Subconvexity for $\rm GL_2 \times GL_2$ $L$-functions in the depth aspect

TL;DR

This work proves a subconvex bound for $L(1/2,f\otimes g\otimes\chi)$ in the depth aspect by combining Munshi's conductor-lowering delta method with GL$_2$ Voronoi summation and a $p$-adic stationary-phase analysis of resulting character sums. The main result is $L(1/2,f\otimes g\otimes\chi)\ll_{f,g,\varepsilon} q^{9/10+\varepsilon}$ for primitive $\chi$ modulo $q=p^n$ with odd prime $p$, improving previous depth-aspect bounds. The proof reduces the problem to estimating oscillatory sums via a delta-method decomposition, transforms the sums through Voronoi to dual lengths, expands the Dirichlet character additively, and achieves square-root cancellations by combining Cauchy--Schwarz with non-archimedean second-derivative techniques. The results illustrate a robust $p$-adic analytic toolkit with potential applicability to broader subconvexity problems in higher rank settings.

Abstract

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ and let $χ$ be a primitive Dirichlet character of prime power conductor $q=p^n$. For any given $\varepsilon>0$, we establish the following subconvexity bound \begin{equation*} L(1/2,f\otimes g \otimes χ)\ll_{f,g,\varepsilon}q^{9/10+\varepsilon}. \end{equation*} The proof employs the DFI circle method with standard manipulations, including the conductor-lowering mechanism, Voronoi summation, and Cauchy--Schwarz inequality. The key input is certain estimates on the resulting character sums, obtained using the $p$-adic version of the van der Corput method.

Theorems & Definitions (28)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Definition 2.1
• Definition 2.2
• Lemma 2.1
• Lemma 2.2: $p$-adic stationary phase
• proof
• Lemma 2.3: Kloosterman sum evaluation
• proof