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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Supersingular elliptic curve ： ウィキペディア英語版 Supersingular elliptic curve In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic ''p'' > 0 with unusually large endomorphism rings. 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. discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that in positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and developed their basic theory. The term "supersingular" has nothing to do with singular points of curves, and all supersingular elliptic curves are non-singular. It comes from the phrase "singular values of the j-invariant" used for values of the j-invariant for which a complex elliptic curve has complex multiplication. The complex elliptic curves with complex multiplication are those for which the endomorphism ring has the maximal possible rank 2. In positive characteristic it is possible for the endomorphism ring to be even larger: it can be an order in a quaternion algebra of dimension 4, in which case the elliptic curve is supersingular. ==Definition== There are many different but equivalent ways of defining supersingular elliptic curves that have been used. Some of the ways of defining them are given below. Let ''K'' be a field with algebraic closure $\backslash overline$ and ''E'' an elliptic curve over ''K''. *The $\backslash overline$-valued points $E(\backslash overline)$ have the structure of an abelian group. For every n, we have a multiplication map $():\; E\backslash to\; E$. Its kernel is denoted by $E()$. Now assume that the characteristic of ''K'' is ''p'' > 0. Then one can show that either : :for ''r'' = 1, 2, 3, ... In the first case, ''E'' is called ''supersingular''. Otherwise it is called ''ordinary''. In other words, an elliptic curve is supersingular if and only if the group of geometric points of order ''p'' is trivial. *Supersingular elliptic curves have many endomorphisms over the algebraic closure $\backslash overline$ in the sense that an elliptic curve is supersingular if and only if its endomorphism algebra (over $\backslash overline$) is an order in a quaternion algebra. Thus, their endomorphism algebra (over $\backslash overline$) has rank 4, while the endomorphism group of every other elliptic curve has only rank 1 or 2. The endomorphism ring of a supersingular elliptic curve can have rank less than 4, and it may be necessary to take a finite extension of the base field ''K'' to make the rank of the endomorphism ring 4. In particular the endomorphism ring of an elliptic curve over a field of prime order is never of rank 4, even if the elliptic curve is supersingular. * Let ''G'' be the formal group associated to ''E''. Since ''K'' is of positive characteristic, we can define its height ht(''G''), which is 2 if and only if E is supersingular and else is 1. *We have a Frobenius morphism $F:\; E\backslash to\; E$, which induces a map in cohomology :$F^$ *: H^1(E, \mathcal_E) \to H^1(E,\mathcal_E). :The elliptic curve ''E'' is supersingular if and only if $F^$ * equals 0. *We have a Frobenius morphism $F:\; E\backslash to\; E$, which induces a map on the global 1-forms :$F^$ *: H^0(E, \Omega^1_E) \to H^0(E,\Omega^1_E). :The elliptic curve ''E'' is supersingular if and only if $F^$ * equals 0. *An elliptic curve is supersingular if and only if its Hasse invariant is 0. *An elliptic curve is supersingular if and only if the group scheme of points of order ''p'' is connected. *An elliptic curve is supersingular if and only if the dual of the Frobenius map is purely inseparable. *An elliptic curve is supersingular if and only if the "multiplication by ''p''" map is purely inseparable and the ''j''-invariant of the curve lies in a quadratic extension of the prime field of ''K'', a finite field of order ''p''2. *Suppose ''E'' is in Legendre form, defined by the equation $y^2\; =\; x(x-1)(x-\backslash lambda)$, and ''p'' is odd. Then ''E'' is supersingular if and only if the sum :$\backslash sum\_^n\; \{n\backslash choose\{i\}\}^2\backslash lambda^i$ :vanishes, where $n\; =\; (p-1)/2$. Using this formula, one can show that there are only finitely many supersingular elliptic curves over ''K'' (up to isomorphism). *Suppose ''E'' is given as a cubic curve in the projective plane given by a homogeneous cubic polynomial ''f''(''x'',''y'',''z''). Then ''E'' is supersingular if and only if the coefficient of (''xyz'')''p''–1 in ''f''''p''–1 is zero. *If the field ''K'' is a finite field of order ''q'', then an elliptic curve over ''K'' is supersingular if and only if the trace of the ''q''-power Frobenius endomorphism is congruent to zero modulo ''p''. :When ''q''=''p'' is a prime greater than 3 this is equivalent to having the trace of Frobenius equal to zero (by the Hasse bound); this does not hold for ''p''=2 or 3. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Supersingular elliptic curve」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.