John Cremona

Systematics of bonding in non-icosahedral carbon clusters

General formulas are presented for the vertex numbers, ν, of pentagon+hexagon polyhedra of icosahedral, tetrahedral or dihedral symmetries. Criteria for uniqueness of representation, isomer counts and grouping of pentagons are established. All polyhedra with 256 vertices or less and belonging to T, D5, D6or their supergroups are listed. With the addition of C3ν to the dihedral and higher groups, at least one pentagon+hexagon cluster is found for all even ν≥20 except for ν = 22 which is unrealisable in any symmetry, and ν = 46 (for which a C3 polyhedron exists). Carbon clusters with closed electronic shells are shown to be generated by a geometrical leapfrog procedure: for all ν = 60+6k (where k is zero or greater than one) at least one closed shell structure is predicted. In dihedral symmetry closed shells also exist for some other values of ν. Separation of the 12 pentagonal faces is not sufficient to ensure a closed electronic shell but appears to be a necessary condition in dihedral or tetrahedral symmetry.

Visualizing elements in the Shafarevich-Tate group

We review a number of ways of “visualizing” the elements of the Shafarevich–Tate group of an elliptic curve Eover a number field K. We are specifically interested in caseswhere the elliptic curves are defined over the rationals, and are subabelian varieties of the new part of the jacobian of a modular curve (specifically, of X0(N), where N is the conductor of the elliptic curve). For a given such E with nontrivial Shafarevich–Tate group, we pose the question: Are all the curves of genus one representing elements of the Shafarevich–Tate group of E isomorphic (over the rationals) to curves contained in a (single) abelian surface A, itself defined over the rationals, containing E as a subelliptic curve, and contained in turn in the new part of the jacobian of a modular curve X0(N)? At first view, one might imagine that there are few E with nontrivial Shafarevich–Tate group for which the answer is yes. Indeed we have a small number of examples where the answer is no, and it is very likely that the answer will be no if the order of the Shafarevich–Tategroup is large enough. Nonetheless, among all (modular) elliptic curves Eas above, with conductors up to 5500 and with no rational point of order 2, we have found the answer to the question to be yes in the vast majority of cases. We are puzzled by this and wonder whether there is some conceptual reason for it. We present a substantial amount of data relating to the curves investigated.

Efficient solution of rational conics

We present efficient algorithms for solving Legendre equations over Q (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known.

Finding All Elliptic Curves with Good Reduction Outside a Given Set of Primes

We describe an algorithm for determining elliptic curves defined over a given number field with a given set of primes of bad reduction. Examples are given over ℚ and over various quadratic fields.

MINIMISATION AND REDUCTION OF 2-, 3- AND 4-COVERINGS OF ELLIPTIC CURVES

In this paper we consider models for genus one curves of degree n for n = 2, 3 and 4, which arise in explicit n-descent on elliptic curves. We prove theorems on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and provide simple algorithms for minimising a given model, valid over general number fields. Finally, for genus one models defined over Q, we develop a theory of reduction and again give explicit algorithms for n = 2, 3 and 4.

Periods of Cusp forms and elliptic curves over imaginary quadratic fields

In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves E defined over an imaginary quadratic field K of class number one, and on the other hand, rational newforms F of weight two for the congruence subgroups , where n is an ideal in the ring of integers R of K. This continues work of the first author and forms part of the Ph.D. thesis of the second author. In each case we compute numerically the value of the L-series at and compare with the value of which is predicted by the Birch-Swinnerton-Dyer conjecture, finding agreement to several decimal places. In particular, we find that whenever has a point of infinite order. Several examples are given in detail from the extensive tables computed by the authors.

Classical invariants and 2-descent on elliptic curves

The classical theory of invariants of binary quartics is applied to the problem of determining the group of rational points of an elliptic curve defined over a field K by 2-descent. The results lead to some simplifications to the method first presented in Birch and Swinnerton-Dyer (1963), and can be applied to give a more efficient algorithm for determining Mordell?Weil groups over Q, as well as being more readily extended to other number fields. In this paper we mainly restrict ourselves to general theory, valid over arbitrary fields of characteristic neither 2 nor 3.

Explicit n-descent on elliptic curves, I. Algebra

Abstract This is the first in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as genus one normal curves of degree n. The methods we describe are practical in the case n = 3 for elliptic curves over the rationals, and have been implemented in MAGMA.

The elliptic curve database for conductors to 130000

Tabulating elliptic curves has been carried out since the earliest days of machine computation in number theory. After some historical remarks, we report on significant recent progress in enlarging the database of elliptic curves defined over ℚ to include all those of conductor N≤130000. We also give various statistics, summarize the data, describe how it may be obtained and used, and mention some recent work regarding the verification of Manin’s “c=1” conjecture.

Minimal Models for 2-coverings of Elliptic Curves

This paper concerns the existence and algorithmic determination of minimal models for curves of genus 1, given by equations of the form y = Q(x) where Q(x) has degree 4. These models are used in the method of 2-descent for computing the rank of an elliptic curve. Our results are complete for unramified extensions ofQ2 and Q3 and for all p-adic fields for p > 5. Our primary motivation is to complete the results of Birch and Swinnerton-Dyer [2], which are incomplete in the case of Q2. Our results in this case (when applied to 2-coverings of elliptic curves over Q) yield substantial improvements in the running times of the 2-descent algorithm implemented in the program mwrank [5]. The paper ends with a section on implementation and examples, and an appendix gives constructive proofs in sufficient detail to be used for implementation.