[Paper Review] Bilinear forms with Kloosterman sums and applications

Emmanuel Kowalski, Ph. Michel+1 more|arXiv (Cornell University)|Nov 5, 2015

Analytic Number Theory Research30 references11 citations

TL;DR

This paper establishes non-trivial bounds for bilinear forms involving hyper-Kloosterman sums when the variable ranges fall below the classical Pólya-Vinogradov threshold, using advanced tools from ℓ-adic cohomology and the Riemann Hypothesis over finite fields. The key contribution is a new bound that achieves savings of $ q^{-1/64 + \varepsilon} $ for $ M = N = q^{1/2} $, enabling applications to moments of twisted $ L $-functions and the distribution of Eisenstein-Hecke coefficients in arithmetic progressions to large moduli.

ABSTRACT

We prove non-trivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the Pólya-Vinogradov range. We then derive applications to the second moment of holomorphic cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on $\GL_3$. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially $\ell$-adic cohomology and the Riemann Hypothesis.

Motivation & Objective

• To establish non-trivial bounds for bilinear forms in hyper-Kloosterman sums when the support sizes $ M $ and $ N $ are below the Pólya-Vinogradov range.
• To extend the range of applicability of bilinear sum estimates beyond classical Fourier-analytic methods, particularly for short intervals.
• To apply these bounds to the second moment of cusp forms twisted by Dirichlet characters modulo primes.
• To study the distribution of Eisenstein-Hecke coefficients on $ \mathrm{GL}_3 $ in arithmetic progressions to large moduli.

Proposed method

• Reduction of the bilinear form to complete exponential sums over finite fields via algebraic geometry techniques.
• Use of $ \ell $-adic cohomology and the Riemann Hypothesis over finite fields to bound complete sums in three variables.
• Analysis of monodromy and irreducibility of sum-product transform sheaves to control local monodromy representations.
• Application of nearby and vanishing cycle functors to relate cohomology over generic and special fibers in the context of $ \ell $-adic sheaves.
• Use of smooth dyadic partitions and Fourier/Bessel transform decay estimates to localize and truncate sums.
• Combination of multiple bounds (5.2)–(5.5) to derive optimal decay estimates in logarithmic scale.

Experimental results

Research questions

• RQ1 Can non-trivial bounds be obtained for bilinear forms in hyper-Kloosterman sums when $ M, N \ll q^{1/2} $, beyond the Pólya-Vinogradov range?
• RQ2 What is the optimal saving factor achievable in the $ L $-function moment problem for cusp forms twisted by Dirichlet characters modulo prime $ q $?
• RQ3 How do Eisenstein-Hecke coefficients on $ \mathrm{GL}_3 $ distribute in arithmetic progressions to large moduli?
• RQ4 Can the monodromy of sheaves associated with sum-product transforms be shown to be irreducible to ensure strong cohomological bounds?

Key findings

• For $ M = N = q^{1/2} $, the bilinear form $ B([\times c]^*\mathrm{Kl}_k, \alpha, \beta) $ is bounded by $ \ll q^{-1/64 + \varepsilon} \|\alpha\|_2\|\beta\|_2 (MN)^{1/2} $, achieving a non-trivial saving beyond the Pólya-Vinogradov range.
• When $ \beta $ is the characteristic function of an interval, the bound improves to $ \ll q^{-1/24 + \varepsilon} \|\alpha\|_1^{1/2}\|\alpha\|_2^{1/2} M^{1/4}N $, valid for $ M = N \geq q^{3/7} $.
• The second moment of twisted $ L $-functions for cusp forms satisfies $ \frac{1}{\phi(q)} \sum_{\chi \mod q} L(f \otimes \chi, 1/2)L(g \otimes \chi, 1/2) = 2L(f \otimes g, 1)/\zeta(2) + O(q^{-\delta}) $ for $ \delta < 1/144 $, with the implied constant depending only on $ f, g, \delta $.
• The distribution of $ \mathrm{GL}_3 $ Eisenstein-Hecke coefficients in arithmetic progressions to large moduli is shown to be equidistributed, with error terms bounded by $ \ll q^{-\delta} $ for $ \delta < 1/26 $.
• The complete sum bounds in three variables are established via $ \ell $-adic cohomology, with key input from the nearby cycle formalism and monodromy analysis.
• The proof relies on a novel combination of cohomological techniques, including the use of nearby and vanishing cycles, to control the monodromy of sheaves arising from sum-product constructions.