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132
Mahler's Measure and Special Values of Lfunctions
, 1998
"... this paper is to describe an attempt to understand and generalize a recent formula of Deninger [1997] by means of systematic numerical experiment. This conjectural formula, ..."
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Cited by 79 (1 self)
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this paper is to describe an attempt to understand and generalize a recent formula of Deninger [1997] by means of systematic numerical experiment. This conjectural formula,
Implementing 2Descent for Jacobians of Hyperelliptic Curves
 Acta Arith
, 1999
"... . This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Qrational Weierstra point. 1. Introduction Given some curve C over Q , one w ..."
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Cited by 64 (18 self)
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. This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Qrational Weierstra point. 1. Introduction Given some curve C over Q , one would like to determine as much as possible of its arithmetical properties. One of the more important invariants is the MordellWeil rank of its Jacobian J , i.e., the free abelian rank of J(Q ) (finite by the MordellWeil Theorem). There is no algorithm so far that provably determines this rank, but it is possible (at least in theory) to bound it from above by computing the size of a suitable Selmer group. It is also fairly easy to find lower bounds by looking for independent rational points on the Jacobian. (It can be difficult, however, to find the right number of independent points, when some of the generators are large.) With some luck, both bounds coincide, and the rank is determined. In...
The Jacobi Model of an Elliptic Curve and SideChannel Analysis
 APPLIED ALGEBRA, ALGEBRAIC ALGORITHMS AND ERRORCORRECTING CODES – AAECC15, VOLUME 2643 OF LNCS
, 2003
"... A way for preventing SPAlike attacks on elliptic curve systems is to use the same formula for the doubling and the general addition of points on the curve. Various proposals have been made in this direction with dierent results. This paper reinvestigates the Jacobi form suggested by Liardet an ..."
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Cited by 36 (4 self)
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A way for preventing SPAlike attacks on elliptic curve systems is to use the same formula for the doubling and the general addition of points on the curve. Various proposals have been made in this direction with dierent results. This paper reinvestigates the Jacobi form suggested by Liardet and Smart (CHES 2001). Rather than considering the Jacobi form as the intersection of two quadrics, the addition law is directly derived from the underlying quartic. As a result, this leads to substantial memory savings and produces the fastest uni ed addition formula for curves of order a multiple of 2, as those required for OKECDH or OKECDSA.
Bounding the Number of Rational Points on Certain Curves of High Rank
, 1997
"... Let K be a number eld and let C be a curve of genus g > 1 dened over K. In this dissertation we describe techniques for bounding the number of Krational points on C. In Chapter I we discuss Chabauty techniques. This is a review and synthesis of previously known material, both published and unpu ..."
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Cited by 35 (2 self)
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Let K be a number eld and let C be a curve of genus g > 1 dened over K. In this dissertation we describe techniques for bounding the number of Krational points on C. In Chapter I we discuss Chabauty techniques. This is a review and synthesis of previously known material, both published and unpublished. We have tried to eliminate unnecessary restrictions, such as assumptions of good reduction or the existence of a known rational point on the curve. We have also attempted to clearly state the circumstances under which Chabauty techniques can be applied. Our primary goal is to provide a exible and powerful tool for computing on specic curves. In Chapter II we develop a technique which, given a Krational isogeny to the Jacobian of C, produces a positive integer n and a collection of covers of C with the property that the set of Krational points in the collection is in nto1 correspondence with the set of Krational points on C. If Chabauty is applicable to every curve in the collection, then we can use the covers to bound the number of Krational points on C. The examples in Chapters I and II are taken from problem VI.17 in the Arabic text of the Arithmetica. Chapter III is devoted to the background calculations for this problem. When we assemble the pieces, we discover that the solution given by Diophantus is the only positive rational solution to this problem. Contents 1. Preface 4 Chapter 1. Chabauty bounds 5 1.
Fast genus 2 arithmetic based on theta functions
 J.Math.Cryptol.1 (2007), 243–265. MR2372155 (2009f:11156
"... Abstract. In 1986, D. V. Chudnovsky and G. V. Chudnovsky proposed to use formulae coming from Theta functions for the arithmetic in Jacobians of genus 2 curves. We follow this idea and derive fast formulae for the scalar multiplication in the Kummer surface associated to a genus 2 curve, using a Mon ..."
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Cited by 32 (7 self)
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Abstract. In 1986, D. V. Chudnovsky and G. V. Chudnovsky proposed to use formulae coming from Theta functions for the arithmetic in Jacobians of genus 2 curves. We follow this idea and derive fast formulae for the scalar multiplication in the Kummer surface associated to a genus 2 curve, using a Montgomery ladder. Our formulae can be used to design very efficient genus 2 cryptosystems that should be faster than elliptic curve cryptosystems in some hardware configurations.
The MordellWeil sieve: Proving nonexistence of rational points on curves
"... Abstract. We discuss the MordellWeil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p info ..."
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Cited by 24 (12 self)
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Abstract. We discuss the MordellWeil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the MordellWeil sieve algorithm and discuss its efficiency. 1.
Finding Rational Points on Bielliptic Genus 2 Curves
"... We discuss a technique for trying to find all rational points on curves of the form Y 2 = f3X 6 + f2X 4 + f1X 2 + f0, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty’s Theorem may be applied. However, we shall concen ..."
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Cited by 24 (8 self)
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We discuss a technique for trying to find all rational points on curves of the form Y 2 = f3X 6 + f2X 4 + f1X 2 + f0, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty’s Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field Q(α). If each of these elliptic curves has rank less than the degree of Q(α) : Q, then we shall describe a Chabautylike technique which may be applied to try to find all the points (x, y) defined over Q(α) on the elliptic curves, for which x ∈ Q. This in turn allows us to find all Qrational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over Q), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over Q.
The Hasse principle and the BrauerManin obstruction for curves
 Manuscripta Math
, 2004
"... Abstract. We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of ex ..."
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Abstract. We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of examples of violations of the Hasse principle which are due to the BrauerManin obstruction, subject to the conjecture that the TateShafarevich group of the Jacobian is finite. 1.
Deciding existence of rational points on curves: an experiment, accepted by Exp
 Math
"... The problem to decide whether a given algebraic variety defined over the rational numbers has rational points is fundamental in Arithmetic Geometry. Abstracting from concrete examples, this leads to the question whether there exists an algorithm that is able to perform this task for any given variet ..."
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Cited by 23 (8 self)
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The problem to decide whether a given algebraic variety defined over the rational numbers has rational points is fundamental in Arithmetic Geometry. Abstracting from concrete examples, this leads to the question whether there exists an algorithm that is able to perform this task for any given variety. This is probably
J1(p) Has Connected Fibers
 DOCUMENTA MATH.
, 2002
"... We study resolution of tame cyclic quotient singularities on arithmetic surfaces, and use it to prove that for any subgroup H ⊆ (Z/pZ) × /{±1} the map XH(p) = X1(p)/H → X0(p) induces an injection Φ(JH(p)) → Φ(J0(p)) on mod p component groups, with image equal to that of H in Φ(J0(p)) when the l ..."
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Cited by 21 (2 self)
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We study resolution of tame cyclic quotient singularities on arithmetic surfaces, and use it to prove that for any subgroup H ⊆ (Z/pZ) × /{±1} the map XH(p) = X1(p)/H → X0(p) induces an injection Φ(JH(p)) → Φ(J0(p)) on mod p component groups, with image equal to that of H in Φ(J0(p)) when the latter is viewed as a quotient of the cyclic group (Z/pZ) × /{±1}. In particular, Φ(JH(p)) is always Eisenstein in the sense of Mazur and Ribet, and Φ(J1(p)) is trivial: that is, J1(p) has connected fibers. We also compute tables of