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17
Cycles of quadratic polynomials and rational points on a genus 2 curve
, 1996
"... 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 ..."
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Cited by 32 (13 self)
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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 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. 1.
Visible evidence in the Birch and SwinnertonDyer Conjecture for modular abelian varieties of analytic rank zero
, 2004
"... This paper provides evidence for the Birch and SwinnertonDyer conjecture for analytic rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af, 1)/ΩA, develop tools for computing with Af, and gather data about f certain arithmetic ..."
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Cited by 24 (16 self)
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This paper provides evidence for the Birch and SwinnertonDyer conjecture for analytic rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af, 1)/ΩA, develop tools for computing with Af, and gather data about f certain arithmetic invariants of the nearly 20, 000 abelian varieties Af of level ≤ 2333. Over half of these Af have analytic rank 0, and for these we compute upper and lower bounds on the conjectural order of �(Af). We find that there are at least 168 such Af for which the Birch and SwinnertonDyer conjecture implies that �(Af) is divisible by an odd prime, and we prove for 37 of these that the odd part of the conjectural order of �(Af) really divides # �(Af) by constructing nontrivial elements of �(Af) using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of ShafarevichTate groups of elliptic curves.
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 16 (1 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
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 ..."
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Cited by 14 (3 self)
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. 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...
Computing torsion points on curves
 Experimental Math
"... Let X be a curve of genus g> 2 over a field k of characteristic 1. Introduction zero Let X ^ ^ A be an Albanese map associated to a point Po 2. Notation on X. The ManinMumford conjecture, first proved by Raynaud, ..."
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Cited by 6 (0 self)
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Let X be a curve of genus g> 2 over a field k of characteristic 1. Introduction zero Let X ^ ^ A be an Albanese map associated to a point Po 2. Notation on X. The ManinMumford conjecture, first proved by Raynaud,
On a Hasse principle for MordellWeil groups
, 712
"... Abstract. In this paper we establish a Hasse principle concerning the linear dependence over Z of nontorsion points in the MordellWeil group of an abelian variety over a number field. 1. Introduction. Let A be an abelian variety over a number field F. Let v be a prime of OF and let kv: = OF/v. Let ..."
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Cited by 5 (1 self)
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Abstract. In this paper we establish a Hasse principle concerning the linear dependence over Z of nontorsion points in the MordellWeil group of an abelian variety over a number field. 1. Introduction. Let A be an abelian variety over a number field F. Let v be a prime of OF and let kv: = OF/v. Let Av denote the reduction of A for a prime v of good reduction and let rv: A(F) → Av(kv)
On arithmetic in Mordell–Weil groups
, 904
"... Abstract. In this paper we investigate linear dependence of points in MordellWeil groups of abelian varieties via reduction maps. In particular we try to determine the conditions for detecting linear dependence in MordellWeil groups via finite number of reductions. 1. Introduction. The main object ..."
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Cited by 4 (0 self)
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Abstract. In this paper we investigate linear dependence of points in MordellWeil groups of abelian varieties via reduction maps. In particular we try to determine the conditions for detecting linear dependence in MordellWeil groups via finite number of reductions. 1. Introduction. The main objective of the paper is to investigate linear dependence of points in the MordellWeil groups of abelian varieties via the reduction maps and the height function. In section 5 we prove the following theorem. Theorem A. Let A/F be an abelian variety defined over a number field F. Assume
RANKS OF ELLIPTIC CURVES WITH PRESCRIBED TORSION OVER NUMBER FIELDS
"... Abstract. We study the structure of the Mordell–Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if T is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup T is empty, or it contains curves of rank 0 as well as curves ..."
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Cited by 3 (2 self)
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Abstract. We study the structure of the Mordell–Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if T is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup T is empty, or it contains curves of rank 0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group T and a quartic field K such that among the elliptic curves over K with torsion subgroup T, there are curves of positive rank, but none of rank 0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call false complex multiplication, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization. 1.
Semistable abelian varieties over
 Q, Manuscripta Math
"... We prove that for N = 6 and N = 10, there do not exist any nonzero semistable abelian varieties over Q with good reduction outside primes dividing N. Our results are contingent on the GRH discriminant bounds of Odlyzko. Combined with recent results of Brumer–Kramer and of Schoof, this result is bes ..."
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Cited by 3 (1 self)
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We prove that for N = 6 and N = 10, there do not exist any nonzero semistable abelian varieties over Q with good reduction outside primes dividing N. Our results are contingent on the GRH discriminant bounds of Odlyzko. Combined with recent results of Brumer–Kramer and of Schoof, this result is best possible: if N is squarefree, there exists a nonzero semistable abelian variety over Q with good reduction outside primes dividing N precisely when N / ∈ {1,2,3,5,6,7,10, 13}. 1 1 Introduction. In 1985, Fontaine [3] proved a conjecture of Shafarevich to the effect that there do not exist any nonzero abelian varieties over Z (or equivalently, abelian varieties A/Q with good reduction everywhere). Fontaine’s approach was via finite group schemes over local fields. In particular, he proved the following theorem:
A REFINEMENT OF KOBLITZ’S CONJECTURE
, 909
"... Abstract. Let E be an elliptic curve over the number field Q. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of Fppoints of E is prime. However, the constant occurring in his asymptotic does not take into account that the distributions o ..."
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Cited by 2 (0 self)
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Abstract. Let E be an elliptic curve over the number field Q. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of Fppoints of E is prime. However, the constant occurring in his asymptotic does not take into account that the distributions of the E(Fp)  need not be independent modulo distinct primes. We shall describe a corrected constant. We also take the opportunity to extend the scope of the original conjecture to ask how often E(Fp)/t is prime for a fixed positive integer t, and to consider elliptic curves over arbitrary number fields. Several worked out examples are provided to supply numerical evidence for the new conjecture. 1.