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

2014 ◽  
Vol 44 (6) ◽  
pp. 2009-2027 ◽  
Author(s):  
Tadahisa Nara
Keyword(s):  
Elliptic Curves ◽  
Lower Bounds ◽  
Canonical Height ◽  
Quadratic Twists

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 LANG'S CONJECTURE AND SHARP HEIGHT ESTIMATES FOR THE ELLIPTIC CURVES y2 = x3 + ax

2013 ◽  
Vol 09 (05) ◽  
pp. 1141-1170 ◽  
Author(s):  
PAUL VOUTIER ◽  
MINORU YABUTA
Keyword(s):  
Lower Bound ◽  
Elliptic Curves ◽  
Lower Bounds ◽  
Rational Points ◽  
Upper And Lower Bounds ◽  
Canonical Height ◽  
The Difference ◽  
Lang's Conjecture ◽  
Lang’S Conjecture ◽  
Logarithmic Height

For elliptic curves given by the equation Ea : y2 = x3 + ax, we establish the best-possible version of Lang's conjecture on the lower bound for the canonical height of non-torsion rational points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.

Complete caps in AG(3, q) from elliptic curves

2014 ◽  
Vol 13 (08) ◽  
pp. 1450050 ◽  
Author(s):  
Irene Platoni
Keyword(s):  
Elliptic Curves ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Complete Caps ◽  
Odd Order ◽  
Galois Space

In a three-dimensional Galois space of odd order q, the known infinite families of complete caps have size far from the theoretical lower bounds. In this paper, we investigate some caps defined from elliptic curves. In particular, we show that for each q between 100 and 350 they can be extended to complete caps, which turn out to be the smallest complete caps known in the literature.

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.

scholarly journals Points of low height on elliptic surfaces with torsion

2010 ◽  
Vol 13 ◽  
pp. 370-387
Author(s):  
Sonal Jain
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Open Subset ◽  
Function Field ◽  
Rational Point ◽  
Torsion Point ◽  
Integral Points ◽  
Elliptic Surfaces ◽  
Canonical Height

AbstractWe determine the smallest possible canonical height$\hat {h}(P)$for a non-torsion pointPof an elliptic curveEover a function field(t) of discriminant degree 12nwith a 2-torsion point forn=1,2,3, and with a 3-torsion point forn=1,2. For eachm=2,3, we parametrize the set of triples (E,P,T) of an elliptic curveE/with a rational pointPandm-torsion pointTthat satisfy certain integrality conditions by an open subset of2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that forn=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.

scholarly journals Ranks of Elliptic Curves in Families of Quadratic Twists

2000 ◽  
Vol 9 (4) ◽  
pp. 583-590 ◽  
Author(s):  
Karl Rubin ◽  
Alice Silverberg
Keyword(s):  
Elliptic Curves ◽  
Quadratic Twists

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

2015 ◽  
Vol 202 (3) ◽  
pp. 1029-1068 ◽  
Author(s):  
Maksym Radziwiłł ◽  
K. Soundararajan
Keyword(s):  
Elliptic Curves ◽  
Quadratic Twists

scholarly journals Quadratic twists of elliptic curves

2014 ◽  
Vol 110 (2) ◽  
pp. 357-394 ◽  
Author(s):  
John Coates ◽  
Yongxiong Li ◽  
Ye Tian ◽  
Shuai Zhai
Keyword(s):  
Elliptic Curves ◽  
Quadratic Twists
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