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284
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 66 (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.
On the practical solution of the Thue equation
 INSTITUTE OF MATHEMATICS, UNIVERSITY OF DEBRECEN
, 1989
"... This paper gives in detail a practical general method for the explicit determination of all solutions of any Thue equation. It uses a combination of Baker’s theory of linear forms in logarithms and recent computational diophantine approximation techniques. An elaborated example is presented. ..."
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Cited by 55 (16 self)
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This paper gives in detail a practical general method for the explicit determination of all solutions of any Thue equation. It uses a combination of Baker’s theory of linear forms in logarithms and recent computational diophantine approximation techniques. An elaborated example is presented.
Searching for Primitive Roots in Finite Fields
, 1992
"... Let GF(p n ) be the finite field with p n elements where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p n ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p n ). ..."
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Cited by 47 (3 self)
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Let GF(p n ) be the finite field with p n elements where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p n ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p n ). We present three results. First, we present a solution to this problem for the case where p is small, i.e., p = n O(1) . Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case where p is large and n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for GF(p) assuming the ERH. Appeared in Mathematics of Computation 58, pp. 369380, 1992. An earlier version of this paper appeared in the 22nd Annual ACM Symposium on Theory of Computing (1990), pp. 546554. 1980 Mathematics Subject Classification (1985 revision): 11T06. 1. Introduction Consider the problem of finding a primitive ...
Invariant Measures for Actions of Higher Rank Abelian Groups
 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS
"... The first part of the paper begins with an introduction into Anosov actions of Z k and R k and an overview of the method of studying invariant measures for such actions based on consideration of conditional measures along various invariant foliations. The main body of that part contains a detailed ..."
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Cited by 44 (21 self)
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The first part of the paper begins with an introduction into Anosov actions of Z k and R k and an overview of the method of studying invariant measures for such actions based on consideration of conditional measures along various invariant foliations. The main body of that part contains a detailed proof of a modified version of the main theorem from [KS3] for actions by toral automorphisms of with applications to rigidity of the measurable structure of such actions with respect to Lebesque measure. In the second part principal technical tools for studying nonuniformly hyperbolic actions of Z k and R k are introduced and developed. These include Lyapunov characteristic exponents, nonstationary normal forms and Lyapunov Hoelder structures. At the end new rigidity results for Z² actions on threedimensional manifolds are outlined.
Fake projective planes
 Inv.Math.168(2007
"... 1.1. A fake projective plane is a smooth compact complex surface P which is not the complex projective plane but has the same first and second Betti numbers as the complex projective plane (b1(P) = 0 and b2(P) = 1). It is wellknown that such a surface is projective algebraic and is the quotient o ..."
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Cited by 44 (3 self)
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1.1. A fake projective plane is a smooth compact complex surface P which is not the complex projective plane but has the same first and second Betti numbers as the complex projective plane (b1(P) = 0 and b2(P) = 1). It is wellknown that such a surface is projective algebraic and is the quotient of the complex two ball in C2 by a cocompact torsionfree discrete subgroup of PU(2, 1). These are surfaces with the smallest EulerPoincaré characteristic among all smooth surfaces of general type. The first fake projective plane was constructed by Mumford [Mu] using padic uniformization, and more recently, two more examples were found by related methods by IshidaKato in [IK]. We have just learnt from Keum that he has an example which may be different from the earlier three. A natural problem in complex algebraic geometry is to determine all fake projective planes. It is proved in [Kl] and [Y] that the fundamental group of a fake projective plane, considered as a lattice of PU(2, 1), is arithmetic. In this paper we make use of this arithmeticity result and the volume formula of [P], together with some number theoretic estimates, to make a complete list of all fake projective planes,
NonKähler compact complex manifolds associated to number fields
 Ann. Inst. Fourier
, 2005
"... Abstract. For algebraic number fields K with s> 0 real and 2t> 0 complex embeddings and ”admissible ” subgroups U of the multiplicative group of integer units of K we construct and investigate certain (s+ t)dimensional compact complex manifolds X(K,U). We show among other things that such ma ..."
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Cited by 36 (6 self)
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Abstract. For algebraic number fields K with s> 0 real and 2t> 0 complex embeddings and ”admissible ” subgroups U of the multiplicative group of integer units of K we construct and investigate certain (s+ t)dimensional compact complex manifolds X(K,U). We show among other things that such manifolds are nonKähler but admit locally conformally Kähler metrics when t = 1. In particular we disprove a conjecture of I. Vaisman. Etant donnés des corps de nombresK avec s> 0 plongements réels et 2t> 0 plongements complexes et des sous groupes ”admissibles ” U du groupe multiplicatif des entiers inversibles, nous construisons et étudions certaines variétés complexes compactes X(K,U). Entre autres, nous montrons que ces variétés ne sont pas kähleriennes, mais admettent des métriques localement conformément kähleriennes lorsque t = 1. En particulier, nous donnons un contreexemple a ̀ une conjecture de I. Vaisman.