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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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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.
Empirical evidence for the Birch and SwinnertonDyer conjectures for modular Jacobians of genus 2 curves
 Math. Comp
, 2001
"... Abstract. This paper provides empirical evidence for the Birch and SwinnertonDyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevic ..."
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Cited by 14 (9 self)
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Abstract. This paper provides empirical evidence for the Birch and SwinnertonDyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the ShafarevichTate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2torsion of the ShafarevichTate group, which we could compute. 1.
Modularity of some potentially BarsottiTate Galois representations
 Compos. Math
"... We prove a portion of a conjecture of B. Conrad, F. Diamond, and R. Taylor, yielding some new cases of the FontaineMazur conjectures, specifically, the modularity of certain potentially BarsottiTate Galois representations. The proof follows the template of Wiles, TaylorWiles, and BreuilConradDi ..."
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Cited by 10 (5 self)
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We prove a portion of a conjecture of B. Conrad, F. Diamond, and R. Taylor, yielding some new cases of the FontaineMazur conjectures, specifically, the modularity of certain potentially BarsottiTate Galois representations. The proof follows the template of Wiles, TaylorWiles, and BreuilConradDiamondTaylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the ring of integers of a local field. This makes crucial use of the filtered φ1modules of C. Breuil. 1. Notation, terminology, and results Throughout this article, we let l be an odd prime, and we fix an algebraic closure Ql of Ql with residue field Fl. The fields K, L, and E will always be finite extensions of Ql inside Ql. We denote by GK the Galois group Gal(Ql/K), by WK the Weil group of K, and by IK the inertia group of K. The group IQl will be abbreviated Il. The character ωn: GQl → Fln ⊂ Fl is defined via u ωn: u ↦→
Curves of genus 2 with real multiplication by a square root of 5
 University of Oxford
, 1998
"... Our aim in this work is to produce equations for curves of genus 2 whose Jacobians have real multiplication (RM) by Q ( √ 5), and to examine the conjecture that any abelian surface with RM by Q ( √ 5) is isogenous to a simple factor of the Jacobian of a modular curve X0(N) for some N. To this end, ..."
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Cited by 5 (0 self)
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Our aim in this work is to produce equations for curves of genus 2 whose Jacobians have real multiplication (RM) by Q ( √ 5), and to examine the conjecture that any abelian surface with RM by Q ( √ 5) is isogenous to a simple factor of the Jacobian of a modular curve X0(N) for some N. To this end, we review previous work in this area, and are able to use a criterion due to Humbert in the last century to produce a family of curves of genus 2 with RM by Q ( √ 5) which parametrizes such curves which have a rational Weierstrass point. We proceed to give a calculation of the ℓadic representations arising from abelian surfaces with RM, and use a special case of this to determine a criterion for the field of definition of RM by Q ( √ 5). We examine when a given polarized abelian surface A defined over a number field k with an action of an order R in a real field F, also defined over k, can be made principally polarized after kisogeny, and prove, in particular, that this is possible when the conductor of R is odd and coprime to the degree of the given polarization.
On the reduction theory of binary forms
"... In [3], a reduction theory for binary forms of degree 3 and 4 with integer coefficients was developed in detail, the motivation in the case of quartics being to improve 2descent algorithms for elliptic curves over Q. In this paper we extend some of these results to forms of higher degree. One appli ..."
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Cited by 3 (3 self)
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In [3], a reduction theory for binary forms of degree 3 and 4 with integer coefficients was developed in detail, the motivation in the case of quartics being to improve 2descent algorithms for elliptic curves over Q. In this paper we extend some of these results to forms of higher degree. One application of this is to the
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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Cited by 2 (0 self)
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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.
Computations on modular Jacobian surfaces
 Lecture Notes in Comput. Sci. (2369
, 2002
"... Abstract. We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties Af attached by Shimura to normalized newforms f ∈ S2(Γ0(N)). We present all the curves corresponding to principally polarized surfaces Af for N ≤ 500. 1 ..."
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Cited by 1 (0 self)
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Abstract. We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties Af attached by Shimura to normalized newforms f ∈ S2(Γ0(N)). We present all the curves corresponding to principally polarized surfaces Af for N ≤ 500. 1
Degrees of polarizations on an abelian surface with real multiplication
, 2000
"... Let F be a real quadratic field, and let R be an order in F. Suppose given a polarized abelian surface (A,λ) defined over a number field k with a symmetric action of R defined over k. This paper considers varying A within the kisogeny class of A to reduce the degree of λ and the conductor of R. It ..."
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Let F be a real quadratic field, and let R be an order in F. Suppose given a polarized abelian surface (A,λ) defined over a number field k with a symmetric action of R defined over k. This paper considers varying A within the kisogeny class of A to reduce the degree of λ and the conductor of R. It is proved, in particular, that there is a kisogenous principally polarized abelian surface with an action of the full ring of integers of F when F has class number 1 and the degree of λ and the conductor of R are odd and coprime. 1