Given an elliptic curve $E$, for any given $p$, there are going to have some number of solutions mod $p$, call this $N_p$ between $0$ and $p^2$. In general it will be around $p$ because of Hasse's Theorem.
I understand that, roughly, the Birch and Swinnerton-Dyer conjecture was based on the idea that the higher the rank of $E$, the larger $N_p/p$ will be. I just don't understand why.
Is it just because more of the arbitrary points of $E$ will be rational, and then end up being integers mod p? If this is the case, it seems like having a few copies of $\mathbb{Q}$ shouldn't change the "probability", is it that the effect is subtle and only seen in the limit?