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Let $p \gg 1$ be a sufficiently large prime. I recently stumbled across a fascinating fact about the number of $\mathbb{F}_p$-points on elliptic curves over finite fields. Specifically, we have: Fact ...

Let $p$ be prime of the form $p=u^2+1$. For $a \in \mathbb{F}_p,a \ne 0$, define $E_a : x^3+a x z^2=y^2 z$ Let $B= \lfloor 2 \sqrt{p}\rfloor$ Must we have $(\#E_a(\mathbb{F}_p) -p - 1) \in \{2,-2,B,-B\...