[Paper Review] Sub-Weyl bounds for $GL(2)$ $L$-functions

Ritabrata Munshi|arXiv (Cornell University)|Jun 19, 2018

Analytic Number Theory Research11 references3 citations

TL;DR

This paper establishes the first sub-Weyl bound for $GL(2)$ $L$-functions in the $t$-aspect by introducing an enhanced $GL(2)$ delta method with spectral averaging, achieving the bound $L(1/2 + it, F) \ll t^{1/3 - 1/1200 + \varepsilon}$ for holomorphic Hecke cusp forms $F$ on $SL(2,\mathbb{Z})$. The method combines Weyl shifts, spectral decomposition, and Kloosterman sum analysis to surpass the classical Weyl exponent $1/3$, marking a breakthrough in subconvexity theory beyond degree one $L$-functions.

ABSTRACT

In this paper we obtain a sub-Weyl bound for $L(1/2+it,f)$ for $f$ a Hecke modular form.

Motivation & Objective

• To establish a sub-Weyl bound for $GL(2)$ $L$-functions in the $t$-aspect, breaking the long-standing Weyl exponent $1/3$ for higher-degree $L$-functions.
• To extend the $GL(2)$ delta method by introducing an additional spectral averaging layer to lower the conductor and improve cancellation in exponential sums.
• To address the subconvexity problem for $L(1/2 + it, F)$ beyond the convexity bound $t^{1/2 + \varepsilon}$, particularly for holomorphic Hecke cusp forms of level one.
• To demonstrate that the method can also yield weak sub-Weyl bounds for the Riemann zeta function and potentially break the Voronoi barrier in the divisor problem for cusp forms.

Proposed method

• Introduces Weyl shifts in the sum $S(N) = \sum_{N < n \leq 2N} \lambda_F(n) n^{it}$ by shifting $n \mapsto n + h$ with $h \sim H \ll \sqrt{N} t^{1/3 - \delta}$ to enhance cancellation.
• Applies the $GL(2)$ delta method with spectral decomposition over Hecke eigenforms $f \in H_k(q, \psi)$, using Petersson trace formula to convert arithmetic sums into Kloosterman sums.
• Implements a new spectral averaging mechanism over $k$, $q$, and $\psi$ to lower the conductor and control the dual off-diagonal terms.
• Analyzes the dual off-diagonal via a refined counting argument for solutions to congruences involving $\bar{h}_2/c_2 \equiv \bar{h}_2'/c_2' \pmod{c_2 c_2'}$, using bounds on $jk$ and $m_2$ to control error terms.
• Employs a non-trivial counting result from [3] to bound the number of solutions in the small $M_2$ regime, ensuring the bound holds when $\delta < 1/1200$.
• Combines estimates from the direct and dual off-diagonals, using smooth cutoffs and exponential sum techniques to control the total contribution to $\mathcal{F}$.

Experimental results

Research questions

• RQ1 Can a sub-Weyl bound be achieved for $GL(2)$ $L$-functions in the $t$-aspect, surpassing the classical Weyl exponent $1/3$?
• RQ2 Can the $GL(2)$ delta method be enhanced with spectral averaging to effectively lower the conductor and improve subconvexity bounds?
• RQ3 Does the method yield non-trivial sub-Weyl bounds for the Riemann zeta function or the divisor problem for cusp forms?
• RQ4 What is the optimal exponent achievable using this spectral averaging technique, and can it break the Voronoi barrier $O(x^{1/3 + \varepsilon})$?

Key findings

• The paper achieves the first sub-Weyl bound for $GL(2)$ $L$-functions: $L(1/2 + it, F) \ll t^{1/3 - 1/1200 + \varepsilon}$ for holomorphic Hecke cusp forms $F$ on $SL(2,\mathbb{Z})$.
• The method introduces a novel spectral averaging layer in the $GL(2)$ delta method, enabling conductor lowering and improved cancellation in exponential sums.
• The bound is derived by analyzing the dual off-diagonal contribution and showing it is acceptable when $\delta < 1/1200$, using refined counting of solutions to congruences involving $m_2 \sim M_2 \ll t^{36\delta}$.
• The same method yields a weak sub-Weyl bound for the Riemann zeta function with exponent $1/6 - 1/2400 + \varepsilon$, though better bounds are expected with classical exponent pair theory.
• The approach is robust enough to potentially break the Voronoi barrier $O(x^{1/3 + \varepsilon})$ for the divisor sum $\sum_{n \leq x} \lambda_F(n)$, as suggested by the method's structure.
• The final bound $\mathcal{F} \ll \sqrt{N} H Q^2 K t^{1/3 - \delta}$ holds under the condition $\delta < 1/1200$, confirming the sub-Weyl exponent.