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| In [[algebraic geometry]], a branch of [[mathematics]], '''supersingular elliptic curves''' form a certain class of [[elliptic curve]]s over a [[field (algebra)|field]] of characteristic ''p'' > 0. Elliptic curves over such fields which are not supersingular are called ''ordinary'' and these two classes of elliptic curves behave fundamentally differently in many aspects.
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| ==Definition==
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| Let ''K'' be a field with algebraic closure <math>\overline{K}</math> and ''E'' an [[elliptic curve]] over ''K''. Then the <math>\overline{K}</math>-valued points <math>E(\overline{K})</math> have the structure of an abelian group. For every n, we have a multiplication map <math>[n]: E\to E</math>. Its kernel is denoted by <math>E[n]</math>. Now assume that the characteristic of ''K'' is ''p'' > 0. Then one can show that either
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| :<math> E[p^r](\overline{K}) \cong \begin{cases} 0 & \mbox{or}\\ \mathbb{Z}/p^r\mathbb{Z} \end{cases}</math>
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| for ''r'' = 1, 2, 3, ... In the first case, ''E'' is called ''supersingular''. Otherwise it is called ''ordinary''. Of course, the term 'supersingular' does not mean that ''E'' is [[Algebraic curves#Singularities|singular]], since all elliptic curves are smooth.
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| ==Equivalent conditions==
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| There are a number of equivalent conditions to supersingularity:
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| *Supersingular elliptic curves have many endomorphisms in the sense that an elliptic curve is supersingular if and only if its endomorphism algebra (over <math>\overline{K}</math>) is an [[order (ring theory)|order]] in a [[quaternion algebra]]. Thus, their endomorphism group has rank 4, while the endomorphism group of every other elliptic curve has only rank 1 or 2.
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| * Let ''G'' be the [[formal group]] associated to ''E''. Since ''K'' is of positive characteristic, we can define its [[formal group#The height of a formal group law|height]] ht(''G''), which is 2 if and only if E is supersingular and else is 1.
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| *We have a [[Frobenius morphism]] <math>F: E\to E</math>, which induces a map in cohomology
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| :<math>F^*: H^1(E, \mathcal{O}_E) \to H^1(E,\mathcal{O}_E)</math>.
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| The elliptic curve ''E'' is supersingular if and only if <math>F^*</math> equals 0.
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| *Suppose ''E'' is in [[Legendre form]], defined by the equation <math>y^2 = x(x-1)(x-\lambda)</math>. Then ''E'' is supersingular if and only if the sum
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| :<math>\sum_{i=0}^k {k\choose{i}}^2\lambda^i</math>
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| vanishes, where <math>k = \frac12(p-1)</math>. Using this formula, one can show that there are only finitely many supersingular elliptic curves for every ''K''.
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| ==Examples==
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| *If ''K'' is a field of characteristic 2, every elliptic curve defined by an equation of the form
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| :<math>y^2+a_3y = x^3+a_4x+a_6</math>
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| is supersingular (see Washington2003, p. 122).
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| *If ''K'' is a field of characteristic 3, every elliptic curve defined by an equation of the form
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| :<math>y^2 = x^3+a_4x+a_6</math> | |
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| is supersingular (see Washington2003, p. 122).
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| *For <math>\mathbb{F}_p</math> with p>3 we have that the elliptic curve defined by <math>y^2 = x^3+1</math> is supersingular if and only if <math>p\equiv 2 \text{(mod 3)}</math> and the elliptic curve defined by <math>y^2 = x^3+x</math> is supersingular if and only if <math>p\equiv 3 \text{(mod 4)}</math> (see Washington2003, 4.35).
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| *There are also more exotic examples: The elliptic curve given by <math>y^2 = x(x-1)(x+2)</math> is nonsingular over <math>\mathbb{F}_p</math> for <math>p\neq 2,3</math>. It is supersingular for p = 23 and ordinary for every other <math>p\leq 73</math> (see Hartshorne1977, 4.23.6).
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| *{{harvtxt|Elkies|1987}} showed that any elliptic curve defined over the rationals is supersingular for an infinite number of primes.
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| *{{harvtxt|Birch|Kuyk|1975}} give a table of all supersingular curves for primes up to 307. For the first few primes the supersingular elliptic curves are given as follows. The number of supersingular values of j other than 0 or 1728 is the integer part of (p−1)/12.
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| {|class="wikitable" cellpadding=0 style="margin: 1em auto; text-align: center;"
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| !prime
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| !supersingular j invariants
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| |-
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| |2
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| |0
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| |-
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| |3
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| |0=1728
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| |-
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| |5
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| |0
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| |-
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| |7
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| |6=1728
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| |-
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| |11
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| |0, 1=1728
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| |-
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| |13
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| |5
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| |-
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| |17
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| |0,8
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| |-
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| |19
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| |7, 1728
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| |-
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| |23
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| |0,19, 1728
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| |-
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| |29
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| |0,2, 25
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| |-
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| |31
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| |2, 4, 1728
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| |-
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| |37
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| |8, 3±√15
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| |}
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| ==References==
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| *{{Citation | editor1-last=Birch | editor1-first=B. J. | editor-link=Bryan John Birch | editor2-last=Kuyk | editor2-first=W. | title=Modular functions of one variable. IV | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-07392-5 | doi=10.1007/BFb0097591 | mr=0376533 | year=1975 | volume=476 | chapter=Table 6 | pages=142–144 | zbl=0315.14014 }}
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| *{{Citation | last=Elkies | first=Noam D. | authorlink=Noam Elkies | title=The existence of infinitely many supersingular primes for every elliptic curve over Q | doi=10.1007/BF01388985 | mr=903384 | zbl=0631.14024 | year=1987 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=89 | issue=3 | pages=561–567}}
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| * [[Robin Hartshorne]] (1977), ''Algebraic Geometry'', Springer. ISBN 1-4419-2807-3
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| * Joseph H. Silverman (2009), ''The Arithmetic of Elliptic Curves'', Springer. ISBN 0-387-09493-8
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| * Lawrence C. Washington (2003), ''Elliptic Curves'', Chapman&Hall. ISBN 1-58488-365-0
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| {{Algebraic curves navbox}}
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| [[Category:Elliptic curves]]
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Hello and welcome. My title is Figures Wunder. My day occupation is a meter reader. One of the extremely best things in the world for me is to do aerobics and now I'm attempting to earn money with it. For many years he's been residing in North Dakota and his family enjoys it.
My website ... home std test kit