Intuition behind supersingular reduction of elliptic curves

Question

I would like to know if there's an obvious (or almost obvious) reason why we should expect elliptic curves over $\mathbb{Q}$ to have supersingular reduction at infinitely many primes. Thank you for your help !

Answer 1

Here is the typical heuristic reasoning. Let $E / \Bbb Q$ be an elliptic curve.

The starting point is that for any prime $p$ of good reduction, Hasse bound tells us that $|a_p| \leq 2 \sqrt p$ where $a_p := p+1 - |E(\Bbb F_p)|$. Recall that if $p \geq 5$, $E$ is supersingular at $p$ iff $a_p = 0$ (this uses Hasse bound implicitly).

Now, if we assume that the values $a_p$ and uniformly distributed and are independent, then the probability that $a_p = 0$ is roughly $\frac{1}{2 \sqrt p}$. So the expected value for the number of supersingular primes $p < X$ should roughly be

$$\sum_{p < X} \frac{1}{2 \sqrt p} \overset{\text{Mertens}}{\sim} \dfrac{\sqrt{X}}{\log(X)} \to \infty$$

when $X \to \infty$, whence we expect infinitely many supersingular primes!

(Actually, there are two caveats. The first one, which is harmless, is that the $a_p$ are distributed along a Sato-Tate measure, not a uniform one. The second one is that the same heuristic would predict also infinitely many anomalous primes, but this blantly fails for some curves, e.g. $E : y^2 = \prod_{i=1}^3 (x-e_i)$ where $e_i$ are pairwise distinct integers: $a_p$ is always even).