Subconvexity for $GL(1)$ twists of Rankin-Selberg $L$-functions

TL;DR

The paper proves a subconvex bound for the central value $L\left(\tfrac12, f \otimes g \otimes \chi\right)$ with $f,g$ cusp forms and $\chi$ a primitive Dirichlet character of prime modulus $p$, showing $L\left(\tfrac12, f \otimes g \otimes \chi\right) \ll p^{\tfrac{27}{28}+\varepsilon}$. The approach reduces the problem to a $GL(2)\times GL(2)$ shifted convolution sum via Jutila's circle method, and then combines Voronoi summation, Poisson summation, and shifted-convolution bounds alongside a careful treatment of twisted character sums. This yields an improvement over the convexity bound and the previous subconvex exponent $1-\frac{1}{1324}$, achieving $1-\frac{1}{28}$ after optimizing parameters. The key innovation is a delta-method circle framework that creates a bilinear structure and leverages short twisted $GL(2)$-sums with conductor $p^2$, enabling subconvexity in the depth aspect for Rankin–Selberg $L$-functions twisted by primitive Dirichlet characters.

Abstract

Let $f$ and $g$ be two holomorphic or Hecke-Maass primitive cusp forms for $SL(2,\mathbb{Z})$ and $χ$ be a primitive Dirichlet character of modulus $p$, an odd prime. A subconvex bound for the central values of the Rankin-Selberg $L$-functions is $L(s, f \otimes g \otimes χ)$ is given by $$L(\frac{1}{2}, f \otimes g \otimes χ) \ll_{f,g,ε}p^{\frac{27}{28}+ε} ,$$ for any $ε> 0$, where the implied constant depends only on the forms $f,g$ and $ε$. Here the convexity bound has exponent $1+ε$, which was improved to $1-\frac{1}{1324}$ (see \cite{HM}). Our bound reduces it further to $1- \frac{1}{28}$. The main ingredients is to reduce the original problem to a $GL(2) \times GL(2)$ shifted convolution sum problem.

Theorems & Definitions (19)

• Theorem 1
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Corollary 1
• proof
• Lemma 3.2.1
• Lemma 3.3.1
• proof