Twists Of Elliptic Curves Research Articles

Conjectures for Moments Associated With Cubic Twists of Elliptic Curves

We extend the heuristic introduced by Conrey, Farmer, Keating, Rubinstein, and Snaith in order to formulate conjectures for the -moments of L-functions of elliptic curves twisted by cubic characters. We also apply the work of Keating and Snaith on the -moments of characteristic polynomials of unitary matrices to extend our conjecture to such that and . Our conjectures are then numerically tested for two families. A novelty of our conjectures is that cubic twists naturally lead us to consider the possibility .

Selmer groups of twists of elliptic curves over K with K-rational torsion points

We generalize a result of Frey on Selmer groups of twists of elliptic curves over \(\mathbf Q \) with \(\mathbf Q \)-rational torsion points to elliptic curves defined over number fields of small degree K with a K-rational torsion point. We also provide examples of elliptic curves coming from Zywina that satisfy the conditions of our Corollary D.

ON POSITIVE PROPORTION OF RANK-ZERO TWISTS OF ELLIPTIC CURVES OVER

AbstractExtending the idea of Dabrowski [‘On the proportion of rank 0 twists of elliptic curves’, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 483–486] and using the 2-descent method, we provide three general families of elliptic curves over $\mathbb{Q}$ such that a positive proportion of prime-twists of such elliptic curves have rank zero simultaneously.

Quadratic twists of elliptic curves with 3-Selmer rank 1

A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve E/ℚ, a positive proportion of its quadratic twists E(d) have rank 1. Using tools from Galois cohomology, we give criteria on E and d which force a positive proportion of the quadratic twists of E to have 3-Selmer rank 1 and global root number -1. We then give four nonisomorphic infinite families of elliptic curves Em,n which satisfy these criteria. Conditional on the rank part of the Birch and Swinnerton-Dyer conjecture, this verifies the aforementioned conjecture for infinitely many elliptic curves. Our elliptic curves are easy to give explicitly and we state precisely which quadratic twists d to use. Furthermore, our methods have the potential of being generalized to elliptic curves over other number fields.

On Quadratic Twists of Elliptic Curves and Some Applications of a Refined Version of Yu's Formula

In this article, we study some cohomology groups and quadratic twists of elliptic curves, and apply Tate local duality and the results of Kramer–Tunnell on local norm cokernel to give a refined version of Yu's formula in the case of elliptic curves. Then, by using this refinement formula, we obtain explicit orders of Shafarevich–Tate groups of some elliptic curves in quadratic number fields, including a few unconditional cases.

Variations on twists of elliptic curves

Variations on twists of elliptic curves

ON SELMER GROUPS AND TATE–SHAFAREVICH GROUPS FOR ELLIPTIC CURVESy2=x3−n3

We study the distribution of the size of Selmer groups and Tate–Shafarevich groups arising from a 2-isogeny and its dual 2-isogeny for elliptic curves En:y2=x3−n3. We show that the 2-ranks of these groups all follow the same distribution. The result also implies that the mean value of the 2-rank of the corresponding Tate–Shafarevich groups for square-free positive integers n≤X is as X→∞. This is quite different from quadratic twists of elliptic curves with full 2-torsion points over ℚ [M. Xiong and A. Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math. 219 (2008), 523–553], where one Tate–Shafarevich group is almost always trivial while the other is much larger.

On the proportion of rank 0 twists of elliptic curves

Let E be an elliptic curve defined over Q, let E d denote its dth quadratic twist, and r E d : = rank E d ( Q ) . We prove, that, for any positive integer k there are pairwise non-isogenous elliptic curves E 1 , … , E k such that r E 1 p = ⋯ = r E k p = 0 for a positive proportion of primes p. To cite this article: A. Dąbrowski, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Rank Computations for the Congruent Number Elliptic Curves

In a companion paper, Rubin and Silverberg relate the question of unboundedness of rank in families of quadratic twists of elliptic curves to the convergence or divergence of certain series. Here we give a computational application of their ideas on counting the rational points in such families; namely, to find curves of high rank in families of quadratic twists. We also observe that the algorithm seems to find as many curves of positive even rank as it does curves of odd rank. Results are given in the case of the congruent number elliptic curves, which are the quadratic twists of the curve y2 = x3 – x; for this family, the highest rank found is 6.

Twists of elliptic curves

If E is an elliptic curve over \(\mathbb{Q}\), then let E(D) denote theD-quadratic twist of E. It is conjectured that there are infinitely many primesp for which E(p) has rank 0, and that there are infinitely many primes\(\ell \) for which \(E(\ell )\) has positive rank. For some special curvesE we show that there is a set S of primes p with density\(\frac{1}{3}\) for which if \(D = \prod {p_j } \) is a squarefree integer where\(p_j \in S\), then E(D) has rank 0. In particular E(p) has rank 0 for every\(p \in S\). As an example let E1 denote the curve

Nonvanishing of L-Series Associated to Cubic Twists of Elliptic Curves

has a rational solution. An extensive compilation of the older history of the problem is given in Dickson [6]. Within the past century, researchers have tried to exploit (either explicitly or unknowingly) the fact that the curve (0.1) is in fact an elliptic curve. During the nineteenth century, Lucas, and later Sylvester, used a descent argument to prove that (0.1) had no solution for infinitely many D in certain congruence classes mod 9 and 18 (see [6], Ch. XXI). Zagier and Kramarz [19] have produced a great deal of numerical evidence about the L-series of the curves; based on these computations, they have argued heuristically that for

Properties of twists of elliptic curves.

Properties of twists of elliptic curves.