scholarly journals GOLDFELD’S CONJECTURE AND CONGRUENCES BETWEEN HEEGNER POINTS

2019 ◽  
Vol 7 ◽  
Author(s):  
DANIEL KRIZ ◽  
CHAO LI
Keyword(s):  
Elliptic Curve ◽  
Analogous Result ◽  
Heegner Points ◽  
Positive Proportion ◽  
Quadratic Twists ◽  
General Elliptic ◽  
Special Cases ◽  
General Bound

Given an elliptic curve$E$over$\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). We show that this conjecture holds whenever$E$has a rational 3-isogeny. We also prove the analogous result for the sextic twists of$j$-invariant 0 curves. For a more general elliptic curve$E$, we show that the number of quadratic twists of$E$up to twisting discriminant$X$of analytic rank 0 (respectively 1) is$\gg X/\log ^{5/6}X$, improving the current best general bound toward Goldfeld’s conjecture due to Ono–Skinner (respectively Perelli–Pomykala). To prove these results, we establish a congruence formula between$p$-adic logarithms of Heegner points and apply it in the special cases$p=3$and$p=2$to construct the desired twists explicitly. As a by-product, we also prove the corresponding$p$-part of the Birch and Swinnerton–Dyer conjecture for these explicit twists.

scholarly journals Hauteur asymptotique des points de Heegner

2008 ◽  
Vol 60 (6) ◽  
pp. 1406-1436 ◽  
Author(s):  
Guillaume Ricotta ◽  
Thomas Vidick
Keyword(s):  
Elliptic Curve ◽  
Quadratic Field ◽  
Theta Series ◽  
Dirichlet Character ◽  
Heegner Points ◽  
Quadratic Twists ◽  
Asymptotic Formulae ◽  
Weight One ◽  
Special Value ◽  
Geometric Intuition

AbstractGeometric intuition suggests that the Néron–Tate height of Heegner points on a rational elliptic curve E should 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 of E with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twisted L-function of E by a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with these L-series and L-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.

scholarly journals Quadratic twists of elliptic curves with 3-Selmer rank 1

2014 ◽  
Vol 10 (05) ◽  
pp. 1191-1217 ◽  
Author(s):  
Zane Kun Li
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Cohomology ◽  
Number Fields ◽  
Root Number ◽  
Positive Proportion ◽  
Quadratic Twists

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.

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

2020 ◽  
Author(s):  
Joachim Petit
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Asymptotic Estimate ◽  
Rational Point ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Quadratic Twists ◽  
The Family ◽  
Minimal Height ◽  
Real Quadratic

Abstract 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.

scholarly journals A Markov model for Selmer ranks in families of twists

2014 ◽  
Vol 150 (7) ◽  
pp. 1077-1106 ◽  
Author(s):  
Zev Klagsbrun ◽  
Barry Mazur ◽  
Karl Rubin
Keyword(s):  
Markov Process ◽  
Markov Model ◽  
Elliptic Curve ◽  
Arbitrary Number ◽  
Equilibrium Distribution ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Quadratic Twists ◽  
The Family ◽  
Positive Integers

AbstractWe study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field $K$. Under the assumption that ${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank $r$ exists for all positive integers $r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of $E$ and $K$. More generally, our results also apply to $p$-Selmer ranks of twists of two-dimensional self-dual ${\bf F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

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

2020 ◽  
Vol 14 (7) ◽  
pp. 1853-1893
Author(s):  
Hung M. Bui ◽  
Alexandra Florea ◽  
Jonathan P. Keating ◽  
Edva Roditty-Gershon
Keyword(s):  
Elliptic Curve ◽  
Function Fields ◽  
Quadratic Twists

scholarly journals Subconvexity and equidistribution of Heegner points in the level aspect

2013 ◽  
Vol 149 (7) ◽  
pp. 1150-1174 ◽  
Author(s):  
Sheng-Chi Liu ◽  
Riad Masri ◽  
Matthew P. Young
Keyword(s):  
Theta Series ◽  
Ideal Class Group ◽  
Heegner Points ◽  
Maass Form ◽  
Quadratic Twists ◽  
Fundamental Discriminant ◽  
Weight One ◽  
Natural Sense ◽  
Large Level ◽  
Quadratic Twist

AbstractLet $q$ be a prime and $- D\lt - 4$ be an odd fundamental discriminant such that $q$ splits in $ \mathbb{Q} ( \sqrt{- D} )$. For $f$ a weight-zero Hecke–Maass newform of level $q$ and ${\Theta }_{\chi } $ the weight-one theta series of level $D$ corresponding to an ideal class group character $\chi $ of $ \mathbb{Q} ( \sqrt{- D} )$, we establish a hybrid subconvexity bound for $L(f\times {\Theta }_{\chi } , s)$ at $s= 1/ 2$ when $q\asymp {D}^{\eta } $ for $0\lt \eta \lt 1$. With this circle of ideas, we show that the Heegner points of level $q$ and discriminant $D$ become equidistributed, in a natural sense, as $q, D\rightarrow \infty $ for $q\leq {D}^{1/ 20- \varepsilon } $. Our approach to these problems is connected to estimating the ${L}^{2} $-restriction norm of a Maass form of large level $q$ when restricted to the collection of Heegner points. We furthermore establish bounds for quadratic twists of Hecke–Maass $L$-functions with simultaneously large level and large quadratic twist, and hybrid bounds for quadratic Dirichlet $L$-functions in certain ranges.

scholarly journals Quadratic Twists of Abelian Varieties With Real Multiplication

2019 ◽  
Author(s):  
Ari Shnidman
Keyword(s):  
Rational Point ◽  
Real Field ◽  
Geometry Of Numbers ◽  
Ring Of Integers ◽  
Real Multiplication ◽  
Positive Proportion ◽  
Quadratic Twists ◽  
Eisenstein Ideal ◽  
Totally Real ◽  
Real Number Field

AbstractLet $F$ be a totally real number field and $A/F$ an abelian variety with real multiplication (RM) by the ring of integers $\mathcal {O}$ of a totally real field. Assuming $A$ admits an $\mathcal {O}$-linear 3-isogeny over $F$, we prove that a positive proportion of the quadratic twists $A_d$ have rank 0. If moreover $A$ is principally polarized and $III(A_d)$ is finite, then a positive proportion of $A_d$ have $\mathcal {O}$-rank $1$. Our proofs make use of the geometry-of-numbers methods from our previous work with Bhargava, Klagsbrun, and Lemke Oliver and develop them further in the case of RM. We quantify these results for $A/\mathbb {Q}$ of prime level, using Mazur’s study of the Eisenstein ideal. For example, suppose $p \equiv 10$ or $19 \pmod {27}$, and let $A$ be the unique optimal quotient of $J_0(p)$ with a rational point $P$ of order 3. We prove that at least $25\%$ of twists $A_d$ have rank 0 and the average $\mathcal {O}$-rank of $A_d(F)$ is at most 7/6. Using the presence of two different 3-isogenies in this case, we also prove that roughly $1/8$ of twists of the quotient $A/\langle P\rangle$ have nontrivial 3-torsion in their Tate–Shafarevich groups.

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