scholarly journals SEQUENCES ASSOCIATED TO ELLIPTIC CURVES WITH NON-CYCLIC TORSION SUBGROUP

2019 ◽  
pp. 1-13
Author(s):  
Betül Gezer
Keyword(s):  
Elliptic Curves ◽  
Torsion Subgroup ◽  
Cyclic Torsion

scholarly journals PARAMETRIZING ELLIPTIC CURVES BY MODULAR UNITS

2015 ◽  
Vol 100 (1) ◽  
pp. 33-41 ◽  
Author(s):  
FRANÇOIS BRUNAULT
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Recent Work ◽  
Short Note ◽  
Torsion Subgroup ◽  
Modular Functions ◽  
Open Questions ◽  
Modular Units

It is well known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by W. Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.

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.

Torsion Points on Certain Families of Elliptic Curves

2003 ◽  
Vol 46 (1) ◽  
pp. 157-160 ◽  
Author(s):  
Małgorzata Wieczorek
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Numerical Calculations ◽  
Torsion Subgroup ◽  
Torsion Points

AbstractFix an elliptic curve y2 = x3 + Ax + B, satisfying A, B ∈ , A ≥ |B| > 0. We prove that the -torsion subgroup is one of (0), /3, /9. Related numerical calculations are discussed.

scholarly journals Arithmetic on certain families of elliptic curves

2000 ◽  
Vol 61 (2) ◽  
pp. 319-327 ◽  
Author(s):  
Andrzej Dabrowski ◽  
Małgorzata Wieczorek
Keyword(s):  
Elliptic Curves ◽  
Torsion Subgroup

Consider a family of elliptic curves (A, A0, d0 fixed integers). We prove that, under certain conditions on A0 and d0, the rational torsion subgroup of E(B) is either cyclic of order ≤ 3 or non-cyclic of order 4. Also, assuming standard conjectures, we establish estimates for the order of the Tate-Shafarevich groups as B varies.

scholarly journals Galois theory, discriminants and torsion subgroup of elliptic curves

2010 ◽  
Vol 214 (8) ◽  
pp. 1340-1346 ◽  
Author(s):  
Irene García-Selfa ◽  
Enrique González-Jiménez ◽  
José M. Tornero
Keyword(s):  
Elliptic Curves ◽  
Galois Theory ◽  
Torsion Subgroup

scholarly journals Elliptic curves with a torsion point

1977 ◽  
Vol 66 ◽  
pp. 99-108 ◽  
Author(s):  
Toshihiro Hadano
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Rational Numbers ◽  
Torsion Subgroup ◽  
Torsion Point ◽  
Weil Group ◽  
Torsion Subgroups

Let E be an elliptic curve defined over the field Q of rational numbers, then the torsion subgroup of the Mordell-Weil group E(Q) is finite and it is known that there exist the elliptic curves whose torsion subgroups E(Q)t are of the following types: (1), (2), (3), (2, 2), (4), (5), (2, 3), (7), (2, 4), (8), (9), (2, 5), (2, 2, 3), (3, 4) and (2, 8). It has been conjectured from various reasons that E(Q)t is exhausted by the above types only. If E has a torsion point of order precisely n, then it is known that E has an n-isogeny, that is to say, an isogeny of degree n.

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