Why is $p$ (or $p + 1$) not the upper bound of the number of solutions of an elliptic curve mod $p$ (finite field)?

Question

In the build-up to the enunciation of the Birch, Swinnerton Dyer conjecture (BSD), the following back-of-the-envelop idea comes up:

Because half of the $1$ to $n-1$ elements mod $p$ are quadratic residues (squares), for the expression $y^2= x^3 +\cdots$ mod a certain prime $p,$ on average $1/2$ of the input $x$ values will return a quadratic residue, and taking the square root will ultimately yield two $y$ values. Now, since the higher the rank, the more rational numbers are going to fall on the curve, regardless of whether the field is $\mathbb Q$ or $\mathbb F_p,$ it follows that in the limit of $\underset{p \leq X}\Pi \frac {Np}p \approx 1$, i.e. an expectation of $p$ points on the curve.

I bet there is some misunderstanding in the way I paraphrased this concept, explaining why I don't see why this $1$ is not the upper bound, regardless of the rank, instead of being the expectation: Given that only half of the integers mod $p$ are quadratic residues, how is it possible to get more than $p$ solutions under any circumstances?

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Yes, it was a silly question... Just visually, there are multiple points at the same height:

Answer 1

Exactly half of the integers $1,\dots,p-1$ are quadratic residues and half are quadratic nonresidues, for sure. However, the values of the cubic polynomial in $x$, as $x$ runs through $0,1,\dots,p-1$, are not just the residue classes $0,1,\dots,p-1$ in a different order; rather, the values of the cubic polynomial will hit some of those residue classes once, some more than once, and some not at all. So it really matters whether the more-than-once/not-at-all images are quadratic residues or nonresidues; a given cubic polynomial can prefer values that are residues or that are nonresidues.

One can see this with specific examples like $y^2 = x^3+x+1$ over $\Bbb F_5$: there are $8$ points on this elliptic curve, namely $(0,\pm1)$, $(2,\pm1)$, $(3,\pm1)$, and $(4,\pm2)$ (and an extra point at infinity if we're counting those), and $8>5$.

Finally, if we were to believe that there can never be more than $p$ (affine) points on an elliptic curve over $\Bbb F_p$, then we would be forced to believe that there can never be fewer than $p$ points as well, simply by multiplying the right-hand side of the equation $y^2=ax^3+bx^2+cx+d$ by any quadratic nonresidue $n$ modulo $p$: if $x$ yielded $2$ or $0$ solutions before the multiplication, it will yield $0$ or $2$ solutions after.