# Number of $\Bbb F_q$-rational points on an elliptic curve

Suppose that $p>2$ be a prime and $q=p^e$, for some integer $e$. Suppose $E: y^2= x^3+ax+b$ be an elliptic curve over $F_q$, ($a,b \in \Bbb F_q$). Let $N_q$ be the number of $\Bbb F_q$-rational points on $E$.
Show that $|N_q - q|\leq‎ \dfrac{q+3}{2}‎$.
• Do you know that $|N_q - (q+1)| \leq 2\sqrt q$ ? – Watson Feb 9 '17 at 12:23
• It seems to be true if $q>13$ : $$|N_q - q| = |N_q - (q + 1) + 1| \leq 2\sqrt q + 1 \leq \dfrac{q+3}2$$ since $$0 \leq q^2-14q+1 \iff 16q \leq q^2+2q+1 \iff 4\sqrt q \leq q+1 \iff 4\sqrt q + 2 \leq q+3$$ and $0 \leq q^2-14q+1$ if $q>13$ is an integer (so $q \geq 14$). – Watson Feb 9 '17 at 12:32
• @Watson, If we consider $f(x) \in \Bbb F_q[x]$, then we know that every solutions of equation $1‎\pm‎ f^\frac{q-1}{2} =0$ in $\Bbb F_q$ are a multiple root of $R(x) = 2f(x)(1 \pm f^\frac{q-1}{2})+ f^\prime (x) (x^q-x)$. With this we can conclusion ? – Masoud Feb 9 '17 at 12:50