# Infinitely many primes $p \in \mathbb{Z}$ where reduced curve $E/\mathbb{F}_p$ has Hasse invariant $1$?

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and fix a Weierstrass equation for $E$ having coefficients in $\mathbb{Z}$. Are there infinitely many primes $p \in \mathbb{Z}$ such that the reduced curve $E/\mathbb{F}_p$ has Hasse invariant $1$?

## 1 Answer

Yes. In fact, Elkies and Serre [1] have independent proofs that for every $\epsilon>0$ there is a constant $C_\epsilon>0$ such that $${\#\{p\le X : E/\mathbb{F}_p \text{ is ordinary}\}} \ge C_\epsilon X^{3/4-\epsilon}.$$ But if you just want to know that there are infinitely many ordinary primes, there is an elementary proof (also suggested by Serre) sketched in [2, Exercise 5.11].

[1] Serre, Quelques applications du theoreme de densite de Chebotarev, IHES Publ. Math. 54 (1981), 323-401.

[2] Silverman, The Arithmetic of Elliptic Curves, GTM 106, Springer.