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Non-vanishing for cubic Hecke $L$-functions

Chantal David, Alexandre de Faveri, Alexander Dunn, Joshua Stucky

Abstract

Let $ω$ be a primitive cubic root of unity. We study the non-vanishing problem for the family of Hecke $L$-functions associated to primitive cubic characters defined over the Eisenstein quadratic number field $\mathbb{Q}(ω)$. We prove unconditionally that a positive proportion of Hecke $L$-functions associated to the cubic residue symbols $χ_q$ with $q \in \mathbb{Z}[ω]$ squarefree and $q \equiv 1 \pmod{9}$ do not vanish at the central point. Our proof goes through the method of first and second mollified moments. The principal new contribution of this paper is the asymptotic evaluation of the mollified second moment with power saving error term. No asymptotic formula for the mollified second moment of a cubic family was known (even over function fields) prior to the writing of this paper. Our new approach makes crucial use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Brown's cubic large sieve, and a Lindelöf-on-average upper bound for the second moment of cubic Dirichlet series that we establish. The significance of our result is that the (unitary) family considered does not satisfy a perfectly orthogonal large sieve bound. This is quite unlike other families of Dirichlet $L$-functions in the literature for which unconditional results are known: the symplectic family of quadratic characters and the unitary family of all Dirichlet characters $χ\pmod{q}$. Consequently, our proof has fundamentally different features from the corresponding works of Soundararajan and of Iwaniec and Sarnak.

Non-vanishing for cubic Hecke $L$-functions

Abstract

Let be a primitive cubic root of unity. We study the non-vanishing problem for the family of Hecke -functions associated to primitive cubic characters defined over the Eisenstein quadratic number field . We prove unconditionally that a positive proportion of Hecke -functions associated to the cubic residue symbols with squarefree and do not vanish at the central point. Our proof goes through the method of first and second mollified moments. The principal new contribution of this paper is the asymptotic evaluation of the mollified second moment with power saving error term. No asymptotic formula for the mollified second moment of a cubic family was known (even over function fields) prior to the writing of this paper. Our new approach makes crucial use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Brown's cubic large sieve, and a Lindelöf-on-average upper bound for the second moment of cubic Dirichlet series that we establish. The significance of our result is that the (unitary) family considered does not satisfy a perfectly orthogonal large sieve bound. This is quite unlike other families of Dirichlet -functions in the literature for which unconditional results are known: the symplectic family of quadratic characters and the unitary family of all Dirichlet characters . Consequently, our proof has fundamentally different features from the corresponding works of Soundararajan and of Iwaniec and Sarnak.
Paper Structure (48 sections, 29 theorems, 489 equations)

This paper contains 48 sections, 29 theorems, 489 equations.

Table of Contents

  1. Introduction
  2. Statement of non-vanishing result
  3. Statement of results on moments
  4. High level sketch
  5. Evaluation of $\mathcal{S}_M(A_2(q);F)$
  6. The polar contribution \ref{['main']} in the unbalanced regime: $N_1$ large and $N_2$ small
  7. The polar contribution \ref{['main']} in the balanced regime: $N_1$ and $N_2$ of moderate size
  8. Endgame for $\mathcal{S}_M(A_2(q);F)$ and the error term in \ref{['secondmomentcor']}
  9. Conventions
  10. Preliminaries
  11. Eisenstein quadratic field and the cubic symbol
  12. Cubic Hecke characters
  13. Cubic Gauss sums and variants
  14. Hecke $L$-functions over $\mathbb{Q}(\omega)$
  15. Poisson summation
  16. ...and 33 more sections

Key Result

Theorem 1.1

For at least $14 \%$ of $q \in \mathbb{Z}[\omega]$ squarefree with $q \equiv 1 \ (\mathrm{mod}\ 9)$, we have $L(1/2,\chi_q) \neq 0$. More precisely, for any fixed $\varepsilon>0$ and sufficiently large $X>0$,

Theorems & Definitions (60)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Corollary 1.7
  • proof
  • Corollary 1.8
  • proof
  • ...and 50 more