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The goal of this paper is the study of simple rank 2 parabolic vector bundles over a 2-punctured elliptic curve C. We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to P1 × P1. We also showcase a special curve Γ isomorphic to C embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over P1 via a modular map which turns out to be the 2:1 cover ramified in Γ. We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles. [ABSTRACT FROM AUTHOR]

Isogenies are the morphisms between elliptic curves and are, accordingly, a topic of interest in the subject. As such, they have been well studied, and have been used in several cryptographic applications. V'elu's formulas show how to explicitly evaluate an isogeny, given a specification of the kernel as a list of points. However, V'elu's formulas only work for elliptic curves specified by a Weierstrass equation. This paper presents formulas similar to V'elu's that can be used to evaluate isogenies on Edwards curves and Huff curves, which are normal forms of elliptic curves that provide an alternative to the traditional Weierstrass form. Our formulas are not simply compositions of V'elu's formulas with mappings to and from Weierstrass form. Our alternate derivation yields efficient formulas for isogenies with lower algebraic complexity than such compositions. In fact, these formulas have lower algebraic complexity than V'elu's formulas on Weierstrass curves. [ABSTRACT FROM AUTHOR]

In this paper we construct infinite families of elliptic curves with given torsion group structures over cyclic cubic number fields. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve defined over ℚ. The aim of this paper is to make it possible to compute Heegner L-functions and anticyclotomic Λ-adic regulators of E, which were studied by Mazur-Rubin and Howard. We generalize results of Cohen and Watkins and thereby compute Heegner points of non-fundamental discriminant. We then prove a relationship between the denominator of a point of E defined over a number field and the leading coefficient of the minimal polynomial of its x-coordinate. Using this relationship, we recast earlier work of Mazur, Stein, and Tate to produce effective algorithms to compute p-adic heights of points of E defined over number fields. These methods enable us to give the first explicit examples of Heegner L-functions and anticyclotomic Λ-adic regulators. [ABSTRACT FROM AUTHOR]

In this paper we give a sufficient condition on the class numbers of distinct quadratic imaginary fields so that on a given CM elliptic curve over Q with fixed modular parameterization, the Heegner points associated to (the maximal orders of) these quadratic imaginary fields are linearly independent. This extends results of Rosen and Silverman from the non-CM to the CM case. [ABSTRACT FROM AUTHOR]