kuznetsov spectral decomposition at level $p$, and a spectral large-sieve bound to control the Maass, holomorphic, and Eisenstein contributions. The analysis reveals a conductor-lowering, short-moment phenomenon in the level aspect by restricting to a subfamily $\{f\otimes(\alpha\cdot\psi)\}$ with $\psi\bmod q^{2/3}$, enabling non-vanishing results and Weyl-type subconvexity within the family. The framework extends to higher prime powers and yields an upper bound that matches the expectations from Lindelöf-on-average, with substantial implications for subconvexity and non-vanishing in families of GL(2) $L$-functions. Thorough harmonic-analysis and spectral techniques underpin the proof, including careful dyadic decompositions, inert-function arguments, and delicate oscillatory integral estimates.
Abstract
We prove a Lindelöf-on-average upper bound for the second moment of the $L$-functions associated to a level 1 holomorphic cusp form, twisted along a coset of subgroup of the characters modulo $q^{2/3}$ (where $q = p^3$ for some odd prime $p$). This result should be seen as a $q$-aspect analogue of Anton Good's (1982) result on upper bounds of the second moment of cusp forms in short intervals. The results generalize easily to higher prime powers as well.