Hasse's theorem on elliptic curves

Remarks

One proof uses the $q$-power Frobenius endomorphism $\phi$ of $E$. The points of $E(\F_q)$ form the kernel of $1 - \phi$, which is separable, so $N = \deg(1 - \phi)$. The degree is a positive definite quadratic form on the endomorphism ring, with $\deg \phi = q$, and the Cauchy–Schwarz inequality for this form gives the bound. Example: $E \colon y^2 = x^3 + 1$ over $\F_5$. The cubes modulo $5$ run through all residues, so $x \mapsto x^3 + 1$ takes each value once. The value $0$ gives one point, the squares $1$ and $4$ give two points each, and the non-squares give none. With the point at infinity, $N = 1 + 2 + 2 + 1 = 6 = q + 1$. The number $a_q = q + 1 - N$ is the trace of Frobenius, and the theorem says $\abs{a_q} \le 2\sqrt{q}$.

Going deeper

For a student

The theorem is the Riemann hypothesis for the zeta function of an elliptic curve over a finite field: the roots of $1 - a_q T + q T^2$ have absolute value $q^{-1/2}$. Weil generalized it to curves of any genus $g$, with bound $2g\sqrt{q}$, and Deligne to all smooth projective varieties. The bound defines the angles in the Sato–Tate conjecture and is used to estimate the running time of algorithms in elliptic curve cryptography.