Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over ℚ {{\mathbb{Q}}} . We prove the existence of explicit infinite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank 1. In addition, we show that these families of quadratic twists satisfy the 2-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.

Ready-made short basis for GLV+GLS on high degree twisted curves

<abstract><p>The crucial step in elliptic curve scalar multiplication based on scalar decompositions using efficient endomorphisms—such as GLV, GLS or GLV+GLS—is to produce a short basis of a lattice involving the eigenvalues of the endomorphisms, which usually is obtained by lattice basis reduction algorithms or even more specialized algorithms. Recently, lattice basis reduction is found to be unnecessary. Benjamin Smith (AMS 2015) was able to immediately write down a short basis of the lattice for the GLV, GLS, GLV+GLS of quadratic twists using elementary facts about quadratic rings. Certainly it is always more convenient to use a ready-made short basis than to compute a new one by some algorithm. In this paper, we extend Smith's method on GLV+GLS for quadratic twists to quartic and sextic twists, and give ready-made short bases for $ 4 $-dimensional decompositions on these high degree twisted curves. In particular, our method gives a unified short basis compared with Hu et al.'s method (DCC 2012) for $ 4 $-dimensional decompositions on sextic twisted curves.</p></abstract>

On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

Rational points on elliptic K3 surfaces of quadratic twist type

In studying rational points on elliptic K3 surfaces of the form $$\begin{equation*} f(t)y^2=g(x), \end{equation*}$$ where f, g are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell–Weil rank, and we relate it to the Hilbert property. Applying to surfaces of Cassels–Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.

On elliptic curves with complex multiplication and root numbers

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text]. Suppose that [Formula: see text] has complex multiplication over [Formula: see text], i.e. [Formula: see text] is an imaginary quadratic field. With the aid of CM theory, we find elliptic curves whose quadratic twists have a constant root number.