Hybrid subconvexity bounds for $L \left(\tfrac{1}{2}, \text{Sym}^2 f \otimes g\right)$

TL;DR

This paper establishes hybrid subconvexity bounds for certain automorphic L-functions involving symmetric squares and primitive forms, in the parameters of the forms' weights and levels, using a first moment method with amplification.

Contribution

It provides the first hybrid subconvexity bounds for $L(1/2, ext{Sym}^2 f imes g)$ in both weight and level aspects, extending previous results to a new parameter range.

Findings

Achieved subconvexity bounds in the specified weight and level ranges.

Applied a first moment method with amplification to obtain bounds.

Extended the range of parameters where subconvexity bounds are known.

Abstract

Fix an integer $\kappa\geqslant 2$. Let $P$ be prime and let $k> \kappa$ be an even integer. For $f$ a holomorphic cusp form of weight $k$ and full level and $g$ a primitive holomorphic cusp form of weight $2 \kappa$ and level $P$, we prove hybrid subconvexity bounds for $L \left(\tfrac{1}{2}, \text{Sym}^2 f \otimes g\right)$ in the $k$ and $P$ aspects when $P^{\frac {13} {64} + \delta} < k < P^{\frac 3 8 - \delta}$ for any $0 < \delta < \frac {11} {128}$. These bounds are achieved through a first moment method (with amplification when $P^{\frac {13} {64}} < k \leqslant P^{\frac 4 {13}}$).