[Paper Review] A new subconvex bound for $ m GL(3)$ $L$-functions in the $t$-aspect

Keshav Aggarwal|arXiv (Cornell University)|Mar 23, 2019

Analytic Number Theory Research12 references4 citations

TL;DR

This paper establishes a new subconvex bound for $\rm GL(3)$ $L$-functions in the $t$-aspect by revisiting Munshi's approach and eliminating the 'conductor lowering' technique, leading to a cleaner stationary phase analysis. The authors achieve the improved bound $ L(1/2+it,\pi) \ll_{\pi,\epsilon} (1+|t|)^{3/4 - 3/40 + \epsilon} $, which holds for all Hecke-Maass cusp forms on $\rm SL(3,\mathbb{Z})$, not just self-dual ones.

ABSTRACT

We revisit Munshi's proof of the $t$-aspect subconvex bound for $ m GL(3)$ $L$-functions, and we are able to remove the `conductor lowering' trick. This simplification along with a more careful stationary phase analysis allows us to improve Munshi's bound to, $$ L(1/2+it, π) \ll_{π, ε} (1+|t|)^{3/4-3/40+ε}. $$

Motivation & Objective

• To improve the subconvex bound for $\rm GL(3)$ $L$-functions in the $t$-aspect for general Hecke-Maass cusp forms on $\rm SL(3,\mathbb{Z})$.
• To eliminate the 'conductor lowering' trick used in Munshi's original proof, thereby simplifying the stationary phase analysis.
• To achieve a stronger subconvexity exponent than Munshi's bound while maintaining generality across all Hecke-Maass cusp forms.
• To refine the application of the delta method and Kloosterman's version of the circle method to enhance oscillatory sum estimates.
• To demonstrate that the improved bound holds uniformly across the critical line, with explicit dependence on $t$ and $\pi$

Proposed method

• Revisit Munshi's proof of the $t$-aspect subconvex bound for $\rm GL(3)$ $L$-functions, removing the 'conductor lowering' technique to streamline the analysis.
• Apply the approximate functional equation to express $ L(1/2+it,\pi) $ in terms of a truncated sum $ S(N) $, which is then bounded via Cauchy-Schwarz and the Ramanujan bound on average.
• Use Kloosterman's version of the circle method to separate variables in the sum, writing $ \delta(n=0) $ via exponential sums over $ a \mod q $, with $ q \leq Q $ and $ a $ coprime to $ q $.
• Apply Voronoi summation to the $ n $- and $ r $-sums to transform the oscillatory sums into more manageable exponential sums with Bessel functions.
• Perform a detailed stationary phase analysis on the resulting oscillatory integrals, carefully estimating the critical points and second derivatives to control the oscillation decay.
• Optimize the choice of the parameter $ Q $, setting $ Q = N^{1/2}/t^{1/5} $, to balance the two main error terms in the bound for $ S(N) $, yielding the final subconvex estimate.

Experimental results

Research questions

• RQ1 Can the 'conductor lowering' trick be removed from Munshi's proof of the $t$-aspect subconvex bound for $\rm GL(3)$ $L$-functions without sacrificing the strength of the bound?
• RQ2 How does the removal of conductor lowering affect the stationary phase analysis and the resulting error estimates in the $t$-aspect subconvexity problem?
• RQ3 What is the optimal bound achievable for $ L(1/2+it,\pi) $ for general Hecke-Maass cusp forms on $\rm SL(3,\mathbb{Z})$ using a refined application of the delta method and circle method?
• RQ4 Can the stationary phase analysis be improved through more careful estimation of the second derivatives of the phase function, leading to a better subconvexity exponent?
• RQ5 Is the improved bound $ O(t^{3/4 - 3/40 + \epsilon}) $ achievable uniformly across all $ \pi $, and does it hold without restricting to self-dual forms?

Key findings

• The paper achieves the subconvex bound $ L(1/2+it,\pi) \ll_{\pi,\epsilon} (1+|t|)^{3/4 - 3/40 + \epsilon} $, improving upon Munshi's original exponent.
• The removal of the 'conductor lowering' trick leads to a cleaner and more transparent stationary phase analysis, reducing technical complications.
• The diagonal contribution in the main term is improved by a factor of $ Q $ compared to Munshi's estimate, which enhances the overall bound.
• The off-diagonal contribution is controlled via a refined estimate of the Bessel function integral and the use of the second derivative condition in the stationary phase lemma.
• The optimal choice of the parameter $ Q = N^{1/2}/t^{1/5} $ balances the two main error terms, yielding $ S(N) \ll N^{3/4} t^{3/10 + \epsilon} $, which beats the trivial bound for $ N \sim t^{6/5} $.
• The final bound holds uniformly for all Hecke-Maass cusp forms $ \pi $ on $ \rm SL(3,\mathbb{Z}) $, not just self-dual ones, extending the range of applicability compared to prior results.