Editing Hasse's theorem on elliptic curves

{{short description|Estimates the number of points on an elliptic curve over a finite field}}
'''[[Helmut Hasse|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]], 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

:<math>|N - (q+1)| \le 2 \sqrt{q}.</math>

The reason is that ''N'' differs from ''q'' + 1, the number of points of the [[projective line]] over the same field, by an 'error term' that is the sum of two [[complex number]]s, each of absolute value <math>\sqrt{q}.</math>

This result had originally been conjectured by [[Emil Artin]] in his thesis.<ref>{{Citation | last1=Artin | first1=Emil | author1-link=Emil Artin | title=Quadratische Körper im Gebiete der höheren Kongruenzen. II. Analytischer Teil | doi=10.1007/BF01181075 | year=1924 | mr=1544652 | jfm=51.0144.05 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | pages=207–246 | volume=19 | issue=1| s2cid=117936362 }}
</ref>  It was proven by Hasse in 1933, with the proof published in a series of papers in 1936.<ref>{{Citation | last1=Hasse | first1=Helmut | author1-link=Helmut Hasse | title=Zur Theorie der abstrakten elliptischen Funktionenkörper. I, II & III | doi=10.1515/crll.1936.175.193 | year=1936 | zbl=0014.14903 | journal=[[Crelle's Journal]] | issn=0075-4102 | volume=1936 | issue=175| s2cid=118733025 }}
</ref>

Hasse's theorem is equivalent to the determination of the [[absolute value]] 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]] for the [[function field of an algebraic variety|function field]] associated with the elliptic curve.

== Hasse–Weil Bound ==

A generalization of the Hasse bound to higher [[Geometric genus|genus]] [[algebraic curves]] 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 <math>\mathbb{F}_q</math> of order ''q'' is <math>\#C(\mathbb{F}_q)</math>, then

:<math>|\#C(\mathbb{F}_q) - (q+1)| \le 2g \sqrt{q}.</math>

This result is again equivalent to the determination of the [[absolute value]] of the roots of the [[local zeta-function]] of ''C'', and is the analogue of the [[Riemann hypothesis]] for the [[function field of an algebraic variety|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.<ref>{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Numbers of solutions of equations in finite fields | url=https://www.ams.org/bull/1949-55-05/S0002-9904-1949-09219-4/home.html | doi=10.1090/S0002-9904-1949-09219-4  | mr=0029393 | year=1949 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=55 | pages=497–508 | issue=5| doi-access=free }}</ref>

==See also==
*[[Sato–Tate conjecture]]
*[[Schoof's algorithm]]
*[[Kloosterman sum#Estimates|Weil's bound]]

== Notes ==
{{Reflist}}

== References ==
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*Chapter V of {{Citation
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*{{Citation
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}}

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[[Category:Elliptic curves]]
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[[Category:Theorems in algebraic number theory]]