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Fourth power moment of twisted Kloosterman sum and Hurwitz class numbers

Neelam Saikia

TL;DR

The paper analyzes the fourth power moment of twisted Kloosterman sums $S(4,\phi)_p$ and links it to weighted sums of Hurwitz class numbers $H^*(D)$ via traces of Frobenius on Legendre elliptic curves and counts of isomorphism classes. It furnishes explicit formulas for $S(4,\phi)_p$ in terms of these class-number sums, with case distinctions according to $p\bmod 4$, and derives sharp asymptotics for the associated weighted Hurwitz sums using harmonic Maass forms, mock modular forms, and holomorphic projection. The authors further connect these moment formulas to twisted Kloosterman-sheaf sums and Gaussian hypergeometric functions over finite fields, obtaining asymptotic averages for two families of $p$-adic hypergeometric functions. The results illuminate deep connections among exponential sum moments, elliptic-curve counts, modular objects, and hypergeometric structures, providing both explicit formulas and robust asymptotic behavior for the fourth moment and its twists. Overall, the work advances understanding of how arithmetic geometry and modular forms govern high-power moments of exponential sums and enriches the toolkit for studying related automorphic phenomena on Calabi–Yau and elliptic-curves frameworks.

Abstract

In this paper, we investigate the fourth power moment of twisted Kloosterman sum and its relationship with Hurwitz class number. We derive an explicit formula expressing this moment in terms of weighted sums involving Hurwitz class numbers. Our approach involves analyzing point counting formulas associated with the resolution of certain Calabi-Yau threefold. Furthermore, we study the asymptotic behaviour of weighted sums of Hurwitz class numbers that appear in the moment formula. To derive these asymptotic formulas, we employ the theory of harmonic Maass forms, mock modular forms and holomorphic projections. As an application of these asymptotic results, we obtain the asymptotic formula for the fourth power moment of twisted Kloosterman sums.

Fourth power moment of twisted Kloosterman sum and Hurwitz class numbers

TL;DR

The paper analyzes the fourth power moment of twisted Kloosterman sums and links it to weighted sums of Hurwitz class numbers via traces of Frobenius on Legendre elliptic curves and counts of isomorphism classes. It furnishes explicit formulas for in terms of these class-number sums, with case distinctions according to , and derives sharp asymptotics for the associated weighted Hurwitz sums using harmonic Maass forms, mock modular forms, and holomorphic projection. The authors further connect these moment formulas to twisted Kloosterman-sheaf sums and Gaussian hypergeometric functions over finite fields, obtaining asymptotic averages for two families of -adic hypergeometric functions. The results illuminate deep connections among exponential sum moments, elliptic-curve counts, modular objects, and hypergeometric structures, providing both explicit formulas and robust asymptotic behavior for the fourth moment and its twists. Overall, the work advances understanding of how arithmetic geometry and modular forms govern high-power moments of exponential sums and enriches the toolkit for studying related automorphic phenomena on Calabi–Yau and elliptic-curves frameworks.

Abstract

In this paper, we investigate the fourth power moment of twisted Kloosterman sum and its relationship with Hurwitz class number. We derive an explicit formula expressing this moment in terms of weighted sums involving Hurwitz class numbers. Our approach involves analyzing point counting formulas associated with the resolution of certain Calabi-Yau threefold. Furthermore, we study the asymptotic behaviour of weighted sums of Hurwitz class numbers that appear in the moment formula. To derive these asymptotic formulas, we employ the theory of harmonic Maass forms, mock modular forms and holomorphic projections. As an application of these asymptotic results, we obtain the asymptotic formula for the fourth power moment of twisted Kloosterman sums.
Paper Structure (9 sections, 34 theorems, 129 equations)

This paper contains 9 sections, 34 theorems, 129 equations.

Table of Contents

  1. Introduction and statement of results
  2. Arithmetic of elliptic curves
  3. Moments of Twisted Kloosterman sums and class numbers
  4. Proof of Theorem \ref{['relation-2']}
  5. Harmonic Maass forms and Mock modular forms
  6. Asymptotic formulas of weighted sums of class numbers
  7. Proof of Corollary \ref{['twisted-moment-1']} and Corollary \ref{['moment-1']}
  8. Hypergeometric functions
  9. Acknowledgements

Key Result

Theorem 1.1

Suppose that $p>3$ be a prime and $-2\sqrt{p}<s<2\sqrt{p}$ be an integer. Then the following are true: (1) If $p\equiv1\pmod{4},$ then (2) If $p\equiv3\pmod{4},$ then

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark
  • Proposition 2.1: Proposition 1.7, Chapter III of Silverman
  • Proposition 2.2: Proposition 3.2 of ahl-ono-1
  • Proposition 2.3: Proposition 3.3 (1) of ahl-ono-1
  • ...and 43 more