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 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 $p$.

Generalized Artin'S Conjecture for Primitive Roots and Cyclicity Mod of Elliptic Curves Over Function Fields

1995 ◽  
Vol 38 (2) ◽  
pp. 167-173 ◽  
Author(s):  
David A. Clark ◽  
Masato Kuwata
Keyword(s):  
Finite Field ◽  
Prime Ideal ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Function Fields ◽  
Artin's Conjecture ◽  
Primitive Roots

AbstractLet k = Fq be a finite field of characteristic p with q elements and let K be a function field of one variable over k. Consider an elliptic curve E defined over K. We determine how often the reduction of this elliptic curve to a prime ideal is cyclic. This is done by generalizing a result of Bilharz to a more general form of Artin's primitive roots problem formulated by R. Murty.

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 Secondary terms in the number of vanishings of quadratic twists of elliptic curve L-functions

2011 ◽  
pp. 215-232 ◽  
Author(s):  
J. B. Conrey ◽  
A. Pokharel ◽  
M. O. Rubinstein ◽  
M. Watkins
Keyword(s):  
Elliptic Curve ◽  
Quadratic Twists

Partial Euler Products on the Critical Line

2005 ◽  
Vol 57 (2) ◽  
pp. 267-297 ◽  
Author(s):  
Keith Conrad
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic Curve ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Function Fields ◽  
Number Fields ◽  
Euler Product ◽  
Second Moments ◽  
Precise Sense ◽  
Euler Products

AbstractThe initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve L-function at s = 1. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the L-function and that the constant in the asymptotics has an unexpected factor of. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical L-function along its critical line. The general phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seemsmuch deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.

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.

On thep-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of

2016 ◽  
Vol 164 (1) ◽  
pp. 67-98 ◽  
Author(s):  
YUKAKO KEZUKA
Keyword(s):  
Lower Bound ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Lower Bounds ◽  
Complex Multiplication ◽  
Ring Of Integers ◽  
Quadratic Twists ◽  
The Family ◽  
Adic Valuation

AbstractWe study infinite families of quadratic and cubic twists of the elliptic curveE=X0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of the algebraic part of the value of the complexL-series ats=1, and, for the family of cubic twists, we establish a lower bound for the 3-adic valuation of the algebraic part of the sameL-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

scholarly journals Ready-made short basis for GLV+GLS on high degree twisted curves

2021 ◽  
Vol 7 (1) ◽  
pp. 306-314
Author(s):  
Bei Wang ◽  
◽  
Songsong Li ◽  
Yi Ouyang ◽  
Honggang Hu ◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Scalar Multiplication ◽  
Lattice Basis Reduction ◽  
Crucial Step ◽  
Quadratic Twists ◽  
Basis Reduction ◽  
High Degree ◽  
Short Basis ◽  
Elliptic Curve Scalar Multiplication

<abstract><p>The crucial step in elliptic curve scalar multiplication based on scalar decompositions using efficient endomorphisms—such as GLV, GLS or GLV+GLS—is to produce a short basis of a lattice involving the eigenvalues of the endomorphisms, which usually is obtained by lattice basis reduction algorithms or even more specialized algorithms. Recently, lattice basis reduction is found to be unnecessary. Benjamin Smith (AMS 2015) was able to immediately write down a short basis of the lattice for the GLV, GLS, GLV+GLS of quadratic twists using elementary facts about quadratic rings. Certainly it is always more convenient to use a ready-made short basis than to compute a new one by some algorithm. In this paper, we extend Smith's method on GLV+GLS for quadratic twists to quartic and sextic twists, and give ready-made short bases for $ 4 $-dimensional decompositions on these high degree twisted curves. In particular, our method gives a unified short basis compared with Hu et al.'s method (DCC 2012) for $ 4 $-dimensional decompositions on sextic twisted curves.</p></abstract>

Sign in / Sign up

Export Citation Format

Share Document