Hasse's theorem on 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 In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

, bounding the value both above and below. If ''N'' is the number of points on the elliptic curve ''E'' over a finite field with ''q'' elements, then Hasse's result states that :

, N - (q+1),  \le 2 \sqrt.

The reason is that ''N'' differs from ''q'' + 1, the number of points of the

projective line In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...

over the same field, by an 'error term' that is the sum of two

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s, each of absolute value

\sqrt.

This result had originally been conjectured by

Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...

in his thesis. It was proven by Hasse in 1933, with the proof published in a series of papers in 1936. Hasse's theorem is equivalent to the determination of the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

of the roots of the local zeta-function of ''E''. In this form it can be seen to be the analogue of the

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...

for the function field associated with the elliptic curve.

Hasse–Weil Bound

A generalization of the Hasse bound to higher

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

algebraic curves In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

is the Hasse–Weil bound. This provides a bound on the number of points on a curve over a finite field. If the number of points on the curve ''C'' of genus ''g'' over the finite field

\mathbb_q

of order ''q'' is

\#C(\mathbb_q)

, then :

, \#C(\mathbb_q) - (q+1),  \le 2g \sqrt.

This result is again equivalent to the determination of the

of the roots of the local zeta-function of ''C'', and is the analogue of the

for the function field associated with the curve. The Hasse–Weil bound reduces to the usual Hasse bound when applied to elliptic curves, which have genus ''g=1''. The Hasse–Weil bound is a consequence of the Weil conjectures, originally proposed by André Weil in 1949 and proved by André Weil in the case of curves.

Notes

References

* * *Chapter V of * {{Algebraic curves navbox Elliptic curves Finite fields Theorems in algebraic number theory

Hasse–Weil Bound

See also

Notes

References