The distribution of Fourier coefficients of half-integral weight forms

Let

\begin{displaymath}f(z)=\sum_{n=1}^\infty a(n) n^{(k-1)/2}e(nz)\end{displaymath}

be a newform of weight $k$ for the full modular group, and let

\begin{displaymath}g(z)=\sum_{n=1}^\infty c(n)^{(k-1)/4}e(nz)\end{displaymath}

be a cusp form of weight $(k+1)/4$ and level 4 which is associated to $f(z)$ by the Shimura map. We normalize $f$ by requiring that $a(1)=1$ and we normalize $g$ by requiring that

\begin{displaymath}\frac{1}{6}\int_{\Gamma_0(4)\backslash \Cal{H}}\vert g(z)\vert^2y^{(k+1)/2} ~d\mu z =1.\end{displaymath}

Then for squarefree $q$ with $\chi_q(-1)=(-1)^k$ we have, by the formula of Kohnen and Zagier,

\begin{displaymath}c(q)^2=\pi^{-\frac k2}\Gamma(\frac k2)L_f(1/2,\chi_q)\langle f,f\rangle ^{-1}\end{displaymath}

where

\begin{displaymath}\langle f,f\rangle =\int_{\Gamma_0(1)\backslash \Cal{H}}\vert f(z)\vert^2y^{k} ~d\mu z .\end{displaymath}

In particular, the Riemann Hypothesis for $L_f(s,\chi_q)$ implies that

\begin{displaymath}c(q)\ll \exp(c \log q / \log \log q)\end{displaymath}

for an appropriate choice of $c$. If the bound

\begin{displaymath}L_f(1/2,\chi_q) \ll \exp (c_1 \sqrt{\log q /\log \log q})\end{displaymath}

holds, then a similar bound for $c(q)$ will hold (but with $c_1$ replaced by $c_1/2$). The question here is to decide which (if either) of these bounds represents the true state of affairs.

If $q$ has a square factor, then $c_q$ can be determined from values of $a_p$ where $p^2\mid q$. It is known, by Deligne's theorem, that

\begin{displaymath}\vert a_n\vert\le d(n) \end{displaymath}

where $d(n)$ is the number of divisors of $n$. Of course, $d(p)=2$ so that we can write

\begin{displaymath}a_p =2\cos \theta_p\end{displaymath}

where $\theta_p$ is real. The conjecture of Sato and Tate about the distribution of the $\theta_p$ asserts that

\begin{displaymath} \lim_{x\to \infty} x^{-1} \sum \Sb p\le x\\ \alpha < \theta_... ...\log p =\frac {2}{\pi}\int_\alpha^\beta \sin^2\theta ~d\theta \end{displaymath}

for $0\le \alpha <\beta \le \pi$. A consequence of the Sato-Tate conjecture is that there are infinitely many $p$ (actually a positive proportion of $p$) for which

\begin{displaymath}2-\epsilon < a_p < 2\end{displaymath}

for any given $\epsilon >0$. The maximum order of $d(n)$ is given by

\begin{displaymath}\limsup_{n\to \infty} \frac{\exp{(\log 2 \log n /\log \log n)}}{d(n)}=1.\end{displaymath}

The Sato-Tate conjecture implies the same assertion with $d(n)$ replaced by $a_n$.

Thus, in particular, we find that

\begin{displaymath}c(p^2) =a(p)\ll \exp(\log 2 \log p/\log \log p)\end{displaymath}

and we expect this bound to be sharp in the sense that the $\log 2 $ cannot be replaced by anything smaller.

Thus, if the ``smaller'' bound is true for $L_f(1/2,\chi_q)$ then there is a great variation between the distribution of the Fourier coefficients $c(n)$ according to whether $n$ is square or squarefree.



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