18 . 783 Elliptic Curves Spring 2013
نویسنده
چکیده
Andrew V. Sutherland A key ingredient to improving the efficiency of elliptic curve primality proving (and many other algorithms) is the ability to directly construct an elliptic curve E/Fq with a specified number of rational points, rather than generating curves at random until a suitable curve is found. To do this we need to develop the theory of complex multiplication. Recall from Lecture 7 that for any elliptic curve E/k, the multiplication-by-n maps [n] form a subring of the endomorphism ring End(E). This subring is isomorphic to Z, and it is notationally convenient to simply identify it with Z.1 Thus the inclusion Z ⊆ End(E) always holds. For curves with complex multiplication, this inclusion is strict.
منابع مشابه
18 . 783 Elliptic Curves Spring 2013 Lecture # 18 04 / 18 / 2013
converge absolutely for any fixed τ ∈ H, by Lemma 16.11, and uniformly over τ in any compact subset of H. The proof of this last fact is straight-forward but slightly technical; see [1, Thm. 1.15] for the details. It follows that g2(τ) and g3(τ) are both holomorphic on H, and therefore ∆(τ) = g2(τ) 3 − 27g3(τ) is also holomorphic on H. Since ∆(τ) is nonzero for all τ ∈ H, by Lemma 16.21, the j-...
متن کامل18 . 783 Elliptic Curves Spring 2015
In Lecture 1 we defined an elliptic curve as a smooth projective curve of genus 1 with a distinguished rational point. An equivalent definition is that an elliptic curve is an abelian variety of dimension one. An abelian variety is a smooth projective variety that is also a group, where the group operation is defined by rational functions (ratios of polynomials). Remarkably, these constraints f...
متن کامل18.783 Elliptic Curves Spring 2013 Lecture #24 05/09/2013
Andrew V. Sutherland In this lecture we give a brief overview of modular forms, focusing on their relationship to elliptic curves. This connection is crucial to Wiles’ proof of Fermat’s Last Theorem [7]; the crux of his proof is that every semistable elliptic curve over Q is modular.1 In order to explain what this means, we need to delve briefly into the theory of modular forms. Our goal in doi...
متن کامل18.783 Elliptic Curves Spring 2013 Lecture #20 04/25/2013 20.1 The Hilbert class polynomial
Let O be an order of discriminant D in an imaginary quadratic field K. In Lecture 19 we saw that there is a one-to-one relationship between isomorphism classes of elliptic curves with complex multiplication by O (the set EllO(C)), and equivalence classes of proper Oideals (the group cl(O)). The first main theorem of complex multiplication states that the elements of EllO(C) are algebraic intege...
متن کاملComplete characterization of the Mordell-Weil group of some families of elliptic curves
The Mordell-Weil theorem states that the group of rational points on an elliptic curve over the rational numbers is a finitely generated abelian group. In our previous paper, H. Daghigh, and S. Didari, On the elliptic curves of the form $ y^2=x^3-3px$, Bull. Iranian Math. Soc. 40 (2014), no. 5, 1119--1133., using Selmer groups, we have shown that for a prime $p...
متن کامل