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About some family of elliptic curves  [PDF]
K. Bugajska
Mathematics , 2010,
Abstract: We examine the moduli space E=T* of complex tori T(t)=C/L(t) where L(t)=cost.n(t)Lt. We find that the Dedekind eta function furnishes a bridge between the euclidean and hyperbolic structures on T*=C-L/L as well as between the doubly periodic Weierstrass function p on T* and the theta function for the lattice E(8). The former one allows us to rewrite the Lame equation for the Bers embedding of T(1,1) in a new form. We show that L has natural decomposition into 8 sublattices (each equivalent to L) together with appropriate half-points and that this leads to some local functions and to a relation with E(8)
Heuristics on pairing-friendly elliptic curves  [PDF]
John Boxall
Mathematics , 2011,
Abstract: We present a heuristic asymptotic formula as $x\to \infty$ for the number of isogeny classes of pairing-friendly elliptic curves with fixed embedding degree $k\geq 3$, with fixed discriminant, with rho-value bounded by a fixed $\rho_0$ such that $1<\rho_0<2$, and with prime subgroup order at most $x$.
Elliptic divisibility sequences and undecidable problems about rational points  [PDF]
Gunther Cornelissen,Karim Zahidi
Mathematics , 2004,
Abstract: Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0) where the \forall-quantifiers run over a total of 8 variables, and where F is a polynomial. This implies that the \Sigma_5-theory of Q is undecidable. We prove that a conjecture about elliptic curves provides an interpretation of Z in Q with quantifier complexity \forall \exists, involving only one universally quantified variable. This improves the complexity of defining Z in Q in two ways, and implies that the \Sigma_3-theory, and even the \Pi_2-theory, of Q is undecidable (recall that Hilbert's Tenth Problem for Q is the question whether the \Sigma_1-theory of Q is undecidable). In short, granting the conjecture, there is a one-parameter family of hypersurfaces over Q for which one cannot decide whether or not they all have a rational point. The conjecture is related to properties of elliptic divisibility sequences on an elliptic curve and its image under rational 2-descent, namely existence of primitive divisors in suitable residue classes, and we discuss how to prove weaker-in-density versions of the conjecture and present some heuristics.
On some congruence properties of elliptic curves  [PDF]
Derong Qiu
Mathematics , 2008, DOI: 10.1007/s00013-009-0100-x
Abstract: In this paper, as a result of a theorem of Serre on congruence properties, a complete solution is given for an open question (see the text) presented recently by Kim, Koo and Park. Some further questions and results on similar types of congruence properties of elliptic curves are also presented and discussed.
Formal groups, elliptic curves, and some theorems of Couveignes  [PDF]
Antonia W. Bluher
Mathematics , 1997,
Abstract: The formal group law of an elliptic curve has seen recent applications to computational algebraic geometry in the work of Couveignes to compute the order of an elliptic curve over finite fields of small characteristic. The purpose of this paper is to explain in an elementary way how to associate a formal group law to an elliptic curve and to expand on some theorems of Couveignes. In addition, the paper serves as background for [J. Number Theory 70 (1998), 127-145]. We treat curves defined over arbitrary fields, including fields of characteristic two or three.
Modularity of some elliptic curves over totally real fields  [PDF]
Bao V. Le Hung
Mathematics , 2013,
Abstract: We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic curves over certain real quadratic fields are modular.
Elliptic curves on some homogeneous spaces  [PDF]
Boris Pasquier,Nicolas Perrin
Mathematics , 2011,
Abstract: Let $X$ be a minuscule homogeneous space, an odd quadric, or an adjoint homogenous space of type different from $A$ and $G_2$. Le $C$ be an elliptic curve. In this paper, we prove that for $d$ large enough, the scheme of degree $d$ morphisms from $C$ to $X$ is irreducible, giving an explicit lower bound for $d$ which is optimal in many cases.
Letter about component groups of elliptic curves  [PDF]
Kenneth A. Ribet
Mathematics , 2001,
Abstract: This is the text of a letter written by K. Ribet to J.-F. Mestre in November, 1987. The letter proves three formulas that had been conjectured by Mestre and Oesterl\'e. These formulas concern component groups of elliptic curves that occur as optimal quotients of certain Jacobians of modular curves, and relations between the component groups and the character group of the toric component of the Jacobian. (The "toric component" and component groups are computed in positive characteristic.)
On Tate-Shafarevich groups of some elliptic curves  [PDF]
Franz Lemmermeyer
Mathematics , 1999,
Abstract: This is an updated version of ANT-0166. Generalizing results of Stroeker and Top we show that the 2-ranks of the Tate-Shafarevich groups of the elliptic curves $y^2 = (x+k)(x^2+k^2)$ can become arbitrarily large. We also present a conjecture on the rank of the Selmer groups attached to rational 2-isogenies of elliptic curves.
Elementary background in elliptic curves  [PDF]
Antonia W. Bluher
Mathematics , 1997,
Abstract: This paper gives additional background in algebraic geometry as an accompaniment to the article, ``Formal Groups, Elliptic Curves, and some Theorems of Couveignes'' [arXiv:math.NT/9708215]. Section 1 discusses the addition law on elliptic curves, and Sections 2 and 3 explain about function fields, uniformizers, and power series expansions with respect to a uniformizer.
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