Boundedness of Mordell-Weil ranks of certain elliptic curves and Lang's conjecture

In this paper, we show that a special case of Lang's conjecture on rational points on surfaces of general type implies that there exist only finitely many elliptic curves, when the x-coordinates of n rational points are specified with n?8.     

On the Tate-Shafarevich groups of certain elliptic curves

The Tate-Shafarevich groups of certain elliptic curves over Fq(t) are related, via étale cohomology, to the group of points of an elliptic curve with complex multiplication. The Cassels-Tate pairing is computed under this identification.     

On construction of certain CM elliptic curves

Let E be a CM elliptic curve defined over an algebraic number field F. In the previous paper [N. Murabayashi, On the field of definition for modularity of CM elliptic curves, J. Number Theory 108 (2004) 268-286], we gave necessary and sufficient conditions for E to be modular over F, i.e. there exists a normalized newform f of weight two on Γ₁(N) for some N such that HomF(E,Jf)≠{0}. We also determined the multiplicity of E as F-simple factor of Jf when HomF(E,Jf)≠{0}. In this process we separated into the three cases. In this paper we construct certain CM elliptic curves which satisfy the conditions of each case. In other words, we show that all three cases certainly occur.     

On standardized models of isogenous elliptic curves

Let be isogenous elliptic curves over given by standardized Weierstrass models. We show that (in the obvious notation)

and, moreover, that there are integers such that

where .

     

Addition law structure of elliptic curves

The study of alternative models for elliptic curves has found recent interest from cryptographic applications, after it was recognized that such models provide more efficiently computable algorithms for the group law than the standard Weierstrass model. Examples of such models arise via symmetries induced by a rational torsion structure. We analyze the module structure of the space of sections of the addition morphisms, determine explicit dimension formulas for the spaces of sections and their eigenspaces under the action of torsion groups, and apply this to specific models of elliptic curves with parametrized torsion subgroups.     

Mod representations of elliptic curves

Explicit equations are given for the elliptic curves (in characteristic ) with mod representation isomorphic to that of a given one.

     

Characterizing defect n-extendable graphs and -critical graphs

A near perfect matching is a matching saturating all but one vertex in a graph. Let G be a connected graph. If any n independent edges in G are contained in a near perfect matching where n is a positive integer and n(|V(G)|-2)/2, then G is said to be defect n-extendable. If deleting any k vertices in G where k|V(G)|-2, the remaining graph has a perfect matching, then G is a k-critical graph. This paper first shows that the connectivity of defect n-extendable graphs can be any integer. Then the characterizations of defect n-extendable graphs and (2k+1)-critical graphs using M-alternating paths are presented.     

On ring class eigenspaces of Mordell-Weil groups of elliptic curves over global function fields

If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional χ-eigenspace (with χ a complex ring class character) provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is non-zero. This represents the analogue in the function field setting of a theorem for elliptic curves over Q due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof employs Kolyvagin-type arguments, and the cohomological machinery is started up by the control on the Galois structure of the torsion of E provided by classical results of Igusa in positive characteristic.     

Pair correlation of torsion points on elliptic curves

We prove the existence of the pair correlation measure associated to torsion points on the real locus E(R) of an elliptic curve E and provide an explicit formula for the limiting pair correlation function.     

Organizing the arithmetic of elliptic curves

Suppose that E is an elliptic curve defined over a number field K, p is a rational prime, and K_∞ is the maximal Z_p-power extension of K. In previous work [B. Mazur, K. Rubin, Elliptic curves and class field theory, in: Ta Tsien Li (Ed.), Proceedings of the International Congress of Mathematicians, ICM 2002, vol. II, Higher Education Press, Beijing, 2002, pp. 185-195; B. Mazur, K. Rubin, Pairings in the arithmetic of elliptic curves, in: J. Cremona et al. (Eds.), Modular Curves and Abelian Varieties, Progress in Mathematics, vol. 224, 2004, pp. 151-163] we discussed the possibility that much of the arithmetic of E over K_∞ (i.e., the Mordell-Weil groups and their p-adic height pairings, the Shafarevich-Tate groups and their Cassels pairings, over all finite extensions of K in K_∞) can be described efficiently in terms of a single skew-Hermitian matrix with entries drawn from the Iwasawa algebra of K_∞/K.In this paper, using work of Nekovár? [J. Nekovár?, Selmer complexes. Preprint available at 〈http://www.math.jussieu.fr/∼nekovar/pu/〉], we show that under not-too-stringent conditions such an “organizing” matrix does in fact exist. We also work out an assortment of numerical instances in which we can describe the organizing matrix explicitly.     

Asymptotics for the number of n-quasigroups of order 4

The asymptotic form of the number of n-quasigroups of order 4 is $3^{n + 1} 2^{2^n + 1} (1 + o(1))$ .     

Bifurcations of travelling wave solutions for the mK(n, n) equation

Using the method of planar dynamical systems to the mK(n, n) equation, the existence of uncountably infinite many smooth and non-smooth periodic wave solutions, solitary wave solutions and kink and anti-kink wave solutions is proved. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All possible exact explicit parametric representations of smooth and non-smooth travelling wave solutions are obtain.     

Finding composite order ordinary elliptic curves using the Cocks-Pinch method

We apply the Cocks-Pinch method to obtain pairing-friendly composite order groups with prescribed embedding degree associated to ordinary elliptic curves, and we show that new security issues arise in the composite order setting.     

Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields

With the goal of producing elliptic curves and higher-dimensional abelian varieties of large rank over function fields, we provide a geometric construction of towers of surfaces dominated by products of curves; in the case where the surface is defined over a finite field our construction yields families of smooth, projective curves whose Jacobians satisfy the conjecture of Birch and Swinnerton-Dyer. As an immediate application of our work we employ known results on analytic ranks of abelian varieties defined in towers of function field extensions, producing a one-parameter family of elliptic curves over Fq(t1/d) whose members obtain arbitrarily large rank as d→∞.     

On the rational topological complexity of coformal elliptic spaces

We establish some upper and lower bounds of the rational topological complexity for certain classes of elliptic spaces. Our techniques permit us in particular to show that the rational topological complexity coincides with the dimension of the rational homotopy for some special families of coformal elliptic spaces.     

Curves in Projective Spaces and Almost MDS Codes

A k-track in PG(n,q) is a set of k points such that every n of them are in general position. Here, we construct a class of n-tracks arising from algebraic curves.     

On the supersingular reduction of elliptic curves

Let a∈Q and denote byE _a the curvey ² = (x ² + l)(x + a). We prove thatE _a(F_p) is cyclic for infinitely many primesp. This fact was known previously only under the assumption of the generalized Riemann hypothesis. Research partially supported by NSERC grant A9418.     

Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Let q be a perfect power of a prime number p and $E({\mathbb{F}}_q)$ be an elliptic curve over ${\mathbb{F}}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer n we denote by $\mathrm{\#}E({\mathbb{F}}_{q^n})$ the number of rational points on E (including infinity) over the extension ${\mathbb{F}}_{q^n}$. Under a mild technical condition, we show that the sequence ${\{\mathrm{\#}E({\mathbb{F}}_{q^n})\}}_{n>0}$ contains at most 10²⁰⁰ perfect squares. If the mild condition is not satisfied, then $\mathrm{\#}E({\mathbb{F}}_{q^n})$ is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range $q<50$ and $n\le 1000$.     

Computing the rank of elliptic curves over real quadratic number fields of class number 1

In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over . Several examples are included.