Ramanujan–Petersson conjecture

Let $\mathbb{H}=\{z\in\mathbb{C}:\operatorname{Im}z>0\}$ and $q=e^{2\pi iz}$.

(1) Coefficients of holomorphic cusp forms. Let $k\geq 1$ be an integer and let $\Gamma\subseteq SL(2,\mathbb{Z})$ be a congruence subgroup containing $\begin{pmatrix}1&1\\0&1\end{pmatrix}$. Let $f$ be a cusp form of weight $k$ for $\Gamma$, i.e. $f:\mathbb{H}\to\mathbb{C}$ is holomorphic, satisfies

f\!\left(\frac{az+b}{cz+d}\right)=(cz+d)^{k}f(z)\qquad\text{for all }\begin{pmatrix}a&b\c&d\end{pmatrix}\in\Gamma,

and is holomorphic and vanishing at every cusp of $\Gamma$, so that its Fourier expansion at $\infty$ is

$$f(z)=\sum_{n=0}^{\infty}a_{n}q^{n},\qquad a_{0}=0.$$

Then for every $\epsilon>0$,

$$a_{n}=O_{\epsilon}\!\left(n^{(k-1)/2+\epsilon}\right).$$

(2) The case of the discriminant form. Let $\eta$ be the Dedekind eta function, let

$$\Delta(z)=q\prod_{n>0}(1-q^{n})^{24}=\eta(z)^{24}=\sum_{n=1}^{\infty}\tau(n)q^{n},$$

a holomorphic cusp form of weight $12$ and level $1$. Then:

• $\tau(mn)=\tau(m)\tau(n)$ whenever $\gcd(m,n)=1$; * $\tau(p^{j+1})=\tau(p)\tau(p^{j})-p^{11}\tau(p^{j-1})$ for every prime $p$ and every $j\geq 2$; * $|\tau(p)|\leq 2p^{11/2}$ for every prime $p$,

and consequently $|\tau(n)|\leq d(n)\,n^{11/2}$ for all $n\geq 1$, where $d(n)$ is the number of divisors of $n$, whence $\tau(n)=O_{\epsilon}\!\left(n^{11/2+\epsilon}\right)$ for every $\epsilon>0$.

Moreover, all nontrivial zeros of the analytic continuation of

$$L(s,\tau)=\sum_{n=1}^{\infty}\frac{\tau(n)}{n^{s}}=\prod_{p\ \mathrm{prime}}\frac{1}{1-\tau(p)p^{-s}+p^{11-2s}}$$

lie on the line $\operatorname{Re}(s)=6$; here $L(s,\tau)$ converges absolutely for $\operatorname{Re}(s)>\tfrac{13}{2}$ and satisfies

$$\frac{\Gamma(s)L(s,\tau)}{(2\pi)^{s}}=\frac{\Gamma(12-s)L(12-s,\tau)}{(2\pi)^{12-s}}.$$

(3) Generalization to automorphic representations. Let $F$ be a global field with ring of adeles $\mathbb{A}_{F}$, let $G$ be a connected reductive group over $F$, and let $\pi\cong\bigotimes_{v}'\pi_{v}$ be a cuspidal automorphic representation of $G(\mathbb{A}_{F})$ that is globally generic, i.e. admits a nonzero Whittaker model with respect to a nondegenerate character of the unipotent radical of a Borel subgroup. Then for every place $v$ of $F$, the local component $\pi_{v}$ is a tempered representation of $G(F_{v})$.

In the special case $G=GL(2)$, the local components of such a $\pi$ lie in the principal series.