. 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 non-degenerate 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 non-degenerate 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 down-to-earth 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...

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

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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 genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve ) = 0, whose Jacobian has 864 rational torsion points.

Abstract. 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 2-descent 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 5-cycles, and show that there exist Gal(Q/Q)-stable N-cycles 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. 1.

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.

Abstract. A curve X over Q is modular if it is dominated by X1(N) for some N; if in addition the image of its jacobian in J1(N) is contained in the new subvariety of J1(N), then X is called a new modular curve. We prove that for each g ≥ 2, the set of new modular curves over Q of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of J0(N) new with N divisible by a prescribed prime, and for modular curves (new or not) with levels in a restricted set. We study new modular hyperelliptic curves in detail. In particular, we find all new modular curves of genus 2 explicitly, and construct what might be the complete list of all new modular hyperelliptic curves of all genera. Finally we prove that for each field k of characteristic zero and g ≥ 2, the set of genus-g curves over k dominated by a Fermat curve is finite and computable. 1. Introduction. Let X1(N) be the usual modular curve over Q; see Section 3.1 for a definition. (All curves and varieties in this paper are smooth, projective, and geometrically integral, unless otherwise specified. When we write an affine equation for a curve, its smooth projective model is implied.) A curve X

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

. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...

If A is an abelian variety defined over Q, its Tate module Tℓ(A) affords an ℓ-adic representation ρ of the absolute Galois group GQ. But under suitable constraints on ramification, such a representation cannot exist. Thus, Fontaine [Fo] proved there do not exist abelian varieties over Z, i.e. with everywhere good reduction. Fontaine