Primes of the form $p=u^2+1$ and number of points on the elliptic curve $x^3+a x z^2=y^2 z$

Question

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 $(\#E_a(\mathbb{F}_p) -p - 1) \in \{2,-2,B,-B\}$?

Answer 1

Look at Section 18.4 in Ireland-Rosen "A classical introduction to modern number theory". Note $p\equiv 1 \pmod{4}$ and $p = \pi\cdot \bar{\pi}$ with $\pi = 1- iu \equiv 1 \pmod{2+2i}$. Let $\lambda: \mathbb{F}_p \to \langle i \rangle$ be the character of order $4$, which is equal to $(\tfrac{\cdot}{p})_4$. Theorem 5 there shows that $$a_p = \overline{\lambda(-a)} \, \pi + \lambda(-a)\,\bar{\pi}$$ where $a_p$ is the negative of your expression $\# E(\mathbb{F}_p)-p-1$. If $\lambda(-a)=1$ then $a_p= 2$. If $\lambda(-a)=i$ then $a_p=-2u$. If $\lambda(-a) = -1$ then $a_p=-2$. If $\lambda(-a)=-i$ then $a_p=-2u$.

So, yes $a_p\in\{-2u,-2,2,2u\}$.