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30
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.
Tamagawa Numbers for Motives with (NonCommutative) Coefficients
 DOCUMENTA MATH.
, 2001
"... Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Qalgebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the Aequivariant Lfunction of M. This conjecture simultaneously generalizes a ..."
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Cited by 37 (11 self)
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Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Qalgebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the Aequivariant Lfunction of M. This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, PerrinRiou et al. and also the central conjectures of classical Galois module theory as developed by Fröhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order A in A for which there exists a ‘projective Astructure ’ on M. The existence of such a structure is guaranteed if A is a maximal order, and also occurs in many natural examples where A is nonmaximal. In each such case the conjecture with respect to a nonmaximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in A by making use of the category of virtual objects introduced by Deligne.
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
LandauSiegel zeroes and black hole entropy,” arXiv:hepth/9903267
"... There has been some speculation about relations of Dbrane models of black holes to arithmetic. In this note we point out that some of these speculations have implications for a circle of questions related to the generalized Riemann hypothesis on the zeroes of Dirichlet Lfunctions. ..."
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Cited by 12 (5 self)
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There has been some speculation about relations of Dbrane models of black holes to arithmetic. In this note we point out that some of these speculations have implications for a circle of questions related to the generalized Riemann hypothesis on the zeroes of Dirichlet Lfunctions.
ALGEBRAIC THETA FUNCTIONS AND THE pADIC INTERPOLATION OF EISENSTEINKRONECKER NUMBERS
, 2007
"... ABSTRACT. We study the properties of EisensteinKronecker numbers, which are related to special values of Hecke Lfunction of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the ..."
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Cited by 6 (3 self)
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ABSTRACT. We study the properties of EisensteinKronecker numbers, which are related to special values of Hecke Lfunction of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the Poincaré bundle of an elliptic curve. We introduce general methods to study the algebraic and padic properties of reduced theta functions for CM abelian varieties. As a corollary, when the prime p is ordinary, we give a new construction of the twovariable padic measure interpolating special values of Hecke Lfunctions of imaginary quadratic fields, originally constructed by ManinVishik and Katz. Our method via theta functions also gives insight for the case when p is supersingular. The method of this paper will be used in subsequent papers in constructing certain twovariable padic distribution for supersingular p interpolating EisensteinKronecker numbers in twovaribales, as well as explicit calculation in twovariables of the padic elliptic polylogarithms for CM elliptic curves.
Nonvanishing of the Central Derivative of Canonical Hecke Lfunctions
, 2000
"... Let K = � ( √ −D) be an imaginary quadratic field of discriminant −D < −4, O its ring of integers, and h its ideal class number. A Hecke character χ of K of conductor � is a called “canonical ” ([Ro1]) if χ(¯�) = χ(�) for each ideal � relatively prime to �. (1.1) χ(αO) = ±α for principal ideals α ..."
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Cited by 5 (1 self)
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Let K = � ( √ −D) be an imaginary quadratic field of discriminant −D < −4, O its ring of integers, and h its ideal class number. A Hecke character χ of K of conductor � is a called “canonical ” ([Ro1]) if χ(¯�) = χ(�) for each ideal � relatively prime to �. (1.1) χ(αO) = ±α for principal ideals αO relatively prime to �. (1.2) The conductor � is divisible only by primes dividing D. (1.3) Every Hecke character of K satisfying (1.1) and (1.2) is actually a quadratic twist of a canonical Hecke character (see Section 2 for a precise description of these characters and which fields have them). Let L(s, χ) denote the Hecke Lfunction of χ, and Λ(s, χ) its completion; Λ(s, χ) satisfies the functional equation Λ(s, χ) = W (χ)Λ(2 − s, χ), where W (χ) = ±1 is the root number. If χ is a canonical Hecke character with W (χ) = 1, then the central value Λ(1, χ) � = 0 by a theorem of Montgomery and Rohrlich [MR]. Of course, it automatically vanishes when W (χ) = −1 by the functional equation. The main result of this paper is Theorem 1.1. Let χ be a canonical Hecke character whose root number W (χ) = −1. Then the central derivative Λ ′ (1, χ) � = 0. In Theorem 2.2 we also prove that Λ ′ (1, χ) � = 0 when χ is a small quadratic twist of a canonical character with W (χ) = −1. When D = p is a prime, canonical Hecke characters are closely connected with the elliptic curves A(p) extensively studied by Gross [Gr]. These curves are defined over F = �(j ( 1+√−p)), where j is the usual modular jfunction, and 2 have complex multiplication by O. Combining Theorem 1.1 and the above result of [MR] with GrossZagier [GZ] and KolyvaginLogachev [KL], one has Corollary 1.2. Let p> 3 be a prime congruent to 3 modulo 4. Then (a) The MordellWeil rank of A(p) is h, p ≡ 3 (mod 8) rank�A(p)(F) =
POINT COUNTING ON REDUCTIONS OF CM ELLIPTIC CURVES
"... Abstract. We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM Qcurves in certain cases. This generalizes earlier results of Gross, Stark, and others. 1. ..."
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Cited by 3 (2 self)
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Abstract. We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM Qcurves in certain cases. This generalizes earlier results of Gross, Stark, and others. 1.
”New” Veneziano amplitudes from ”old” Fermat (hyper)surfaces
, 2003
"... The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its ma ..."
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Cited by 3 (2 self)
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The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This work aims to bridge the gap between the mathematical and physical treatments. Using some results of recent publications (e.g. J.Geom.Phys.38 (2001) 81; ibid 43 (2002) 45) new topological, algebrogeometric, numbertheoretic and combinatorial treatment of the multiparticle Veneziano amplitudes is developed. As a result, an entirely new physical meaning of these amplitudes is emerging: they are periods of differential forms associated with homology cycles on Fermat (hyper)surfaces. Such (hyper)surfaces are considered as complex projective varieties of Hodge type. Although the computational formalism developed in this work resembles that used in mirror symmetry calculations, many additional results from mathematics are used along with their suitable physical interpretation. For instance, the Hodge spectrum of the Fermat (hyper)surfaces is in onetoone correspondence with the possible spectrum of particle masses. The formalism also allows us to obtain correlation functions of both conformal field theory and particle physics using the same type of the PicardFuchs equations whose solutions are being interpreted in terms of periods.