Results 1  10
of
18
Computational Aspects of Curves of Genus at Least 2
 Algorithmic number theory. 5th international symposium. ANTSII
, 1996
"... . 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 per ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
(Show Context)
. 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...
Covering Collections and a Challenge Problem of Serre
"... We answer a challenge of Serre by showing that every rational point on the projective curve X 4 + Y 4 = 17Z 4 is of the form (±1, ±2, 1) or (±2, ±1, 1). Our approach builds on recent ideas from both Nils Bruin and the authors on the application of covering collections and Chabauty arguments to curve ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
We answer a challenge of Serre by showing that every rational point on the projective curve X 4 + Y 4 = 17Z 4 is of the form (±1, ±2, 1) or (±2, ±1, 1). Our approach builds on recent ideas from both Nils Bruin and the authors on the application of covering collections and Chabauty arguments to curves of high rank. This is the only value of c ≤ 81 for which the Fermat quartic X 4 +Y 4 = cZ 4 cannot be solved trivially, either by local considerations or maps to elliptic curves of rank 0, and it seems likely that our approach should give a method of attack for other nontrivial values of c.
Five squares in arithmetic progression over quadratic fields
, 2009
"... We give several criteria to show over which quadratic number fields Q ( √ D) there should exists a nonconstant arithmetic progressions of five squares. This is done by translating the problem to determining when some genus five curves CD defined over Q have rational points, and then using a Morde ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
(Show Context)
We give several criteria to show over which quadratic number fields Q ( √ D) there should exists a nonconstant arithmetic progressions of five squares. This is done by translating the problem to determining when some genus five curves CD defined over Q have rational points, and then using a MordellWeil sieve argument among others. Using a elliptic Chabautylike method, we prove that the only nonconstant arithmetic progressions of five squares over Q ( √ 409), up to equivalence, is 7 2
Torsion in rank 1 Drinfeld modules and the uniform boundedness conjecture, Mathematische Annalen 308
, 1997
"... Abstract. It is conjectured that for fixed A, r ≥ 1, and d ≥ 1, there is a uniform bound on the size of the torsion submodule of a Drinfeld Amodule of rank r over a degree d extension L of the fraction field K of A. We verify the conjecture for r = 1, and more generally for Drinfeld modules having ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
Abstract. It is conjectured that for fixed A, r ≥ 1, and d ≥ 1, there is a uniform bound on the size of the torsion submodule of a Drinfeld Amodule of rank r over a degree d extension L of the fraction field K of A. We verify the conjecture for r = 1, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of K. Moreover, we show that within an Lisomorphism class, there are only finitely many Drinfeld modules up to isomorphism over L which have nonzero torsion. For the case A = Fq[T], r = 1, and L = Fq(T), we give an explicit description of the possible torsion submodules. We present three methods for proving these cases of the conjecture, and explain why they fail to prove the conjecture in general. Finally, an application of the Mordell conjecture for characteristic p function fields proves the uniform boundedness for the pprimary part of the torsion for rank 2 Drinfeld Fq[T]modules over a fixed function field. 1. Conjectures and Theorems In a 1977 paper, Mazur [13] proved that if E is an elliptic curve over Q, its torsion subgroup is one of the following fifteen groups:
Rational points on certain families of curves of genus at least 2
 Proc. London Math. Soc
, 1987
"... A conjecture of Mordell, recently proven by Faltings [7], states that a curve of genus at least 2 has only finitely many rational points. Faltings ' proof is not effective, although a careful reworking of his proof, combined with some further ideas of Faltings, Mumford, Parshin, and Raynaud, al ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
A conjecture of Mordell, recently proven by Faltings [7], states that a curve of genus at least 2 has only finitely many rational points. Faltings ' proof is not effective, although a careful reworking of his proof, combined with some further ideas of Faltings, Mumford, Parshin, and Raynaud, allows one to give an upper bound for the number of rational points. (See [19, XI, §2].) Unfortunately, the resulting bound depends in quite a nasty manner on the set of primes at which the curve has bad reduction. Prior to Faltings ' proof of Mordell's conjecture, there were two methods which in certain rather restrictive cases could be used to prove finiteness of the number of rational points. The first was due to Chabauty [3], and the second to Dem'janenko [5], generalized by Manin [12]. Recently, Coleman [4] has analysed Chabauty's method and used it to give relatively small upper bounds in those cases where it can be applied. For example, he proves that if C/Q is a smooth curve of genus g 3 = 2, and if the Jacobian variety J of C satisfies rank/(Q)<g, then
On uniform lower bound of the Galois images associated to elliptic curves, Journal de théorie des nombres de Bordeaux 20
, 2008
"... Abstract. Let p be a prime and K be a number field. Let ρE,p: GK −→ Aut(TpE) ∼ = GL2(Zp) be the Galois representation given by the Galois action on the padic Tate module of an elliptic curve E over K. Serre showed that the image of ρE,p is open if E has no complex multiplication. For an elliptic ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Let p be a prime and K be a number field. Let ρE,p: GK −→ Aut(TpE) ∼ = GL2(Zp) be the Galois representation given by the Galois action on the padic Tate module of an elliptic curve E over K. Serre showed that the image of ρE,p is open if E has no complex multiplication. For an elliptic curve E over K whose jinvariant does not appear in an exceptional finite set, we give an explicit uniform lower bound of the size of the image of ρE,p. 1.
Computing Rational Points on Curves
 In Number theory for the millennium, III (Urbana, IL, 2000), 149–172, A K Peters
, 2000
"... We give a brief introduction to the problem of explicit determination of rational points on curves, indicating some recent ideas that have led to progress. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
We give a brief introduction to the problem of explicit determination of rational points on curves, indicating some recent ideas that have led to progress.
ON THE NUMBER OF RATIONAL ITERATED PREIMAGES OF THE ORIGIN UNDER QUADRATIC DYNAMICAL SYSTEMS
, 2008
"... For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article “Uniform Bounds on PreImages Under Quadratic Dynamical Systems,” by the present authors ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article “Uniform Bounds on PreImages Under Quadratic Dynamical Systems,” by the present authors and five others, it was shown that the number of rational preimages of the origin is bounded as one varies the morphism in a certain onedimensional family. To provide a conditional bound, we use the ideas from that paper and a number of modern tools for locating rational points on high genus curves. We also provide further insight into the geometry of the “preimage curves.”
CURVES OVER EVERY GLOBAL FIELD VIOLATING THE LOCALGLOBAL PRINCIPLE
"... Abstract. There is an algorithm that takes as input a global field k and produces a curve over k violating the localglobal principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k) = n. 1. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. There is an algorithm that takes as input a global field k and produces a curve over k violating the localglobal principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k) = n. 1.