Naive point count underestimates the number of mod $p$ points of an elliptic curve for infinitely many primes Let $E$ be an elliptic curve over $\mathbb{Z}[1/N]$ where $N$ is some non-zero integer. Can one show that that the integer $n_p-p-1$ (where $n_p$ is the number of points of $E$ mod $p$) is positive for infinitely many primes $p$ (not dividing $N$) without invoking the modularity theorem or the Sato-Tate conjecture?
 A: Here's a weak variant that should really be a comment: For any twist $E^d: dy^2 = f(x)$ of the elliptic curve $y^2 = f(x)$, for either $E$ or $E^d$, there exist infinitely many primes such that your required condition is met.
Note that if $d$ is not a square mod $p$, that is, $E^d$ is a non trivial twist mod $p$, then $$E^d(\mathbb F_p)+E(\mathbb F_p) = 2(p+1).$$
This is because for every value of $x \in \mathbb F_p$, $f(x)$ is either a square in which case this contributes two points to $E(\mathbb F_p)$ or it is not a square, in which case it contributes two points to $E^d(\mathbb F_p)$. This shows that one of the two sets $E(\mathbb F_p)$ and $E^d(\mathbb F_p)$ has to be $\geq p+1$ and the other one has to be $\leq p+1$. Since there are infinitely many primes for which $d$ is not a square mod $p$, this proves the claim. 
A: I don't think that anyone has any idea how to prove such a theorem without also being able to prove that $E$ is (potentially) modular. For example you can ask the same question for curves of higher genus. For curves of genus $2$, this problem appears to have been open before 2018, and for curves of higher genus it is still open, see the introduction to http://www.math.uchicago.edu/~fcale/papers/CDM.pdf
