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Quartic Gauss sums over primes and metaplectic theta functions

Chantal David, Alexander Dunn, Alia Hamieh, Hua Lin

TL;DR

The paper advances the distribution theory of quartic Gauss sums over primes in $\mathbb{Z}[i]$ by proving an improved bound for twisted sums with smooth cutoffs, namely $\ll X^{5/6+\varepsilon}+X^{3/4+\varepsilon}|\ell|^{3/2+\varepsilon}$, and by formulating a conjecture predicting a main term of size $X^{3/4}$ for the first moment. The authors decompose sums via Vaughan’s identity into Type-I and Type-II components; Type-II sums are controlled through Onodera–Goldmakher–Louvel quadratic large sieve over $\mathbb{Q}(i)$, while Type-I analyses hinge on Dirichlet series $\psi^{(4)}_{\beta}(s,\nu,\ell)$ tied to metaplectic Eisenstein series on the 4-fold cover of $\mathrm{GL}_2$ and to residues of quartic theta functions. The work builds a comprehensive analytic framework for metaplectic objects, including Dirichlet-series with level structure, functional equations, and Voronoi-type transforms, to achieve subconvex-type bounds in the $X$-aspect and bound the arithmetic contributions via theta-function residues. These results sharpen Patterson-type estimates in the quartic setting and illuminate the role of metaplectic theta coefficients, while also outlining precise bottlenecks that prevent extending the $X^{5/6}$ bound to higher orders. Overall, the paper blends automorphic methods, large-sieve bounds over number fields, and Voronoi analysis to push forward understanding of higher-order Gauss sums and their moments.

Abstract

We improve 1987 estimates of Patterson for sums of quartic Gauss sums over primes. Our Type-I and Type-II estimates feature new ideas, including use of the quadratic large sieve over $\mathbb{Q}(i)$, and Suzuki's evaluation of the Fourier-Whittaker coefficients of quartic theta functions at squares. We also conjecture asymptotics for certain moments of quartic Gauss sums over primes.

Quartic Gauss sums over primes and metaplectic theta functions

TL;DR

The paper advances the distribution theory of quartic Gauss sums over primes in by proving an improved bound for twisted sums with smooth cutoffs, namely , and by formulating a conjecture predicting a main term of size for the first moment. The authors decompose sums via Vaughan’s identity into Type-I and Type-II components; Type-II sums are controlled through Onodera–Goldmakher–Louvel quadratic large sieve over , while Type-I analyses hinge on Dirichlet series tied to metaplectic Eisenstein series on the 4-fold cover of and to residues of quartic theta functions. The work builds a comprehensive analytic framework for metaplectic objects, including Dirichlet-series with level structure, functional equations, and Voronoi-type transforms, to achieve subconvex-type bounds in the -aspect and bound the arithmetic contributions via theta-function residues. These results sharpen Patterson-type estimates in the quartic setting and illuminate the role of metaplectic theta coefficients, while also outlining precise bottlenecks that prevent extending the bound to higher orders. Overall, the paper blends automorphic methods, large-sieve bounds over number fields, and Voronoi analysis to push forward understanding of higher-order Gauss sums and their moments.

Abstract

We improve 1987 estimates of Patterson for sums of quartic Gauss sums over primes. Our Type-I and Type-II estimates feature new ideas, including use of the quadratic large sieve over , and Suzuki's evaluation of the Fourier-Whittaker coefficients of quartic theta functions at squares. We also conjecture asymptotics for certain moments of quartic Gauss sums over primes.
Paper Structure (18 sections, 14 theorems, 187 equations)

This paper contains 18 sections, 14 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Background and statement of results
  3. Overall strategy of the paper
  4. Acknowledgements
  5. Conventions
  6. Quartic Gauss sums
  7. Metaplectic Eisenstein series and theta functions
  8. Dirichlet series
  9. Group action on $\mathbb{H}^3$ and Laplacian
  10. Congruence subgroup $\Gamma_1(\lambda^4)$ and Kubota multiplier
  11. Metaplectic Eisenstein series
  12. Theta functions
  13. Onodera's and Goldmakher--Louvel's quadratic large sieve over $\mathbb{Q}(i)$
  14. Preliminary reductions and Vaughan's identity
  15. Type-II estimates
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $R:(0,\infty) \rightarrow \mathbb{C}$ be a smooth function with compact support in $[1,2]$. Then for any $\ell \in \mathbb{Z}$, $\beta \in \{1,1+\lambda^3 \} \space\pmod{4}$, and $\varepsilon>0$, we have as $X \rightarrow \infty$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Proposition 4.3
  • proof
  • Corollary 4.5
  • Theorem 5.1
  • Lemma 6.2
  • Proposition 7.1
  • proof
  • Lemma 8.1
  • proof
  • ...and 14 more