For an elliptic curve $E$ over a finite field $\mathbb F_q$, let $a_q(E) = q+1 – |E(\mathbb F_q)|$. Hasse–Weil bound tells us that $|a_q(E)| \leq 2 \sqrt{q}$.
I am interesting in knowing how large/small can $2 \sqrt{q} – a_q$ be. For instance, is there a sequence of primes $(p_k)$ and elliptic curves $E_k$ over $\mathbb F_{p_k}$ such that
$$b(E_k) := (2 \sqrt{p_k} – a_{p_k}(E_k)) / \log(p_k) \to 0$$ ?
I know that 1) Sato–Tate distribution describes the distribution of $a_p(E) / (2 \sqrt{p})$, but here we are looking at the difference between $a_p$ and $2 \sqrt{p}$, not the quotient, and 2) supersingular curves show that $b(E_k)$ can go to infinity.
Best Answer
Given a fixed prime power $q = p^n$ (with $p$ prime), Waterhouse (1969) gives a complete description of which numbers can occur as $\#E(\mathbb{F}_q)$ for some elliptic curve $E$ over the finite field $\mathbb{F}_q$. In particular, the extremal cases $a_q = \pm 2 \sqrt{q}$ do occur (for some $E$) whenever $q$ is a perfect square (i.e., whenever $n$ is even).
Rück (1987) goes further and describes all possible group structures for $E(\mathbb{F}_q)$. These papers were brought to my attention by this answer of Joe Silverman to a related question on MathOverflow.
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