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59
Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
Faltings, Degeneration of abelian varieties
, 1990
"... An abelian variety A defined over a finite field Fq admits sufficiently many complex multiplications, as Tate showed in [27]. For some details about complex multiplication, see §1.1. Is A the reduction of an abelian variety with sufficiently many complex multiplications in characteristic zero? We fo ..."
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Cited by 109 (6 self)
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An abelian variety A defined over a finite field Fq admits sufficiently many complex multiplications, as Tate showed in [27]. For some details about complex multiplication, see §1.1. Is A the reduction of an abelian variety with sufficiently many complex multiplications in characteristic zero? We formulate several versions of this “CMlifting problem ” in §1.2. Honda
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 55 (2 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.
Applications of Arithmetical Geometry to Cryptographic Constructions
 Proceedings of the Fifth International Conference on Finite Fields and Applications
"... Public key cryptosystems are very important tools for data transmission. Their performance and security depend on the underlying crypto primitives. In this paper we describe one such primitive: The Discrete Logarithm (DL) in cyclic groups of prime order (Section 1). To construct DLsystems we use me ..."
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Cited by 40 (1 self)
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Public key cryptosystems are very important tools for data transmission. Their performance and security depend on the underlying crypto primitives. In this paper we describe one such primitive: The Discrete Logarithm (DL) in cyclic groups of prime order (Section 1). To construct DLsystems we use methods from algebraic and arithmetic geometry and especially the theory of abelian varieties over finite fields. It is explained why Jacobian varieties of hyperelliptic curves of genus 4 are candidates for cryptographically "good" abelian varieties (Section 2). In the third section we describe the (constructive and destructive) role played by Galois theory: Local and global Galois representation theory is used to count points on abelian varieties over finite fields and we give some applications of Weil descent and Tate duality.
Heegner points, padic Lfunctions, and the CerednikDrinfeld uniformisation
 Invent. Math
, 1998
"... ..."
Hypersymmetric Abelian Varieties
, 2006
"... We introduce the notion of a hypersymmetric abelian variety over a field of positive characteristic p. We show that every symmetric Newton polygon admits a hypersymmetric abelian variety having that Newton polygon; see 2.5 and 4.8. Isogeny classes of absolutely simple hypersymmetric abelian variet ..."
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Cited by 20 (12 self)
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We introduce the notion of a hypersymmetric abelian variety over a field of positive characteristic p. We show that every symmetric Newton polygon admits a hypersymmetric abelian variety having that Newton polygon; see 2.5 and 4.8. Isogeny classes of absolutely simple hypersymmetric abelian varieties are classified in terms of their endomorphism algebras and Newton polygons. We also discuss connections with abelian varieties of PELtype, i.e. abelian varieties with extra symmetries, especially abelian varieties with real multiplications.
Abelian Varieties over Q and modular forms
 Progress in Math. 224, Birkhäusser
, 2004
"... conjecture asserts that there is a nonconstant map of algebraic curves Xo(N) → C which is defined over Q. Here, Xo(N) is the standard modular curve associated with the problem of classifying elliptic curves E together with cyclic subgroups of E ..."
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Cited by 16 (0 self)
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conjecture asserts that there is a nonconstant map of algebraic curves Xo(N) → C which is defined over Q. Here, Xo(N) is the standard modular curve associated with the problem of classifying elliptic curves E together with cyclic subgroups of E
Galois groups and complex multiplication
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 1978
"... ..."
Rational conformal field theories and complex multiplication,” arXiv:hepth/0203213
"... We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on CalabiYau manifolds. We perform a detailed study of RCFT’s corresponding to T 2 target and identify the Cardy branes with geometric branes. The T 2 ’s leading to RCFT’s admit ..."
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Cited by 15 (2 self)
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We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on CalabiYau manifolds. We perform a detailed study of RCFT’s corresponding to T 2 target and identify the Cardy branes with geometric branes. The T 2 ’s leading to RCFT’s admit “complex multiplication ” which characterizes Cardy branes as specific D0branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary CalabiYau nfolds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for CalabiYau nfolds for n> 2. RCFT’s on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of CalabiYau nfolds in connection with freezing geometric moduli. March
Oort – Moduli of abelian varieties and pdivisible groups: density of Hecke orbits and a conjecture by Grothendieck
 In: Arithmetic Geometry. Clay Mathematics Summer School, Arithmetic Geometry, July 17–August 11, 2006, Göttingen July (Editors
"... In the week 7 – 11 August 2006 we gave a course, and here are notes for that course. Our main topic is: geometry and arithmetic of Ag ⊗ Fp, the moduli space of polarized abelian varieties of dimension g in positive characteristic. We illustrate properties, and some of the available techniques by tre ..."
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Cited by 13 (10 self)
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In the week 7 – 11 August 2006 we gave a course, and here are notes for that course. Our main topic is: geometry and arithmetic of Ag ⊗ Fp, the moduli space of polarized abelian varieties of dimension g in positive characteristic. We illustrate properties, and some of the available techniques by treating two topics: and