[Paper Review] Subconvexity for L-functions of non-spherical cusp forms on GL(3)

Valentin Blomer, Jack Buttcane|arXiv (Cornell University)|Jan 23, 2018

Analytic Number Theory Research4 references3 citations

TL;DR

This paper establishes a subconvexity bound for L-functions associated with non-spherical cusp forms on GL(3) arising from generalized principal series representations. Using a refined version of the Kuznetsov formula with amplification and Poisson summation across four variables, the authors achieve the bound $ L(1/2,f) \ll T^{3/4 - 1/140000} $ for forms with spectral parameter $ r \asymp d \asymp T $, marking the first such result for non-spherical forms in this family.

ABSTRACT

Let f be a cusp form for SL(3, Z) associated with a generalized principal series representation of minimal weight d, spectral parameter r and associated L-function L(s, f). For $r \asymp d \asymp T$ the subconvexity bound $L(1/2, f) \ll T^{3/4 - 1/140000}$ is proved.

Motivation & Objective

• To establish subconvex bounds for L-functions of non-spherical cusp forms on GL(3) associated with generalized principal series representations.
• To extend the analytic theory of Maaß forms beyond spherical and symmetric square lifts.
• To address the spectral density and distribution of such forms in the family with $ d \asymp T $, $ r \asymp T $.
• To develop a framework for analyzing L-functions in the non-spherical setting using spectral methods and Kloosterman sum techniques.

Proposed method

• Apply the Kuznetsov formula for the spectral family of non-spherical cusp forms on GL(3) with generalized principal series parameters $ (d,r) $.
• Use amplification to enhance the contribution of the central L-value in the fourth moment, focusing on forms with $ d \asymp T $, $ r \asymp T $.
• Perform Poisson summation on all four variables of the Weyl element Kloosterman sums, transforming them into congruence conditions.
• Analyze the 4-fold Fourier transform of Whittaker-type kernels in the Kuznetsov formula via delicate estimates and case distinctions.
• Distinguish and bound three types of contributions: central, mixed, and generic terms, using tailored lemmas for each case.
• Employ hidden cancellation in kernel functions and variable-length savings via congruence conditions and dyadic decomposition to optimize the final bound.

Experimental results

Research questions

• RQ1 Can subconvexity bounds be established for L-functions of non-spherical cusp forms on GL(3) in the generalized principal series family?
• RQ2 What is the optimal subconvexity exponent achievable for such L-functions when the spectral parameter and weight are both $ \asymp T $?
• RQ3 How does the analytic conductor $ \mathcal{C}(f) \asymp T^3 $ influence the size of the subconvexity savings?
• RQ4 What role does the non-spherical nature of the forms play in the complexity of the spectral analysis compared to spherical or symmetric square lifts?
• RQ5 Can the method of amplification and Kuznetsov formula be adapted to handle the non-tempered, non-spherical Whittaker functions arising from generalized principal series?

Key findings

• The paper establishes the subconvexity bound $ L(1/2,f) \ll T^{3/4 - 1/140000} $ for non-spherical cusp forms on GL(3) with $ d \asymp T $, $ r \asymp T $, improving on the convexity bound.
• The bound is achieved through a refined application of the Kuznetsov formula with amplification and Poisson summation across four variables, leading to a decomposition into central, mixed, and generic terms.
• The central term contributes an off-diagonal main term, bounded by $ T^{3+\varepsilon}/L $, where $ L $ is the amplification parameter.
• The generic and mixed terms are controlled via case analysis based on congruence conditions and savings from Lemma 6, yielding a net saving of $ T^{-1/2380} $ in the main term.
• The final bound is optimized by choosing $ \lambda = 1/35000 $, $ \eta = 1/100 $, resulting in the exponent $ 3/4 - 1/140000 $.
• This result provides the first subconvexity bound for L-functions of non-spherical cusp forms on GL(3), opening a new direction in the spectral theory of autom forms.