Ramanujan–Petersson conjecture

Remarks

Ramanujan conjectured the bound for $\tau(p)$ in 1916, and Petersson extended the conjecture to other modular forms. Deligne proved it in 1974 as a consequence of his proof of the Weil conjectures. Earlier work of Eichler, Shimura and Deligne had identified $a_p$ with the trace of Frobenius on an étale cohomology group, whose eigenvalues have absolute value $p^{(k-1)/2}$ by the Weil conjectures. Example: $\tau(2) = -24$ and $2 \cdot 2^{11/2} = 2^{13/2} \approx 90.5$; $\tau(3) = 252$ and $2 \cdot 3^{11/2} \approx 841.8$. The normalization $a_1 = 1$ and the eigenform hypothesis are needed: the cusp form $100\Delta$ has $a_2 = -2400$, which breaks the bound. The analogous statement for Maass forms is still open.

Going deeper

For a student

For a normalized eigenform, the Euler factor at $p \nmid N$ is $1 - a_p p^{-s} + p^{k-1-2s}$. Writing it as $(1 - \alpha_p p^{-s})(1 - \beta_p p^{-s})$, the bound is equivalent to $\abs{\alpha_p} = \abs{\beta_p} = p^{(k-1)/2}$. So the theorem is an analogue of the Riemann hypothesis for the local factors of the L-function of $f$. Together with multiplicativity, it gives $\abs{a_n} \le d(n)\, n^{(k-1)/2}$ for $n$ prime to $N$, where $d(n)$ is the number of divisors of $n$. The Sato–Tate conjecture describes how the normalized values $a_p / (2 p^{(k-1)/2}) \in [-1, 1]$ are distributed.