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
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
, 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
:
The reason is that ''N'' differs from ''q'' + 1, the number of points of the
projective line
In mathematics, 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 resultant elimination of special cases; ...
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 form ...
s, each of absolute value .
This result had originally been conjectured by
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
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 is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the roots of the
local zeta-function In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as
:Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^m\right)
where is a non-singular -dimensional projective algebr ...
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 ...
for the
function field associated with the elliptic curve.
Hasse-Weil Bound
A generalization of the Hasse bound to higher
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
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 c ...
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
of order ''q'' is
, then
:
This result is again 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 is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the roots of the
local zeta-function In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as
:Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^m\right)
where is a non-singular -dimensional projective algebr ...
of ''C'', and is 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 ...
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
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
, originally proposed by
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
in 1949 and proved by André Weil in the case of curves.
See also
*
Sato–Tate conjecture
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves ''Ep'' obtained from an elliptic curve ''E'' over the rational numbers by reduction modulo almost all prime numbers ''p''. Mikio Sato and J ...
*
Schoof's algorithm Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields. The algorithm has applications in elliptic curve cryptography where it is important to know the number of points to judge the difficulty of solving t ...
*
Weil's bound
Notes
References
*
*
*Chapter V of
*
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