Hasse's theorem on elliptic curves

Hasse's theorem on elliptic curves, also called the Hasse bound, helps us understand how many points can exist on a special type of mathematical shape called an elliptic curve. Elliptic curves are important in many areas of mathematics and computer science, especially in cryptography, where they help keep information safe.

The theorem gives us a way to estimate the number of points on an elliptic curve over a finite field. A finite field is a set of numbers where arithmetic works a bit differently than with the numbers we use every day. Hasse's theorem tells us that the number of points on the curve is close to the number of elements in the field plus one. The difference between these numbers is small and can be measured using special numbers called complex numbers.

This idea was first suggested by Emil Artin in his thesis, and later proven by Hasse in 1933. His proof was published in several papers in 1936. Hasse's theorem is also connected to another big idea in mathematics called the Riemann hypothesis, but for a different kind of number system called a function field. This connection shows how different areas of mathematics can relate to each other in surprising ways.

Hasse–Weil Bound

The Hasse–Weil Bound is a way to estimate the number of points on special shapes called algebraic curves over finite fields. It builds on Hasse's theorem and works for curves of any genus, not just elliptic curves. The bound helps us understand how many points a curve can have by looking at the properties of its related zeta-function.

This idea is linked to the Riemann hypothesis but for function fields instead of regular numbers. When we use it for elliptic curves, which have a genus of 1, it simplifies to Hasse's original bound. The Hasse–Weil Bound comes from the Weil conjectures, which were suggested by André Weil and later proven for curves.

Main article: Weil conjectures
Further information: Riemann hypothesis