Quadratic Twists Of Elliptic Curves Research Articles

Elliptic curves with all cubic twists of the same root number

Let E/K be an elliptic curve with j-invariant 0 defined over a number field K. In this paper, we give a simple condition on K which determines whether all cubic twists of E/K have the same root number or not. This is a cubic twist analogue to the work [D-D] of Dokchitser and Dokchitser on quadratic twists of elliptic curves.

Disparity in Selmer ranks of quadratic twists of elliptic curves

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolute Galois group of K by characters of order p.

ON SELMER GROUPS OF QUADRATIC TWISTS OF ELLIPTIC CURVES WITH A TWO‐TORSION OVER

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial 2-torsion point over . This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.219 (2008), 523–553] which studied the same subject for elliptic curves with full 2-torsions over and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves . Mathematika 58 (2012), 236–274.] for the special elliptic curves . It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is for square-free positive integers as .

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.

Periods of quadratic twists of elliptic curves

In this paper we prove a relation between the period of an elliptic curve and the period of its real and imaginary quadratic twists. This relation is often misstated in the literature.

Ranks of twists of elliptic curves and Hilbert’s tenth problem

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert's Tenth Problem has a negative answer over the ring of integers of every number field.

2-Selmer groups of quadratic twists of elliptic curves

In this paper we investigate families of quadratic twists of elliptic curves. Addressing a speculation of Ono, we identify a large class of elliptic curves for which the parities of the “algebraic parts” of the central values $L(E^{(d)}/\mathbb {D}{Q},1)$, as $d$ varies, have essentially the same multiplicative structure as the coefficients $a_d$ of $L(E/\mathbb {D}{Q},s)$. We achieve this by controlling the 2-Selmer rank (à la Mazur and Rubin) when the Tamagawa numbers do not already dictate the parity.

Quadratic twists of elliptic curves with small Selmer rank

Given an elliptic curve E over the rational with no rational 2-torsion points, we prove the existence of a quadratic twist of E for which the 2-Selmer rank is less than or equal to 1. By the author's earlier result, we establish a lower bound on the number of D's for which the twists E(D) have 2-Selmer rank <= 1. We include in the introduction our (brief) opinion about why it is supposed to be hard to push our technique to make the Selmer group trivial.

Hauteur asymptotique des points de Heegner

AbstractGeometric intuition suggests that the Néron–Tate height of Heegner points on a rational elliptic curveEshould be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross–Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin–Selberg convolution ofEwith a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twistedL-function ofEby a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with theseL-series andL-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves.

Distribution of Selmer groups of quadratic twists of a family of elliptic curves

We study the distribution of the size of the Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with full 2-torsion points in Q . We show that one of these Selmer groups is almost always bounded, while the 2-rank of the other follows a Gaussian distribution. This provides us with a small Tate–Shafarevich group and a large Tate–Shafarevich group. When combined with a result obtained by Yu [G. Yu, On the quadratic twists of a family of elliptic curves, Mathematika 52 (1–2) (2005) 139–154 (2006)], this shows that the mean value of the 2-rank of the large Tate–Shafarevich group for square-free positive integers n less than X is 1 2 log log X + O ( 1 ) , as X → ∞ .

Skew-Frobenius Maps on Hyperelliptic Curves

Scalar multiplication methods using the Frobenius maps are known for efficient methods to speed up (hyper)elliptic curve cryptosystems. However, those methods are not efficient for the cryptosystems constructed on fields of small extension degrees due to costs of the field operations. Iijima et al. showed that one can use certain automorphisms on the quadratic twists of elliptic curves for fast scalar multiplications without the drawback of the Frobenius maps. This paper shows an extension of the automorphisms on the Jacobians of hyperelliptic curves of arbitrary genus.

Rank Frequencies for Quadratic Twists of Elliptic Curves

We give explicit examples of infinite families of elliptic curves E over Q with (nonconstant) quadratic twists over Q(t) of rank at least 2 and 3. We recover some results announced by Mestre, as well as some additional families. Suppose D is a squarefree integer and let rE(D) denote the rank of the quadratic twist of E by D. We apply results of Stewart and Top to our examples to obtain results of the form for all sufficiently large x.

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.

Selmer groups of quadratic twists of elliptic curves

(1.1) E : y + a1xy + a3y = x 3 + a2x 2 + a4x+ a6 where a1, a2, a3, a4, a6 ∈ Z. Let N(E) denote the conductor of E, j(E) the j-invariant of E, and L(E, s) = ∑∞ n=1 a(n)n −s the Hasse-Weil L-function of E. If E is modular, then let FE(z) = ∑∞ n=1 aE(n)q n ∈ S2(N(E), χ1) be the associated weight 2 cusp form. Here χ1 denotes the trivial Dirichlet character. Throughout, D will denote a square-free integer, and χD shall denote the Kronecker character for the field Q( √ D). Let h(D) denote the order of Cl(D), the ideal class group of Q( √ D). If l is prime, then define h(D)l by

Quadratic twists of elliptic curves associated to the simplest cubic fields

Quadratic twists of elliptic curves associated to the simplest cubic fields

Bounded ranks and Diophantine error terms

We show that Lang's conjecture on error terms in Diophantine approximation implies Honda's conjecture on ranks of elliptic curves over number fields. We also show that even a very weak version of Lang's error term conjecture would be enough to deduce boundedness of ranks for quadratic twists of elliptic curves over number fields. This can be seen as evidence for boundedness of ranks not relying on probabilistic heuristics on elliptic curves.