[Paper Review] Non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms

Bin Chen, Jianliang Wu|arXiv (Cornell University)|Dec 28, 2015

Advanced Algebra and Geometry23 references4 citations

TL;DR

This paper investigates the sign changes, non-vanishing, and distribution of Hecke eigenvalues for half-integral weight cusp forms, focusing on the sequence $\mathfrak{a}_{\mathfrak{f}}(tn^2)$ for square-free $t$. Using the Shimura lift and multiplicative function theory, it establishes an upper bound $n_{\mathfrak{f}} \ll (k^2N)^{9/20}$ for the first negative coefficient, proves positive density for non-vanishing and sign-changed coefficients, and shows infinitely many non-vanishing terms in short intervals and arithmetic progressions via $\mathscr{B}$-free number theory.

ABSTRACT

In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form $\\mathfrak{f}$ with half-integral weight:\\begin{itemize}\\item[--]The first negative coefficient of the sequence $\\{\\mathfrak{a}\\_{\\mathfrak{f}}(tn^2)\\}\\_{n\\in \\N}$,\\item[--]The number of coefficients $\\mathfrak{a}\\_{\\mathfrak{f}}(tn^2)$ of same signs,\\item[--]Non-vanishing of coefficients $\\mathfrak{a}\\_{\\mathfrak{f}}(tn^2)$ in short intervals and in arithmetic progressions,\\end{itemize}where $\\mathfrak{a}\\_{\\mathfrak{f}}(n)$ is the $n$-th Fourier coefficient of $\\mathfrak{f}$ and $t$ is a square-free integersuch that $\\mathfrak{a}\\_{\\mathfrak{f}}(t)\ ot=0$.

Motivation & Objective

• To determine the first negative coefficient in the sequence $\{\mathfrak{a}_{\mathfrak{f}}(tn^2)\}_{n \in \mathbb{N}}$ for half-integral weight cusp forms.
• To quantify the number of coefficients of the same sign in the sequence $\{\mathfrak{a}_{\mathfrak{f}}(tn^2)\}$.
• To establish non-vanishing results for $\mathfrak{a}_{\mathfrak{f}}(tn^2)$ in short intervals and arithmetic progressions.
• To extend prior work on sign changes by providing effective bounds and density estimates for eigenvalue behavior.

Proposed method

• Utilizes the Shimura lift to relate half-integral weight forms to integral weight modular forms, enabling analytic techniques from the latter.
• Applies the method of [12, 13] to derive an upper bound for the first negative coefficient $n_{\mathfrak{f}}$ in terms of $k$ and $N$.
• Employs multiplicative function theory and exponential sum estimates to analyze the density of non-vanishing and sign-changed coefficients.
• Introduces a $\mathscr{B}_{\mathfrak{f}}$-free number system to model non-vanishing conditions, leveraging sieve methods and multiple exponential sum estimates.
• Establishes that the set $\mathscr{B}_{\mathfrak{f}}$ satisfies the conditions for $\mathscr{B}$-free number theory, including convergence of reciprocal sums and pairwise coprimality.
• Uses integration by parts and bounds on the number of primes $p$ with $\mathfrak{a}_{\mathfrak{f}}(tp^2) = 0$ to verify the convergence condition $\sum_{b \in \mathscr{B}_{\mathfrak{f}}} b^{-1} < \infty$.

Experimental results

Research questions

• RQ1 What is the smallest $n$ such that $\mathfrak{a}_{\mathfrak{f}}(tn^2) < 0$ and $(n, N/2) = 1$?
• RQ2 How many terms in the sequence $\{\mathfrak{a}_{\mathfrak{f}}(tn^2)\}$ have the same sign, and what is their natural density?
• RQ3 Do the coefficients $\mathfrak{a}_{\mathfrak{f}}(tn^2)$ remain non-zero in short intervals and arithmetic progressions?
• RQ4 Can the distribution of non-vanishing coefficients be modeled using $\mathscr{B}$-free numbers?

Key findings

• The first negative coefficient $n_{\mathfrak{f}}$ in the sequence $\{\mathfrak{a}_{\mathfrak{f}}(tn^2)\}$ satisfies $n_{\mathfrak{f}} \ll (k^2N)^{9/20}$, with an implied constant independent of $k$ and $N$.
• The number of non-vanishing coefficients $\mathfrak{a}_{\mathfrak{f}}(tn^2)$ up to $x$ is asymptotically $\rho_{\mathfrak{f}} x \{1 + O_{\mathfrak{f},\varepsilon}((\log x)^{-1/4 + \varepsilon})\}$, where $\rho_{\mathfrak{f}} > 0$ is a positive density constant.
• The number of positive and negative coefficients $\mathfrak{a}_{\mathfrak{f}}(tn^2)$ up to $x$ each have density $\frac{1}{2}\rho_{\mathfrak{f}} x \{1 + O_{\mathfrak{f},\varepsilon}((\log x)^{-1/4 + \varepsilon})\}$.
• For any $\varepsilon > 0$, there are $\gg_{\mathscr{B}_{\mathfrak{f}},\varepsilon} y$ $\mathscr{B}_{\mathfrak{f}}$-free integers $n \in (x, x+y]$ with $y \gg x^{7/17 + \varepsilon}$, ensuring non-vanishing of $\mathfrak{a}_{\mathfrak{f}}(tn^2)$ in short intervals.
• In arithmetic progressions $n \equiv a \pmod{q}$ with $q \leq x^{\varepsilon}$ and $\gcd(a,q)$ coprime to all $b \in \mathscr{B}_{\mathfrak{f}}$, there are $\gg_{\mathscr{B}_{\mathfrak{f}},\varepsilon} y/q$ such $n$ with $\mathfrak{a}_{\mathfrak{f}}(tn^2) \neq 0$ for $y \gg x^{17/38 + 100\varepsilon}$.
• The set $\mathscr{B}_{\mathfrak{f}}$ satisfies both $\sum_{b \in \mathscr{B}_{\mathfrak{f}}} b^{-1} < \infty$ and pairwise coprimality, validating its use in $\mathscr{B}$-free number theory.