Subconvex bound for Rankin-Selberg $L$-functions in prime power level

TL;DR

The paper addresses subconvexity in the level aspect for Rankin-Selberg L-functions $L(\tfrac{1}{2}, f \otimes g)$ where $f$ is $p$-primitive with level $p^{4r}$ and supercuspidal at $p$, and $g$ is a $\mathrm{SL}(2,\mathbb{Z})$ cusp form. It employs Munshi's delta-method combined with Jutila’s circle method to construct a bilinear structure, then uses GL(2) Voronoi summation on both the $n$- and $m$-sums and careful off-diagonal analysis (via Cauchy–Schwarz and Poisson) to obtain cancellations. The main result is the subconvex bound $L(\tfrac{1}{2}, f \otimes g) \ll_{g,\varepsilon} p^{\tfrac{23r}{12}+\varepsilon}$, improving prior level-aspect bounds (e.g., KMV) for prime-power levels and illustrating the efficacy of the delta-method in this context. The techniques illuminate subconvexity for $GL(2)\times GL(2)$ L-functions with $p$-adic supercuspidal input and offer a framework potentially adaptable to more general conductor-lowering scenarios and Maass-forms as well as holomorphic forms.

Abstract

Let $f$ be a $p$-primitive cusp form of level $p^{4r}$, where local representation of $f$ be supercuspidal at $p$, $p$ being an odd prime, $r\geq 1$ and $g$ be a Hecke-Maass or holomorphic primitive cusp form for $\mathrm{SL}(2,\mathbb{Z})$. A subconvex bound for the central values of the Rankin-Selberg $L$-functions $L(s, f \otimes g )$ is given by $$ L (\frac{1}{2}, f \otimes g ) \ll_{g,ε}p^{\frac{23r}{12} +ε}.$$

Theorems & Definitions (15)

• Theorem 1
• Remark 1
• Remark 2
• Corollary 1
• proof
• Lemma 2.1.1
• Lemma 2.2.1
• Lemma 3.2.1
• proof
• Lemma 5.1.1