Non-Vanishing of Quadratic Twists of Modular L-Functions

1997 ◽  
pp. 133-176
Author(s):  
M. Ram Murty ◽  
V. Kumar Murty
Keyword(s):  
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 $p$.

scholarly journals Disparity in Selmer ranks of quadratic twists of elliptic curves

2013 ◽  
Vol 178 (1) ◽  
pp. 287-320 ◽  
Author(s):  
Zev Klagsbrun ◽  
Barry Mazur ◽  
Karl Rubin
Keyword(s):  
Elliptic Curves ◽  
Quadratic Twists

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

Points on quadratic twists of X0(N)

2012 ◽  
Vol 152 (4) ◽  
pp. 323-348 ◽  
Author(s):  
Ekin Ozman
Keyword(s):  
Quadratic Twists

scholarly journals Discretisation for odd quadratic twists

2011 ◽  
pp. 201-214 ◽  
Author(s):  
J. B. Conrey ◽  
M. O. Rubinstein ◽  
N. C. Snaith ◽  
M. Watkins
Keyword(s):  
Quadratic Twists

scholarly journals DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS

2019 ◽  
pp. 1-42
Author(s):  
Joachim König ◽  
François Legrand
Keyword(s):  
Hyperelliptic Curves ◽  
Branch Points ◽  
Abc Conjecture ◽  
Galois Cover ◽  
Quadratic Twists ◽  
Large Genus ◽  
Galois Covers ◽  
The One ◽  
Almost All ◽  
Global Principle

We provide evidence for this conclusion: given a finite Galois cover $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$ , almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$ . We show that this holds if the number of branch points of $f$ is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of $\mathbb{Q}$ of given group and bounded discriminant. This widely extends a result of Granville on the lack of $\mathbb{Q}$ -rational points on quadratic twists of hyperelliptic curves over $\mathbb{Q}$ with large genus, under the abc-conjecture (a diophantine reformulation of the case $G=\mathbb{Z}/2\mathbb{Z}$ of our result). As a further evidence, we exhibit a few finite groups $G$ for which the above conclusion holds unconditionally for almost all covers of $\mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$ . We also introduce a local–global principle for specializations of Galois covers $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ and show that it often fails if $f$ has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local–global conclusion underscores the ‘smallness’ of the specialization set of a Galois cover of $\mathbb{P}_{\mathbb{Q}}^{1}$ . On the other hand, it allows to generate conditionally ‘many’ curves over $\mathbb{Q}$ failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.

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 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.

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