[Paper Review] The Fifth Moment of modular L-functions

Eren Mehmet Kıral, Matthew P. Young|arXiv (Cornell University)|Jan 25, 2017

Analytic Number Theory Research26 references7 citations

TL;DR

This paper establishes a sharp upper bound for the fifth moment of central L-values of holomorphic Hecke eigenforms of fixed small weight and prime level q, using spectral theory and the Bruggeman-Kuznetsov formula to handle cancellation in multivariable sums involving divisor functions and Kloosterman sums. The key result is $\sum_{f\in H_{\kappa}(q)} L(1/2,f)^5 \ll q^{1+\theta+\varepsilon}$, where $\theta = 7/64$ is the best-known progress towards the Ramanujan-Petersson conjecture, improving prior subconvexity bounds.

ABSTRACT

We prove a sharp bound on the fifth moment of modular L-functions of fixed small weight, and large prime level.

Motivation & Objective

• To establish a sharp upper bound for the fifth moment of central L-values of holomorphic Hecke eigenforms of fixed small weight and large prime level.
• To overcome the challenge of significant cancellation in multivariable sums involving divisor functions and Kloosterman sums.
• To avoid solving shifted convolution problems, taking a different route from prior works on amplified or mollified fourth moments.
• To demonstrate that the 'fake' main terms in the spectral decomposition cancel sufficiently under appropriate weight function choices, enabling a non-trivial bound.

Proposed method

• Apply the Bruggeman-Kuznetsov formula to transform the fifth moment into a spectral sum involving twisted fourth moments of GL₂ cuspidal L-functions.
• Use a Kuznetsov/Motohashi-type duality to relate the original moment problem to a dual moment in another family of L-functions.
• Employ a family of weight functions with controlled stationary phase behavior to manage oscillatory integral transforms with multiple variables.
• Identify and analyze Kloosterman sums associated with Atkin-Lehner cusps of $\Gamma_0(q)$ using a 'correct' scaling matrix to preserve multiplicativity.
• Treat the continuous spectrum by shifting contour integrals past poles of Dirichlet L-functions, carefully tracking savings from non-principal characters.
• Use a partition of unity in dyadic intervals to sum over variables and ensure absolute convergence, especially when $G(1/2) = 0$ is exploited to avoid divergent sums.

Experimental results

Research questions

• RQ1 Can a sharp bound be established for the fifth moment of central L-values in the family of holomorphic cusp forms of fixed small weight and prime level?
• RQ2 How can cancellation in multivariable sums with divisor functions and Kloosterman sums be effectively harnessed to avoid trivial bounds?
• RQ3 What role do 'fake' main terms—appearing in spectral expansions—play, and can they be controlled without explicit cancellation?
• RQ4 To what extent can the method avoid solving shifted convolution problems, as in previous works on fourth moments?
• RQ5 How does the choice of weight functions in approximate functional equations affect the behavior of spurious main terms in the spectral analysis?

Key findings

• The fifth moment satisfies $\sum_{f\in H_{\kappa}(q)} L(1/2,f)^5 \ll q^{1+\theta+\varepsilon}$ as $q \to \infty$ through primes, with $\theta = 7/64$ from Kim and Sarnak.
• The bound implies the subconvexity estimate $L(1/2,f) \ll_{\varepsilon} q^{(1+\theta)/5 + \varepsilon}$, improving on the previous best-known result.
• The method avoids shifted convolution problems and instead relies on spectral theory and careful contour integration with weight functions.
• The contribution from the continuous spectrum is shown to be $O(q^{\varepsilon})$ after proper contour shifts and summing over dyadic intervals.
• The analysis of 'fake' main terms is resolved by choosing weight functions such that their combined contribution remains consistent with the main bound.
• The final bound is sharp and consistent with the expectation that moments one beyond the 'barrier' (here, the fourth moment) can yield subconvexity via self-amplification.