---
tag: 7Y26317GJ0
kind: statement
role: theorem
name: Hasse's theorem on elliptic curves
msc: [11G20, 14H52, 11G05]
uses: [427N6S6QQW, 0ZXZ82FTX2]
about: [427N6S6QQW]
sources:
  - url: https://en.wikipedia.org/wiki/Hasse%27s_theorem_on_elliptic_curves
same_as:
  wikidata: Q3502015
  wikipedia: Hasse's theorem on elliptic curves
---

## Statement

Let $\F_q$ be a [[finite field|0ZXZ82FTX2]] with $q$ elements, and let $E$ be an [[elliptic curve|427N6S6QQW]] over $\F_q$.
Let $N = \# E(\F_q)$ be the number of points of $E$ over $\F_q$, including the point at infinity.
Then
$$\abs{N - (q + 1)} \le 2\sqrt{q}.$$

## 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 $\abs{a_q} \le 2\sqrt{q}$.

## Explanations

### L1 human

An elliptic curve over a finite field with $q$ elements has about $q + 1$ points.
Hasse's theorem says the error is at most $2\sqrt{q}$, which is small compared to $q$.

### L2 human

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|TRGBJYPBXH]] and is used to estimate the running time of algorithms in elliptic curve cryptography.