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139
Arithmetic and Attractors
, 2003
"... We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the “attractor mechanism ” of N = 2 supergravity. In IIB string compactification this mechanism singles out certain “attractor varieties. ” We show that these attractor varieties are ..."
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Cited by 73 (3 self)
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We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the “attractor mechanism ” of N = 2 supergravity. In IIB string compactification this mechanism singles out certain “attractor varieties. ” We show that these attractor varieties are constructed from products of elliptic curves with complex multiplication for N = 4, 8 compactifications. The heterotic dual theories are related to rational conformal field theories. In the case of N = 4 theories Uduality inequivalent backgrounds with the same horizon area are counted by the class number of a quadratic imaginary field. The attractor varieties are defined over fields closely related to class fields of the quadratic imaginary field. We discuss some extensions to more general CalabiYau compactifications and explore further connections to arithmetic including connections to Kronecker’s Jugendtraum and the theory of modular heights. The paper also includes a short review of the attractor mechanism. A much shorter version of the paper summarizing the main points is the companion note entitled “Attractors and Arithmetic,” hepth/9807056.
Supersingular abelian varieties in cryptology
 Advances in Cryptology  CRYPTO 2002
"... Abstract. For certain security applications, including identity based encryption and short signature schemes, it is useful to have abelian varieties with security parameters that are neither too small nor too large. Supersingular abelian varieties are natural candidates for these applications. This ..."
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Cited by 51 (7 self)
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Abstract. For certain security applications, including identity based encryption and short signature schemes, it is useful to have abelian varieties with security parameters that are neither too small nor too large. Supersingular abelian varieties are natural candidates for these applications. This paper determines exactly which values can occur as the security parameters of supersingular abelian varieties (in terms of the dimension of the abelian variety and the size of the finite field), and gives constructions of supersingular abelian varieties that are optimal for use in cryptography. 1
Constructing hyperelliptic curves of genus 2 suitable for cryptography
 Math. Comp
, 2003
"... Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1. ..."
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Cited by 39 (2 self)
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Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.
Homomorphisms of Abelian varieties
 J. REINE ANGEW. MATH
, 1998
"... It is wellknown that an abelian variety is (absolutely) simple or is isogenous to a selfproduct of an (absolutely) simple abelian variety if and only if the center of its endomorphism algebra is a field. In this paper we prove that the center is a field if the field of definition of points of prim ..."
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Cited by 30 (7 self)
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It is wellknown that an abelian variety is (absolutely) simple or is isogenous to a selfproduct of an (absolutely) simple abelian variety if and only if the center of its endomorphism algebra is a field. In this paper we prove that the center is a field if the field of definition of points of prime order ℓ is “big enough”. The paper is organized as follows. In §1 we discuss Galois properties of points of order ℓ on an abelian variety X that imply that its endomorphism algebra End 0 (X) is a central simple algebra over the field of rational numbers. In §2 we prove that similar Galois properties for two abelian varieties X and Y combined with the linear disjointness of the corresponding fields of definitions of points of order ℓ imply that X and Y are nonisogenous (and even Hom(X, Y) = 0). In §3 we give applications to endomorphism algebras of hyperelliptic jacobians. In §4 we prove that if X admits multiplications by a number field E and the dimension of the centralizer of E in End 0 (X) is “as large as possible ” then X is an abelian variety of CMtype isogenous to a selfproduct of an absolutely simple abelian variety. Throughout the paper we will freely use the following observation [21, p. 174]: if an abelian variety X is isogenous to a selfproduct Z d of an abelian variety Z then a choice of an isogeny between X and Z d defines an isomorphism between End 0 (X) and the algebra Md(End 0 (Z)) of d × d matrices over End 0 (Z). Since the center of End 0 (Z) coincides with the center of Md(End 0 (Z)), we get an isomorphism
The 2adic CM method for genus 2 curves with application to cryptography
 in ASIACRYPT ‘06, Springer LNCS 4284
, 2006
"... Abstract. The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method ..."
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Cited by 22 (2 self)
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Abstract. The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method as far as possible. We have thus designed a new algorithm for the construction of CM invariants of genus 2 curves, using 2adic lifting of an input curve over a small finite field. This provides a numerically stable alternative to the complex analytic method in the first phase of the CM method for genus 2. As an example we compute an irreducible factor of the Igusa class polynomial system for the quartic CM field Q(i p 75 + 12 √ 17), whose class number is 50. We also introduce a new representation to describe the CM curves: a set of polynomials in (j1, j2, j3) which vanish on the precise set of triples which are the Igusa invariants of curves whose Jacobians have CM by a prescribed field. The new representation provides a speedup in the second phase, which uses Mestre’s algorithm to construct a genus 2 Jacobian of prime order over a large prime field for use in cryptography. 1
LINEAR INDEPENDENCE OF GAMMA VALUES IN POSITIVE CHARACTERISTIC
, 2001
"... We investigate the arithmetic nature of special values of Thakur’s function field Gamma function at rational points. Our main result is that all linear dependence relations over the field of algebraic functions are consequences of the AndersonDeligneThakur bracket relations. ..."
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Cited by 17 (7 self)
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We investigate the arithmetic nature of special values of Thakur’s function field Gamma function at rational points. Our main result is that all linear dependence relations over the field of algebraic functions are consequences of the AndersonDeligneThakur bracket relations.
Anticyclotomic Main Conjectures
 DOCUMENTA MATH.
, 2006
"... In this paper, we prove many cases of the anticyclotomic main conjecture for general CM fields with pordinary CM type. ..."
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Cited by 17 (10 self)
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In this paper, we prove many cases of the anticyclotomic main conjecture for general CM fields with pordinary CM type.
Nonvanishing modulo p of Hecke L–values and application
 LFUNCTIONS AND GALOIS REPRESENTATIONS EDITED BY DAVID BURNS , KEVIN BUZZARD , JAN NEKOVÁR
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