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157
The CasselsTate Pairing On Polarized Abelian Varieties
 Ann. of Math
, 1998
"... . Let (A; ) be a principally polarized abelian variety dened over a global eld k, and let X(A) be its Shafarevich{Tate group. Let X(A) nd denote the quotient of X(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing X(A) nd X(A) nd ! Q=Z : If A is an ellip ..."
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Cited by 69 (16 self)
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. Let (A; ) be a principally polarized abelian variety dened over a global eld k, and let X(A) be its Shafarevich{Tate group. Let X(A) nd denote the quotient of X(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing X(A) nd X(A) nd ! Q=Z : If A is an elliptic curve, then by a result of Cassels' the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent denitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on X(A) nd . These criteria are expressed in terms of an element c 2 X(A) nd that is canonically associated to the polarization . In the case that A is the Jacobian of some curve, a downtoearth version of the result allows us to determine eectively whether #X(A) (if nite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperell...
Cycles of quadratic polynomials and rational points on a genus 2 curve
, 1996
"... It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), who ..."
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Cited by 45 (14 self)
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It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), whose rational points had been previously computed. We prove there are none with N = 5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal(Q/Q)stable 5cycles, and show that there exist Gal(Q/Q)stable Ncycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N = 6.
ON THE BIRCH–SWINNERTONDYER QUOTIENTS MODULO SQUARES
, 2006
"... Let A be an abelian variety over a number field K. An identity between the Lfunctions L(A/Ki, s) for extensions Ki of K induces a conjectural relation between the Birch–SwinnertonDyer quotients. We prove these relations modulo finiteness of X, and give an analogous statement for Selmer groups. Ba ..."
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Cited by 45 (9 self)
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Let A be an abelian variety over a number field K. An identity between the Lfunctions L(A/Ki, s) for extensions Ki of K induces a conjectural relation between the Birch–SwinnertonDyer quotients. We prove these relations modulo finiteness of X, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of X[6 ∞ ] and some restrictions on the reduction at primes above 2 and 3: the parity of the MordellWeil rank of E/K agrees with the parity of the analytic rank, as determined by the root number.
A cryptographic application of Weil descent
 CODES AND CRYPTOGRAPHY, LNCS 1746
, 1999
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Large Torsion Subgroups Of Split Jacobians Of Curves Of Genus Two Or Three
 FORUM MATH
, 1998
"... We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus2 curves over Q whose Jac ..."
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Cited by 40 (8 self)
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We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus3 curve ) = 0, whose Jacobian has 864 rational torsion points.
Construction of secure random curves of genus 2 over prime fields
 Advances in Cryptology – EUROCRYPT 2004, volume 3027 of Lecture Notes in Comput. Sci
, 2004
"... Abstract. For counting points of Jacobians of genus 2 curves defined over large prime fields, the best known method is a variant of Schoof’s algorithm. We present several improvements on the algorithms described by Gaudry and Harley in 2000. In particular we rebuild the symmetry that had been broken ..."
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Cited by 39 (16 self)
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Abstract. For counting points of Jacobians of genus 2 curves defined over large prime fields, the best known method is a variant of Schoof’s algorithm. We present several improvements on the algorithms described by Gaudry and Harley in 2000. In particular we rebuild the symmetry that had been broken by the use of Cantor’s division polynomials and design a faster division by 2 and a division by 3. Combined with the algorithm by Matsuo, Chao and Tsujii, our implementation can count the points on a Jacobian of size 164 bits within about one week on a PC. 1
Speeding Up the Discrete Log Computation on Curves With Automorphisms
, 1999
"... We show how to speed up the discrete log computations on curves having automorphisms of large order, thus generalizing the attacks on ABC elliptic curves. This includes the first known attack on CM (hyper)elliptic curves, as well as most of the hyperelliptic curves described in the literature. ..."
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Cited by 39 (2 self)
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We show how to speed up the discrete log computations on curves having automorphisms of large order, thus generalizing the attacks on ABC elliptic curves. This includes the first known attack on CM (hyper)elliptic curves, as well as most of the hyperelliptic curves described in the literature.
Torsion points on elliptic curves defined over quadratic fields
 Nagoya Mathematical Journal
, 1988
"... Let He a quadratic field and E an elliptic curve defined over k. The authors [8,12, 13] [23] discussed the ^rational points on E of prime power order. For a prime number p, let n = n{k,p) be the least non negative integer such that E p~(k) = U ker (p»: E> E)(k) c E pn for all elliptic curves E ..."
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Cited by 33 (0 self)
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Let He a quadratic field and E an elliptic curve defined over k. The authors [8,12, 13] [23] discussed the ^rational points on E of prime power order. For a prime number p, let n = n{k,p) be the least non negative integer such that E p~(k) = U ker (p»: E> E)(k) c E pn for all elliptic curves E defined over a quadratic field k ([15]). For prime
Finding large Selmer rank via an arithmetic theory of local constants
 Annals of Math. http://arxiv.org/abs/math/0512085
"... Abstract. We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K − denote the maximal abelian pextension of K that is unramified ..."
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Cited by 31 (7 self)
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Abstract. We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K − denote the maximal abelian pextension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(K − /K) by −1). We prove (under mild hypotheses on p) that if the Zprank of the prop Selmer group Sp(E/K) is odd, then rankZp Sp(E/F) ≥ [F: K] for every finite extension F of K in K −.
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 27 (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