Results 1  10
of
13
Class invariants for quartic CM fields
, 2004
"... Abstract. One can define class invariants for a quartic primitive CM field K as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to K. Such constructions were given in [DSG] and [Lau]. We provide explicit bounds on the primes appearing in the denominators of ..."
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Abstract. One can define class invariants for a quartic primitive CM field K as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to K. Such constructions were given in [DSG] and [Lau]. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct Sunits in certain abelian extensions of a reflex field of K, where S is effectively determined by K, and to bound the primes appearing in the denominators of the Igusa class polynomials arising in the construction of genus 2 curves with CM, as conjectured in [Lau]. 1.
Rational curves and points on K3 surfaces
 Amer. J. Math
"... Abstract. — We study the distribution of algebraic points on K3 surfaces. ..."
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Cited by 5 (4 self)
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Abstract. — We study the distribution of algebraic points on K3 surfaces.
Abelian varieties isogenous to a Jacobian
, 2004
"... Introduction (0.1) Question Given an abelian variety A; does there exist an algebraic curve C such that there is an isogeny between A and the Jacobian of C? * If the dimension of A is at most three, such a curve exists; see (1.3). ..."
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Introduction (0.1) Question Given an abelian variety A; does there exist an algebraic curve C such that there is an isogeny between A and the Jacobian of C? * If the dimension of A is at most three, such a curve exists; see (1.3).
NONHYPERELLIPTIC MODULAR JACOBIANS OF DIMENSION 3
, 2008
"... Abstract. We present a method to solve in an efficient way the problem of constructing the curves given by Torelli’s theorem in dimension 3 over the complex numbers: For an absolutely simple principally polarized abelian threefold A over C given by its period matrix Ω, compute a model of the curve o ..."
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Abstract. We present a method to solve in an efficient way the problem of constructing the curves given by Torelli’s theorem in dimension 3 over the complex numbers: For an absolutely simple principally polarized abelian threefold A over C given by its period matrix Ω, compute a model of the curve of genus three (unique up to isomorphism) whose Jacobian, equipped with its canonical polarization, is isomorphic to A as a principally polarized abelian variety. We use this method to describe the nonhyperelliptic modular Jacobians of dimension 3. We investigate all the nonhyperelliptic new modular Jacobians Jac(Cf) of dimension 3 which are isomorphic to Af,wheref∈Snew 2 (X0(N)), N ≤ 4000.
Computing Igusa class polynomials
, 2008
"... We give an algorithm that computes the genus two class polynomials of a primitive quartic CM field K, and we give a runtime bound and a proof of correctness of this algorithm. This is the first proof of correctness and the first runtime bound of any algorithm that computes these polynomials. Our alg ..."
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Cited by 2 (1 self)
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We give an algorithm that computes the genus two class polynomials of a primitive quartic CM field K, and we give a runtime bound and a proof of correctness of this algorithm. This is the first proof of correctness and the first runtime bound of any algorithm that computes these polynomials. Our algorithm uses complex analysis and runs in time e O( ∆ 7/2), where ∆ is the discriminant of K. 1
ON A CONJECTURE OF SERRE ON ABELIAN THREEFOLDS
, 710
"... En genre 3, le théorème de Torelli s’applique de façon moins satisfaisante: on doit extraire une mystérieuse racine carrée (J.P.S., Collected Papers, n o 129) Abstract. In this article, we give a reformulation of a result from Howe, Leprevost and Poonen on a three dimensional family of abelian thre ..."
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En genre 3, le théorème de Torelli s’applique de façon moins satisfaisante: on doit extraire une mystérieuse racine carrée (J.P.S., Collected Papers, n o 129) Abstract. In this article, we give a reformulation of a result from Howe, Leprevost and Poonen on a three dimensional family of abelian threefolds. We also link their result to a conjecture of Serre on a precise form of Torelli theorem for genus 3 curves.
Holomorphic rank2 vector bundles on primary Kodaira surfaces
"... Introduction Let X be a smooth compact complex manifold of dimension n and consider V ß \Gamma! X be a topological complex vector bundle on X . A classical problem demands to determine the conditions to be satisfied by V ensuring that V is the underlying topological bundle of a holomorphic vector ..."
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Introduction Let X be a smooth compact complex manifold of dimension n and consider V ß \Gamma! X be a topological complex vector bundle on X . A classical problem demands to determine the conditions to be satisfied by V ensuring that V is the underlying topological bundle of a holomorphic vector bundle on X . A priori there is no reasonable way to control the existence of holomorphic structures. Then one needs firstly to classify all the topological vector bundles by using some suitable invariants, and then to produce a set of holomorphic vector bundles whose corresponding invariants cover as much as possible. In the case of surfaces, Wu has proved (cf. [Wu]) that topological vector bundles are completely characterized by the rank and the Chern classes (for threefolds this stays no longer true as seen in [AR], [BaP]). There
GENUS TWO CURVES WITH QUATERNIONIC MULTIPLICATION AND MODULAR JACOBIAN
"... Abstract. We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces Af with quaternionic multiplication attached to a normalized newform f without complex multiplication. We include an example of Af with quaternionic multiplication for w ..."
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Abstract. We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces Af with quaternionic multiplication attached to a normalized newform f without complex multiplication. We include an example of Af with quaternionic multiplication for which we find numerically a curve C whose Jacobian is Af up to numerical approximation, and we prove that it has quaternionic multiplication and is isogenous to Af. 1.
The Schottky Problem: An Update
"... Abstract. The aim of these lecture notes is to update Beauville’s beautiful 1987 Séminaire Bourbaki talk on the same subject. The Schottky problem is the problem of finding characterizations of Jacobians among all principally polarized abelian varieties. We review the numerous approaches to this pro ..."
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Abstract. The aim of these lecture notes is to update Beauville’s beautiful 1987 Séminaire Bourbaki talk on the same subject. The Schottky problem is the problem of finding characterizations of Jacobians among all principally polarized abelian varieties. We review the numerous approaches to this problem. In the “analytical approach”, one tries to find polynomials in the thetaconstants that define the Jacobian locus in the moduli space of principally polarized abelian varieties. We review van Geemen’s (1984) and Donagi’s (1987) work in that direction. The loci they get contain the Jacobian locus as an irreducible component. In the “geometrical approach”, one tries to give geometric properties that are satisfied only by Jacobians. We review the following: singularities of the theta divisor (Andreotti and Mayer 1967); reducibility of intersections of a theta divisor with a translate (Welters 1984) and trisecants to the Kummer variety (Welters 1983, Beauville and Debarre 1986, Debarre 1992); the K–P equation and Novikov’s conjecture (Shiota 1986, Arbarello and De Concini 1984); double translation hypersurfaces (Little 1989); the van Geemen–van der Geer conjectures on the base locus of the set of second order theta functions that vanish with multiplicity ≥ 4 at the origin (Beauville and Debarre 1989, Izadi 1993); subvarieties with minimal class (Ran 1981, Debarre 1992); the Buser–Sarnak approach (1993). For g ≥ 2, the moduli space Ag