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A taxonomy of pairingfriendly elliptic curves
, 2006
"... Elliptic curves with small embedding degree and large primeorder subgroup are key ingredients for implementing pairingbased cryptographic systems. Such “pairingfriendly” curves are rare and thus require specific constructions. In this paper we give a single coherent framework that encompasses all ..."
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Cited by 78 (10 self)
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Elliptic curves with small embedding degree and large primeorder subgroup are key ingredients for implementing pairingbased cryptographic systems. Such “pairingfriendly” curves are rare and thus require specific constructions. In this paper we give a single coherent framework that encompasses all of the constructions of pairingfriendly elliptic curves currently existing in the literature. We also include new constructions of pairingfriendly curves that improve on the previously known constructions for certain embedding degrees. Finally, for all embedding degrees up to 50, we provide recommendations as to which pairingfriendly curves to choose to best satisfy a variety of performance and security requirements.
Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevich’s conjecture
"... ABSTRACT. We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. As a corollary we show that a direct generalization of the ..."
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Cited by 8 (5 self)
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ABSTRACT. We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. As a corollary we show that a direct generalization of the geometric version of Shafarevich’s original conjecture holds for infinitesimally rigid families of canonically polarized varieties. CONTENTS
THE SELMER GROUP, THE SHAFAREVICHTATE GROUP, AND THE WEAK MORDELLWEIL THEOREM
"... Abstract. This is an introduction to classical descent theory, in the context of abelian varieties over number fields. 1. Further reading We begin by suggesting reference for readers who want to see more details than are presented in this article. Here are some references for group cohomology, rough ..."
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Cited by 1 (0 self)
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Abstract. This is an introduction to classical descent theory, in the context of abelian varieties over number fields. 1. Further reading We begin by suggesting reference for readers who want to see more details than are presented in this article. Here are some references for group cohomology, roughly in order of increasing depth: Appendix B to [Sil99], the articles by Atiyah & Wall and Gruenberg in [CF86], and the books [Ser79] and [Ser02]. Here are some references for the MordellWeil Theorem, and for the Selmer and ShafarevichTate groups, again roughly in order of increasing depth: Chapters 8 and 10 of [Sil99], the book [Ser97], the “Abelian varieties ” article by Milne in [CS86], and the book [Mil86]. Also, many of these topics are covered in lecture notes of courses given by Milne, available at www.jmilne.org at no cost. 2. Group cohomology: H 0 Let G be a profinite group. Let A be a (discrete, left) Gmodule. This means that A is an abelian group on which G acts, and that the map G × A → A is continuous when A is given the discrete topology. Define A G and H 0 (G, A) by A G = H 0 (G, A): = { a ∈ A: ga = a for all g ∈ G}. The subgroup A G is known as the subgroup of Ginvariants of A. The following example demonstrates why this concept is important to us. Let k be a number field. Let G = Gk: = Gal(k/k) be the absolute Galois group of k. Let A be an abelian variety 1 over k. Then Gk acts on the abelian group A(k). The abbreviation H 0 (k, A) is commonly used for H 0 (Gk, A(k)). By Galois theory, H 0 (k, A) = A(k), where A(k) is the group of krational points on A, also known as the MordellWeil group of A. MordellWeil Theorem. The group A(k) is a finitely generated abelian group. In other words A(k) � Z r ⊕ T, where r is a nonnegative integer, and T is a finite abelian group. The integer r is called the rank of A over k. The group T is called the torsion subgroup, because it consists of the set of elements of A(k) of finite order. There exists an algorithm for computing T in theory, and this algorithm is practical at least when A is an Date: February 20, 2002. The writing of this article was supported by NSF grant DMS9801104, and a Packard Fellowship. It is based on a series of two lectures given at the Arizona Winter School on March 13–14, 1999. 1 Readers unfamiliar with abelian varieties can replace “abelian variety
unknown title
, 2002
"... A proof of the non existence of Frey curves without using TSW theorem ..."
Heights for line bundles on arithmetic surfaces
, 1995
"... For line bundles on arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of an arithmetic surface, generically of genus g, for line bundles of degree g equivalence is shown to the height on the Jacobian defined by Θ. 1 ..."
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For line bundles on arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of an arithmetic surface, generically of genus g, for line bundles of degree g equivalence is shown to the height on the Jacobian defined by Θ. 1