On twists of the Fermat cubic x^3+y^3=2 (original) (raw)

Height estimates on cubic twists of the Fermat elliptic curve

Bulletin of the Australian Mathematical Society 09/2005; 72(02):177 - 186.

We give bounds for the canonical height of rational and integral points on cubic twists of the Fermat elliptic curve. As a corollary we prove that there is no integral arithmetic progression on certain curves in this family

A Note on Higher Twists of Elliptic Curves

Glasgow Mathematical Journal, 2010

We show that for any pair of elliptic curves E 1 , E 2 over ‫ޑ‬ with jinvariant equal to 0, we can find a polynomial D ∈ ‫[ޚ‬u, v] such that the cubic twists of the curves E 1 , E 2 by D(u, v) have positive rank over ‫(ޑ‬u, v). We also prove that for any quadruple of pairwise distinct elliptic curves E i , i = 1, 2, 3, 4, with j-invariant j = 0, there exists a polynomial D ∈ ‫[ޚ‬u] such that the sextic twists of E i , i = 1, 2, 3, 4, by D(u) have positive rank. A similar result is proved for quadruplets of elliptic curves with j-invariant j = 1, 728.

Cubic forms, powers of primes and classification of elliptic curves

2021

We prove that for almost all primes p the equation (x+y)(x 2 +Bxy+y 2) = p α z n with B ∈ {0, 1, 4, 6} has no solution in pairwise coprime nonzero integers x, y, z = ±1, integer α > 0 and prime n ≥ p 16p. Proving this we classify primes p for which there exists an elliptic curve over Q with conductor 48p, 192p or 384p and with nontrivial rational 2-torsion.

On the Distribution of Analytic Values on Quadratic Twists of Elliptic Curves

Experimental Mathematics, 2006

The aim of this paper is to analyze the distribution of analytic (and signed) square roots of X values on imaginary quadratic twists of elliptic curves. Given an elliptic curve E of rank zero and prime conductor N , there is a weight-3 2 modular form g associated with it such that the d-coefficient of g is related to the value at s = 1 of the L-series of the (−d)-quadratic twist of the elliptic curve E. Assuming the Birch and Swinnerton-Dyer conjecture, we can then calculate for a large number of integers d the order of X of the (−d)-quadratic twist of E and analyze their distribution.

High rank quadratic twists of pairs of elliptic curves

Journal of Number Theory, 2017

Given a pair of elliptic curves E 1 and E 2 over the rational field Q whose jinvariants are not simultaneously 0 or 1728, Kuwata and Wang proved the existence of infinitely many square-free rationals d such that the d-quadratic twists of E 1 and E 2 are both of positive rank. We construct infinite families of pairs of elliptic curves E 1 and E 2 over Q such that for each pair there exist infinitely many square-free rationals d for which the d-quadratic twists of E 1 and E 2 are both of rank at least 2.

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

Inventiones mathematicae, 2010

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. This material is based upon work supported by the National Science Foundation under grants DMS-0700580 and DMS-0757807.

Automorphic forms and cubic twists of elliptic curves

arXiv: Number Theory, 1994

This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^{th} Whittaker--Fourier coefficient is essentially the L--series of the curve E_{m^2n}. One may obtain information about the collective behavior the curves E_D by exploiting this connection; for example, one can prove: Theorem: Fix any prime p \ne 3, and any congruence class c mod p. Then there are infinitely many D congruent to c mod p such that the curve E_D has no rational solutions. This paper is fairly self-contained; no prior knowledge of algebraic number theory, analytic number theory or metaplectic forms is assumed. On the other hand, this paper is a survey, no proofs are included.

Rank Zero Quadratic Twists of Modular Elliptic Curves

1996

In (11) L. Mai and M. R. Murty proved that if E is a modular elliptic curve with conductor N, then there exists infinitely many square-free integers D 1 mod 4N such that ED, the D quadratic twist of E, has rank 0. Moreover assuming the Birch and Swinnerton-Dyer Conjecture, they obtain analytic estimates on the lower bounds for the orders of their Tate-Shafarevich groups. However regarding ranks, simply by the sign of functional equations, it is not expected that there will be infinitely many square-free D in every arithmetic progression r (mod t) where gcd(r,t) is square-free such that ED has rank zero. Given a square-free positive integer r, under mild conditions we show that there exists an integer tr and a positive integer N where tr r mod Q ◊ 2

Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor

arXiv (Cornell University), 2022

Watkins' conjecture asserts that the rank of an elliptic curve is upper bounded by the 2-adic valuation of its modular degree. We show that this conjecture is satisfied when E is any quadratic twist of an elliptic curve with rational 2-torsion and prime power conductor. Furthermore, we give a lower bound of the congruence number for elliptic curves of the form y 2 = x 3 − dx, with d a biquadratefree integer.