Explore: Curves Over Finite And Local Fields

hypothesis for curves over finite fields states | ω i | = q 1 / 2   . {\displaystyle |\omega _{i}|=q^{1/2}\ .} For example, for the elliptic curve case there

elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding

enough to include all non-singular cubic curves; see § Elliptic curves over a general field below.) An elliptic curve is an abelian variety – that is, it has

field K. Mordell's theorem (generalized to arbitrary number fields by André Weil) says the group of rational points on an elliptic curve has a finite

global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields: Algebraic

the curve is called a parametrization, and the curve is a parametric curve. In this article, these curves are sometimes called topological curves to distinguish

It is of particular interest for local fields and global fields, such as algebraic number fields. For K a finite field, Friedrich Karl Schmidt (1931) proved

cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The theory of fields proves that angle trisection and squaring the circle

algebraic curves: those curves that cannot be written as the union of two smaller curves. Up to birational equivalence, the irreducible curves over a field F

which is a generalisation of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that