Question

Hi I am curious about the growth of coefficients of cusp forms. I am aware of Ramanujan-Petersson conjecture/theorem in general terms but I was hoping for a more detailed description of precise statements on the growth of all the cusp form coefficients and whether these were tight. Also, for the Fourier coefficients of Hasse-Weil L-series of elliptic curves (meaning what can we say about these without making use to modularity). Any good references would be very useful also. Thanks!

Answer 1

It's worth distinguishing between the prime coefficients $a_p$, and the coefficients $a_n$ for general $n$. Let's look at $a_p$ first.

Firstly: for elliptic curves, it is fairly easy and elementary to prove that $|a_p| < 2\sqrt{p}$ (Hasse's inequality). This should be in any decent textbook e.g. Silverman. And this is the best bound you're going to get, because for any fixed elliptic curve the set $\{ a_p / (2\sqrt{p}) : p\ prime\}$ is dense in $(-1, 1)$, and in fact obeys a very specific distribution. (This distribution depends on whether or not E has complex multiplication. If E does not have CM, it the Sato-Tate distribution, a long-standing conjecture recently settled by Barnet-Lamb, Geraghty, Harris and Taylor. In the CM case you get a different, simpler distribution; thanks KConrad for pointing this out.)

For general modular eigenforms of weight $k$, the "right" bound is $|a_p| \le 2p^{(k-1)/2}$, but this is a very deep theorem (it follows from Deligne's work on the Weil conjectures). There is an easy elementary argument that gives $a_p = O(p^{k/2})$: this is in Miyake's book (corollary 2.1.6 if I recall correctly). By purely analytic methods you can push this a bit further, e.g. Rankin proved $a_p = O(p^{k/2 - 1/5})$ if I remember correctly, but you can't get the "right" bound this way.

For general $n$, the relations giving the $a_n$ in terms of the $a_p$ mean that the Deligne bound $|a_p| \le 2p^{(k-1)/2}$ turns into something like $a_n \le n^{(k-1)/2} d(n)$, where $d(n)$ is the number of divisors of $n$; this is $O(n^{(k-1)/2 + \epsilon})$ for any $\epsilon > 0$ (but it is not $O(n^{(k-1)/2})$).

(EDIT: For Hasse's inequality it's obvious that we have strict inequality $|a_p| < 2\sqrt{p}$ simply because $a_p \in \mathbf{Z}$ and $2\sqrt{p}$ isn't. But for general modular forms one only gets the non-strict inequality $|a_p| \le 2p^{(k-1)/2}$, not a strict inequality as I originally claimed (thanks to François for pointing this out); strict inequality is conjectured to hold if $k > 1$, and Coleman and Edixhoven have shown that this would follow from some standard conjectures on varieties over finite fields. If $k = 1$ then in fact $a_p = 2$ for a positive density set of primes, so equality definitely can occur in this case.)

(EDIT: Just to emphasize, if you have an elliptic curve over $\mathbf{Q}$, the Hasse bound $|a_p| < 2\sqrt{p}$ "follows from" the Deligne bound and the fact that $E$ is modular, but this would be a ridiculously laborious way of proving that: the direct elementary proof of Hasse's inequality is vastly easier than using modularity. In fact (some parts of) Deligne's proof can be interpreted as "trying to adapt Hasse's proof to a general algebraic variety", so the flow of information here is the other way.)

Answer 2

Let me focus on how to deduce a bound for all coefficients $a_n$ assuming the Deligne bound on $a_p$ (this is a standard argument).

Let $f$ be a newform of weight $k$, level $N$ and Nebentypus character $\psi$ modulo $N$. Assume the Deligne bound $|a_p| \leq 2p^{(k-1)/2}$, and let us prove the inequality $|a_n| \leq d(n) n^{(k-1)/2}$ for every $n \geq 1$ (here $d(n)$ is the sum of the positive divisors of $n$).

Since both sides are multiplicative in $n$, it suffices to consider the case $n=p^m$, where $p$ is prime. Put $u_m=a_{p^m}$. By looking at the Euler factor of $f$ at $p$, we know the following formal identity :

$\sum_{m \geq 0} u_m X^m = \frac{1}{1-a_p X +\psi(p) p^{k-1} X^2}$.

Now there are two main cases :

1) $p$ divides $N$. Then $\psi(p)=0$ and $u_m = a_p^m$ for every $m \geq 0$. It can be shown, using purely analytical arguments, that $|a_p| \leq p^{(k-1)/2}$ (see Theorem 3 in Li's article "Newforms and functional equations", Math. Ann., 1975). Thus we get $|u_m| \leq p^{m(k-1)/2}$ which gives the desired inequality (and in fact, a stronger one).

2) $p$ doesn't divide $N$. Put $1-a_p X +\psi(p) p^{k-1} X^2 = (1-\alpha X)(1-\beta X)$. In the case $k \geq 2$, the Weil conjectures proved by Deligne imply that $|\alpha|=|\beta|=p^{(k-1)/2}$ (this still holds if $k=1$, by a theorem of Deligne and Serre). There are two subcases :

2a) $\alpha \neq \beta$. Solving the linear recurrence relation satisfied by $u_m$ gives

$u_m = \frac{\alpha^{m+1}-\beta^{m+1}}{\alpha-\beta} = \alpha^m+\alpha^{m-1} \beta + \ldots + \beta^m$.

In particular $|u_m| \leq (m+1) p^{m(k-1)/2}$ which is what we want.

2b) $\alpha = \beta = \frac{a_p}{2}$. In this case $u_m = (m+1) \alpha^m$ for every $m \geq 0$, and the inequality also follows.

Remarks : It is conjectured that case 2b never happens if $k \geq 2$ (semi-simplicity of crystalline Frobenius). Note also that in case 2a, we can write $u_m = \beta^m (1+\lambda+ \ldots + \lambda^m)$, where $\lambda := \alpha/\beta$ satisfies $|\lambda|=1$ and $\lambda \neq 1$. Thus in fact we get a bound of the form $|u_m| \leq C \cdot p^{m(k-1)/2}$, where the constant $C$ depends only on the argument of $\alpha/\beta$. Finally, note that we also have the strict bound $|a_p|<2p^{(k-1)/2}$ in case 2a.

EDIT : case 2b can indeed happen in the case $k=1$ (thanks to David for pointing this out).