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- {{Short description|Mathematical curves that are isomorphic over algebraic closures}} ...curves is an [[isogeny]] of degree 1, that is an invertible isogeny. Some curves have higher order twists such as '''cubic''' and '''quartic twists'''. The ...8 KB (1,250 words) - 04:50, 30 November 2024
- ...er fields of large characteristic, in particular ''p'' > 3. For curves over fields of small characteristic more efficient algorithms based on ''p' ==Approaches to counting points on elliptic curves== ...14 KB (2,262 words) - 21:37, 30 December 2023
- ...ce spans over a century because of the various generalizations of elliptic curves as the field has developed. All of these generalizations are contained in < ...over <math>\text{Spec}(\mathbb{Z})</math>, but is not a scheme as elliptic curves have non-trivial automorphisms. ...14 KB (2,264 words) - 21:44, 22 September 2024
- ...ts for this group addition and certain related operations that are used in elliptic curve cryptography algorithms. ...nal operations. The rows of the table are for different models of elliptic curves. These are the operations considered: ...6 KB (855 words) - 10:50, 29 September 2024
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- ...orphisms, necessitating the construction of the [[Moduli stack of elliptic curves]]. | title = Arithmetic moduli of elliptic curves ...2 KB (343 words) - 12:52, 8 November 2024
- ...rch and Swinnerton-Dyer conjecture]] for a positive proportion of elliptic curves. ...2 KB (303 words) - 00:31, 15 May 2024
- ...te elliptic integral of the second kind]] and sn, cn and dn are [[Jacobi's elliptic functions]].<ref name="oprea" /> [[Category:Plane curves]] ...1,019 bytes (163 words) - 21:26, 29 December 2024
- {{Short description|1-form on an elliptic curve}} ...ndré Néron]], is an almost canonical choice of [[One-form|1-form]] on an [[elliptic curve]] or [[abelian variety]] defined over a [[local field]] or [[global f ...1 KB (188 words) - 06:15, 23 December 2024
- {{Short description|Type of elliptic curve}} ...stein|Bernstein]], Lange, and Kohel.<ref>{{cite web |title=Twisted Hessian curves |url=http://hyperelliptic.org/EFD/g1p/auto-twistedhessian.html |accessdate= ...7 KB (1,032 words) - 20:45, 23 December 2024
- ...ts. They play a central role in the study of [[counting points on elliptic curves]] in [[Schoof's algorithm]]. * The division polynomials form a generic [[elliptic divisibility sequence]] over the ring <math>\mathbb{Q}[x,y,A,B]/(y^2-x^3-Ax ...5 KB (844 words) - 14:19, 28 December 2023
- ...ts for this group addition and certain related operations that are used in elliptic curve cryptography algorithms. ...nal operations. The rows of the table are for different models of elliptic curves. These are the operations considered: ...6 KB (855 words) - 10:50, 29 September 2024
- ...)|pencil]] (one-dimensional family) of [[Cubic plane curve|cubic plane]] [[elliptic curve]]s in the [[complex projective plane]], defined by the equation ...ivalent]] to a curve of the Hesse pencil; this is the [[Hessian form of an elliptic curve]]. However, the parameters (<math>\lambda,\mu</math>) of the Hessian ...3 KB (435 words) - 16:27, 6 January 2020
- ...|first=Richard|date=2014-03-25|title=Lectures on Moduli Spaces of Elliptic Curves|class=math.AG|eprint=0812.1803}}</ref> there is a moduli space of principal ...s the upper half plane, which appears prominently in the study of elliptic curves. In general, any point <math>\Omega \in H_g</math> gives a complex torus < ...5 KB (824 words) - 17:32, 19 February 2025
- ...ntroduces the notion of a level structure. The classic one is that on an [[elliptic curve]] (see [[#Example: an abelian scheme]]). There is a level structure a == Level structures on elliptic curves == ...5 KB (801 words) - 17:06, 13 December 2020
- {{Short description|Mathematical curves that are isomorphic over algebraic closures}} ...curves is an [[isogeny]] of degree 1, that is an invertible isogeny. Some curves have higher order twists such as '''cubic''' and '''quartic twists'''. The ...8 KB (1,250 words) - 04:50, 30 November 2024
- ...The genus–degree formula is a generalization of this fact to higher genus curves. The basic idea would be to use higher degree equations. Consider the quart ...ion formula]]; for details, see {{slink|Adjunction formula|Applications to curves}}.<ref>''Algebraic geometry'', [[Robin Hartshorne]], Springer GTM 52, {{ISB ...4 KB (612 words) - 11:27, 10 December 2024
- {{Short description|Type of elliptic curve}} ...ierstrass form]] and it is also important in [[elliptic curve cryptography|elliptic-curve cryptography]] because the doubling speeds up considerably (computing ...7 KB (1,133 words) - 04:43, 30 November 2024
- {{Short description|Curves in algebraic geometric}} ...ic curves are important in [[public key cryptography]] and twisted Edwards curves are at the heart of an electronic signature scheme called [[EdDSA]] that of ...10 KB (1,557 words) - 07:15, 7 February 2025
- ...coordinates of the varying point on this curve are given by the [[Jacobian elliptic function]]s. ...tic Function]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Jacobi Elliptic Functions |url=https://mathworld.wolfram.com/ |access-date=2023-01-31 |webs ...3 KB (473 words) - 17:42, 26 July 2024
- ...of finite order in the [[Elliptic curve#The group law|group law]] of the [[elliptic curve]]. The primes involved in the conductor are precisely the primes of The conductor of an elliptic curve over a local field was implicitly studied (but not named) by {{harvtx ...7 KB (1,093 words) - 10:16, 16 July 2024
- ...rve]] <math>C/K</math> for a [[Generic point|generic fiber]] and unions of curves (possibly [[Irreducible component|reducible]], [[Singular point of an algeb ...ame=SAT311>Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, p. 311.</ref> ...8 KB (1,239 words) - 01:24, 14 May 2024
- {{Short description|Prime number with a certain relationship to an elliptic curve}} ...od reduction#G|reduction]] of ''E'' modulo ''p'' is a [[supersingular elliptic curve]] over the [[residue field]] '''F'''<sub>''p''</sub>. ...3 KB (432 words) - 11:11, 15 September 2024
- ...Over the open subscheme where ''q'' is invertible, the Tate curve is an [[elliptic curve]]. The Tate curve can also be defined for ''q'' as an element of a [[ ...995}} in a 1959 manuscript originally titled "Rational Points on Elliptic Curves Over Complete Fields"; he did not publish his results until many years lat ...6 KB (1,007 words) - 19:15, 13 December 2024
- ...]], [[number theory]], and the connections between [[modular form]]s and [[elliptic curve]]s.<ref name="Shimura">{{citation | last1=Shimura | first1=Goro | ti ...riod of mathematics, there was renewed interest in the theory of [[modular curves]] due to the work of Taniyama and [[Goro Shimura]].<ref name="Lang"/> Durin ...8 KB (1,126 words) - 08:04, 31 December 2024