107,063 research outputs found
Local root numbers of elliptic curves over dyadic fields
We consider an elliptic curve over a dyadic field with additive, potentially
good reduction. We study the finite Galois extension of the dyadic field
generated by the three-torsion points of the elliptic curve. As an application,
we give a formula to calculate the local root number of the elliptic curve over
the dyadic field.Comment: 11 page
Legendre elliptic curves over finite fields
We show that every elliptic curve over a finite field of odd characteristic
whose number of rational points is divisible by 4 is isogenous to an elliptic
curve in Legendre form, with the sole exception of a minimal respectively
maximal elliptic curve. We also collect some results concerning the
supersingular Legendre parameters
Elliptic nets and elliptic curves
An elliptic divisibility sequence is an integer recurrence sequence
associated to an elliptic curve over the rationals together with a rational
point on that curve. In this paper we present a higher-dimensional analogue
over arbitrary base fields. Suppose E is an elliptic curve over a field K, and
P_1, ..., P_n are points on E defined over K. To this information we associate
an n-dimensional array of values in K satisfying a nonlinear recurrence
relation. Arrays satisfying this relation are called elliptic nets. We
demonstrate an explicit bijection between the set of elliptic nets and the set
of elliptic curves with specified points. We also obtain
Laurentness/integrality results for elliptic nets.Comment: 34 pages; several minor errors/typos corrected in v
Torsion points with multiplicatively dependent coordinates on elliptic curves
In this paper, we study the finiteness problem of torsion points on an
elliptic curve whose coordinates satisfy some multiplicative dependence
relations. In particular, we prove that on an elliptic curve defined over a
number field there are only finitely many torsion points whose coordinates are
multiplicatively dependent. Moreover, we produce an effective result when the
elliptic curve is defined over the rational numbers or has complex
multiplication.Comment: 11 page
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