Quadratic Twists Of Elliptic Curves Research Articles

Extreme central L-values of almost prime quadratic twists of elliptic curves

Extreme central L-values of almost prime quadratic twists of elliptic curves

Erratum to "Disparity in Selmer ranks of quadratic twists of elliptic curves"

Erratum to "Disparity in Selmer ranks of quadratic twists of elliptic curves"

A note on rank one quadratic twists of elliptic curves and the non-degeneracy of 𝑝-adic regulators at Eisenstein primes

We show that for certain non-CM elliptic curves E / Q E_{/\mathbb {Q}} such that 3 3 is an Eisenstein prime of good reduction, a positive proportion of the quadratic twists E ψ E_{\psi } of E E have Mordell–Weil rank one and the 3 3 -adic height pairing on E ψ ( Q ) E_{\psi }(\mathbb {Q}) is non-degenerate. We also show similar but weaker results for other Eisenstein primes. The method of proof also yields examples of middle codimensional algebraic cycles over number fields of arbitrarily large dimension (generalized Heegner cycles) that have non-zero p p -adic height. It is not known – though expected – that the archimedian height of these higher-codimensional cycles is non-zero.

Congruences of elliptic curves arising from nonsurjective mod 𝑁 Galois representations

We study N N -congruences between quadratic twists of elliptic curves. If N N has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular curves in question correspond to the normaliser of a Cartan subgroup of GL 2 ⁡ ( Z / N Z ) \operatorname {GL}_2(\mathbb {Z}/N\mathbb {Z}) . By computing explicit models for these double covers we find all pairs ( N , r ) (N, r) such that there exist infinitely many j j -invariants of elliptic curves E / Q E/\mathbb {Q} which are N N -congruent with power r r to a quadratic twist of E E . We also find an example of a 48 48 -congruence over Q \mathbb {Q} . We make a conjecture classifying nontrivial ( N , r ) (N,r) -congruences between quadratic twists of elliptic curves over Q \mathbb {Q} . Finally, we give a more detailed analysis of the level 15 15 case. We use elliptic Chabauty to determine the rational points on a modular curve of genus 2 2 whose Jacobian has rank 2 2 and which arises as a double cover of the modular curve X ( n s 3 + , n s 5 + ) X(\mathrm {ns\,} 3^+, \mathrm {ns\,} 5^+) . As a consequence we obtain a new proof of the class number 1 1 problem.

On class numbers, torsion subgroups, and quadratic twists of elliptic curves

The Mordell-Weil groups $E(\mathbb{Q})$ of elliptic curves influence the structures of their quadratic twists $E_{-D}(\mathbb{Q})$ and the ideal class groups $\mathrm{CL}(-D)$ of imaginary quadratic fields. For appropriate $(u,v) \in \mathbb{Z}^2$, we define a family of homomorphisms $\Phi_{u,v}: E(\mathbb{Q}) \rightarrow \mathrm{CL}(-D)$ for particular negative fundamental discriminants $-D:=-D_E(u,v)$, which we use to simultaneously address questions related to lower bounds for class numbers, the structures of class groups, and ranks of quadratic twists. Specifically, given an elliptic curve $E$ of rank $r$, let $\Psi_E$ be the set of suitable fundamental discriminants $-D 0$, we show that as $X \to \infty$, we have $$\#\, \left\{-X < -D < 0: -D \in \Psi_E \right \} \, \gg_{\varepsilon} X^{\frac{1}{2}-\varepsilon}.$$ In particular, if $\ell \in \{3,5,7\}$ and $\ell \mid |E_{\mathrm{tor}}(\mathbb{Q})|$, then the number of such discriminants $-D$ for which $\ell \mid h(-D)$ is $\gg_{\varepsilon} X^{\frac{1}{2}-\varepsilon}.$ Moreover, assuming the Parity Conjecture, our results hold with the additional condition that the quadratic twist $E_{-D}$ has rank at least 2.

Quadratic twists of elliptic curves and class numbers

For positive rank r elliptic curves E(Q), we employ ideal class pairingsE(Q)×E−D(Q)→CL(−D), for quadratic twists E−D(Q) with a suitable “small y-height” rational point, to obtain explicit class number lower bounds that improve on earlier work by the authors. For the curves E(a):y2=x3−a, with rank r(a), this givesh(−D)≥110⋅|Etor(Q)|RQ(E)⋅πr(a)22r(a)Γ(r(a)2+1)⋅log⁡(D)r(a)2log⁡log⁡D, representing a general improvement to the classical lower bound of Goldfeld, Gross and Zagier when r(a)≥3. We prove that the number of twists E−D(a)(Q) with such a suitable point (resp. with such a point and rank ≥2 under the Parity Conjecture) is ≫a,εX12−ε. We give infinitely many cases where r(a)≥6. These results can be viewed as an analogue of the classical estimate of Gouvêa and Mazur for the number of rank ≥2 quadratic twists, where in addition we obtain “log-power” improvements to the Goldfeld-Gross-Zagier class number lower bound.

On $2$-Selmer groups and quadratic twists of elliptic curves

Let $K$ be a number field and $E/K$ be an elliptic curve with no $2$-torsion points. In the present article we give lower and upper bounds for the $2$-Selmer rank of $E$ in terms of the $2$-torsion of a narrow class group of a certain cubic extension of $K$ attached to $E$. As an application, we prove (under mild hypotheses) that a positive proportion of prime conductor quadratic twists of $E$ have the same $2$-Selmer group.

Moments of quadratic twists of elliptic curve L-functions over function fields

We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over F_q[x], asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving derivatives of L-functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves.

On the 2‐part of the Birch and Swinnerton‐Dyer conjecture for quadratic twists of elliptic curves

In the present paper, we prove, for a large class of elliptic curves defined over Q, the existence of an explicit infinite family of quadratic twists with analytic rank 0. In addition, we establish the 2-part of the conjecture of Birch and Swinnerton-Dyer for many of these infinite families of quadratic twists. Recently, Xin Wan has used our results to prove for the first time the full Birch–Swinnerton-Dyer conjecture for some explicit infinite families of elliptic curves defined over Q without complex multiplication.

A lower bound result for the central L-values of elliptic curves

In the present paper, we prove a stronger lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the family of certain quadratic twists of elliptic curves with analytic rank zero and non-trivial rational 2-torsion.

On $2$-Selmer ranks of quadratic twists of elliptic curves

We study the $2$-Selmer ranks of elliptic curves. We prove that for an arbitrary elliptic curve $E$ over an arbitrary number field $K$, if the set $A_E$ of 2-Selmer ranks of quadratic twists of $E$ contains an integer $c$, it contains all integers larger than $c$ and having the same parity as $c$. We also find sufficient conditions on $A_E$ such that $A_E$ is equal to $\Z_{\ge t_E}$ for some number $t_E$. When all points in $E[2]$ are rational, we give an upper bound for $t_E$.

Selmer ranks of quadratic twists of elliptic curves with partial rational two-torsion

This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]), then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r_2, the number of complex places of K. We also obtain results about the minimum number of twists of E with rank 0, and subject to standard conjectures, the number of twists with rank 1, provided E does not have a cyclic 4-isogeny defined over K.

Non-vanishing theorems for quadratic twists of elliptic curves

In this paper, we show that, by applying some results on modular symbols, for a family of certain elliptic curves defined over $\mathbb Q$, there is a large class of explicit quadratic twists whose complex $L$-series does not vanish at $s=1$, and for which the $2$-part of Birch-Swinnerton-Dyer conjecture holds.

Ranks of quadratic twists of elliptic curves

Nous donnons un compte rendu d’un projet de grande envergure sur les rangs des courbes elliptiques dans une famille de tordues quadratiques en nous focalisant sur les courbes associées aux nombres congruents. Afin d’exclure certaines courbes, nos méthodes incluent des tests sur les 2,4,8-groupes de Selmer, l’utilisation de la formule explicite de Guinand-Weil et également des 3-descentes dans quelques cas. Nous constatons que les tordues quadratiques de rang 6 sont assez répandues (bien que toujours assez difficile à trouver), alors que celles de rang 7 semblent bien plus rares. Nous décrivons aussi notre incapacité à obtenir des tordues quadratiques de rang 8 et expliquons en quoi nos résultats peuvent se comparer à certaines prédictions sur la croissance du rang en fonction du conducteur. Enfin, nous expliquons une heuristique due à Granville, qui, lorsqu’elle est interprétée judicieusement, pourrait prédire que le rang maximal pour cette famille est en effet égal à 7.

Moments and distribution of central $$L$$ L -values of quadratic twists of elliptic curves

We show that if one can compute a little more than a particular moment for some family of L-functions, then one has upper bounds of the conjectured order of magnitude for all smaller (positive, real) moments and a one-sided central limit theorem holds. We illustrate our method for the family of quadratic twists of an elliptic curve, obtaining sharp upper bounds for all moments below the first. We also establish a one sided central limit theorem supporting a conjecture of Keating and Snaith. Our work leads to a conjecture on the distribution of the order of the Tate-Shafarevich group for rank zero quadratic twists of an elliptic curve, and establishes the upper bound part of this conjecture (assuming the Birch-Swinnerton-Dyer conjecture).

Quadratic twists of elliptic curves

The paper generalizes, for a wide class of elliptic curves defined over Q, the celebrated classical lemma of Birch and Heegner about quadratic twists with prime discriminants, to quadratic twists by discriminants having any prescribed number of prime factors. In addition, it proves stronger results for the family of quadratic twists of the modular elliptic curve X 0 ( 49 ) , including showing that there is a large class of explicit quadratic twists whose complex L-series does not vanish at s = 1 , and for which the full Birch–Swinnerton-Dyer conjecture is valid.

Lower bounds of the canonical height on quadratic twists of elliptic curves

We compute a lower bound of the canonical height on quadratic twists of elliptic curves over $\Q$. Also, we show a simple method for constructing families of quadratic twists with an explicit rational point. Using the above lower bound, we show that the explicit rational point is primitive as an element of the Mordell-Weil group.

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.

Bounded gaps between primes in Chebotarev sets

A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes $p_1,p_2$ with $|p_1-p_2|\leq 600$ as a consequence of the Bombieri-Vinogradov Theorem. In this paper, we apply his general method to the setting of Chebotarev sets of primes. We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over $\mathbb{Q}$, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms.

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.