prime index coefficients of $L$-functions of Elliptic Curves over $\mathbb{Q}$ and analytical rank

Question

I have searched without success if there was any known relationships or conjectures between the rank and the prime index coefficients of the $L$-functions of a modular elliptic curves. Anyone has a suggestion or some references ?

Thanks !

Answer 1

Let $E$ be a non-CM elliptic curve. By the Sato–Tate conjecture, which has been proven for elliptic curves over $\mathbb Q$, for any two real numbers $\alpha, \beta$ with $0\le \alpha<\beta\le\pi$, we have $$\lim _{N\to\infty}\frac{\#\{p\le N:2\alpha\sqrt p \le a_p\le 2\beta\sqrt p\}}{\#\{p\le N\}}=\frac2\pi\int^\beta_\alpha\sin^2\theta \ d\theta.$$

In particular, the $a_p$'s are distributed according to a measure that does not depend on the rank.

A refinement of the conjecture (and still a conjecture), due to Lang and Trotter, predicts that, for a given non-zero integer $a$, we have $$\lim_{N\to\infty}\frac{\#\{p\le N : a_p = a\}}{\#\{p\le N\}} = c(E, a)\frac{\sqrt X}{\log X},$$ where $c(E, a)$ is an explicit constant. Again, there is no dependence on the rank.

These conjectures cover the case that $E$ is fixed, and we vary $p$. Your question is slightly different, in that you fix a prime $p$ and vary over all elliptic curves, but I think the answer should be the same: the specific value of $a_p$ should not affect the rank.

As for your observation: rank $2$ elliptic curves over $\mathbb Q$ are just much rarer than rank $0$ and $1$ ones in general. In fact, it is conjectured that $100\%$ of elliptic curves have rank $0$ or $1$.