TITLE: Naive point count underestimates the number of mod $p$ points of an elliptic curve for infinitely many primes
QUESTION [13 upvotes]: 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?

REPLY [4 votes]: 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