[Paper Review] Burgess-like subconvex bounds for $GL_2 imes GL_1$

Han Wu|arXiv (Cornell University)|Sep 26, 2012

Advanced Algebra and Geometry2 references3 citations

TL;DR

This paper establishes explicit Burgess-like subconvex bounds for the central value of $L$-functions associated with $\mathrm{GL}_2 \times \mathrm{GL}_1$ automorphic $L$-functions over number fields. By combining unipotent translation techniques with amplification methods, the author derives a sharp subconvexity exponent $\delta = \frac{1 - 2\theta}{8}$, where $\theta$ quantifies the Ramanujan-Petersson conjecture, achieving the optimal $\delta = \frac{1}{8}$ under the conjecture.

ABSTRACT

We give a Burgess-like subconvex bound for $L(s, π\otimes χ)$ in terms of the analytical conductor of $χ$, where $π$ is a $GL_2$ cuspidal representation and $χ$ is a Hecke character.

Motivation & Objective

• To provide an explicit, quantitative subconvex bound for $L(1/2, \pi \otimes \chi)$, where $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A}_F)$ with unitary central character and $\chi$ is a Hecke character of conductor $Q$.
• To refine previous subconvexity results by making the exponent $\delta$ in the bound $Q^{1/2 - \delta + \epsilon}$ explicit in terms of the Ramanujan-Petersson parameter $\theta$.
• To extend the method of unipotent translation—originated by Sarnak and developed by Michel and Venkatesh—into a systematic amplification framework for $\mathrm{GL}_2 \times \mathrm{GL}_1$ $L$-functions.
• To establish the first explicit Burgess-type bound in the hybrid aspect for number fields, even when $F \neq \mathbb{Q}$.

Proposed method

• The proof employs the amplification method to localize the $L$-function central value using carefully chosen test vectors $\varphi_0$ in the automorphic representation space.
• Local analysis at archimedean and non-archimedean places is conducted to estimate matrix coefficients and Whittaker functions, particularly in the context of Kirillov models and spectral decomposition.
• The method uses unipotent translation via $n(T)$ to exploit cancellation in the spectral expansion, inspired by Sarnak’s idea and refined by Michel and Venkatesh.
• Global estimates are derived by truncating the spectral expansion and bounding contributions from cuspidal and Eisenstein series using Sobolev norms and decay estimates of matrix coefficients.
• The analysis distinguishes cases based on the ramification and position of places $v_1, v_1', v_2, v_2'$, and uses a refined basis choice to control local norms and support.
• The final bound is obtained by combining convexity bounds for $L$-functions in the Eisenstein contribution with spectral trace estimates and $L^2$-norm control via the Laplacian trace.

Experimental results

Research questions

• RQ1 What is the optimal explicit subconvexity exponent $\delta$ for $L(1/2, \pi \otimes \chi)$ in terms of the Ramanujan-Petersson parameter $\theta$?
• RQ2 Can the amplification method be adapted to $\mathrm{GL}_2 \times \mathrm{GL}_1$ $L$-functions to yield explicit bounds beyond the existence of $\delta > 0$?
• RQ3 How does the unipotent translation technique interact with the spectral decomposition to produce cancellation in the global $L$-function sum?
• RQ4 What is the role of the Eisenstein series contribution in the amplification framework, and how can it be controlled under varying ramification conditions?
• RQ5 To what extent can the method be extended to the hybrid aspect, where both the level and the character conductor grow?

Key findings

• The paper establishes the subconvex bound $L(1/2, \pi \otimes \chi) \ll Q^{1/2 - \delta + \epsilon}$ with $\delta = \frac{1 - 2\theta}{8}$, where $Q$ is the analytic conductor of $\chi$.
• Under the Ramanujan-Petersson conjecture ($\theta = 0$), the bound achieves $\delta = \frac{1}{8}$, matching the classical Burgess bound in the level aspect.
• The result is new in the hybrid aspect, even for $F = \mathbb{Q}$, and improves upon prior work by Munshi and Blomer-Harcos.
• The amplification method successfully controls the Eisenstein contribution by showing it is dominated by the cuspidal term, with the bound scaling as $Q^{(\kappa - 1)/2 + \epsilon} E$ for certain parameters.
• The method achieves uniformity in $\pi$ and $F$, with all implied constants depending only on $F$, $\epsilon$, and $\pi$, not on the conductor $Q$.
• The analysis confirms that the typical case (Type 1) dominates the contribution, and more complicated ramification types (Types 3, 6, 8) contribute less due to cancellation from the denominator $E^{-k}$.