The torsion subgroup of the elliptic curve $Y^2 = X^3 + AX$ over the maximal abelian extension of $\mathbb{Q}$

2020 ◽  
Vol 63 (2) ◽  
pp. 137-149
Author(s):  
Jerome T. Dimabayao
Keyword(s):  
Elliptic Curve ◽  
Abelian Extension ◽  
Torsion Subgroup

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.

Generators and number fields for torsion points of a special elliptic curve

2019 ◽  
Vol 26 (1/2) ◽  
pp. 227-231
Author(s):  
Hasan Sankari ◽  
Mustafa Bojakli
Keyword(s):  
Elliptic Curve ◽  
Prime Number ◽  
Number Fields ◽  
Torsion Subgroup ◽  
Torsion Points

Let E be an elliptic curve with Weierstrass form y2=x3−px, where p is a prime number and let E[m] be its m-torsion subgroup. Let p1=(x1,y1) and p2=(x2,y2) be a basis for E[m], then we prove that ℚ(E[m])=ℚ(x1,x2,ξm,y1) in general. We also find all the generators and degrees of the extensions ℚ(E[m])/ℚ for m=3 and m=4.

scholarly journals Algebraicity of L-values for elliptic curves in a false Tate curve tower

2007 ◽  
Vol 142 (2) ◽  
pp. 193-204 ◽  
Author(s):  
THANASIS BOUGANIS ◽  
VLADIMIR DOKCHITSER
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Abelian Extension ◽  
Deligne's Conjecture ◽  
Special Value ◽  
Positive Integers ◽  
Tate Curve

AbstractLet E be an elliptic curve over ${\mathbb Q}$, and τ an Artin representation over ${\mathbb Q}$ that factors through the non-abelian extension ${\mathbb Q}(\sqrt[p^n]{m},\mu_{p^n})/{\mathbb Q}$, where p is an odd prime and n, m are positive integers. We show that L(E,τ,1), the special value at s=1 of the L-function of the twist of E by τ, divided by the classical transcendental period Ω+d+|Ω−d−|ε(τ) is algebraic and Galois-equivariant, as predicted by Deligne's conjecture.

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 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.

Ramification of the Kummer extension generated from torsion points of elliptic curves

2015 ◽  
Vol 11 (06) ◽  
pp. 1725-1734
Author(s):  
Masaya Yasuda
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Torsion Subgroup ◽  
Torsion Point ◽  
Torsion Points ◽  
Root Of Unity

For a prime p, let ζp denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field k with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over k(ζp). In this paper, for p = 5 and 7, we study the ramification of such Kummer extensions using explicit Kummer generators directly computed by Verdure in 2006.

KUMMER GENERATORS AND TORSION POINTS OF ELLIPTIC CURVES WITH BAD REDUCTION AT SOME PRIMES

2013 ◽  
Vol 09 (07) ◽  
pp. 1743-1752 ◽  
Author(s):  
MASAYA YASUDA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Quadratic Fields ◽  
Torsion Subgroup ◽  
Torsion Point ◽  
Torsion Points ◽  
Root Of Unity ◽  
Fundamental Units

For a prime p, let ζp denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field K with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over K(ζp), and in this paper, we study the ramification of such Kummer extensions using the Kummer generators directly computed by Verdure in 2006. For quadratic fields K, we also give unramified Kummer extensions over K(ζp) generated from elliptic curves over K having a p-torsion point with bad reduction at certain primes. Many of these unramified Kummer extensions have not appeared in the previous work using fundamental units of quadratic fields.

Elliptic surfaces over ℙ1 and large class groups of number fields

2019 ◽  
Vol 15 (10) ◽  
pp. 2151-2162
Author(s):  
Jean Gillibert ◽  
Aaron Levin
Keyword(s):  
Elliptic Curve ◽  
Large Class ◽  
Class Group ◽  
Number Fields ◽  
Torsion Subgroup ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Elliptic Surfaces ◽  
Cubic Fields

Given a non-isotrivial elliptic curve over [Formula: see text] with large Mordell–Weil rank, we explain how one can build, for suitable small primes [Formula: see text], infinitely many fields of degree [Formula: see text] whose ideal class group has a large [Formula: see text]-torsion subgroup. As an example, we show the existence of infinitely many cubic fields whose ideal class group contains a subgroup isomorphic to [Formula: see text].

scholarly journals NON-COMMUTATIVE p-ADIC L-FUNCTIONS FOR SUPERSINGULAR PRIMES

2012 ◽  
Vol 08 (08) ◽  
pp. 1813-1830
Author(s):  
ANTONIO LEI
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Extension ◽  
Abelian Extension ◽  
Supersingular Primes ◽  
Cyclotomic Extension

Let E/ℚ be an elliptic curve with good supersingular reduction at p with ap(E) = 0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the ℤp-cyclotomic extension of a finite Galois extension of ℚ where p is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for p-ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.

scholarly journals COMPUTING IMAGES OF GALOIS REPRESENTATIONS ATTACHED TO ELLIPTIC CURVES

2016 ◽  
Vol 4 ◽  
Author(s):  
ANDREW V. SUTHERLAND
Keyword(s):  
Monte Carlo ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Monte Carlo Algorithm ◽  
Quadratic Fields ◽  
Complete List ◽  
Torsion Subgroup ◽  
Probabilistic Algorithms

Let $E$ be an elliptic curve without complex multiplication (CM) over a number field $K$, and let $G_{E}(\ell )$ be the image of the Galois representation induced by the action of the absolute Galois group of $K$ on the $\ell$-torsion subgroup of $E$. We present two probabilistic algorithms to simultaneously determine $G_{E}(\ell )$ up to local conjugacy for all primes $\ell$ by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine $G_{E}(\ell )$ up to one of at most two isomorphic conjugacy classes of subgroups of $\mathbf{GL}_{2}(\mathbf{Z}/\ell \mathbf{Z})$ that have the same semisimplification, each of which occurs for an elliptic curve isogenous to $E$. Under the GRH, their running times are polynomial in the bit-size $n$ of an integral Weierstrass equation for $E$, and for our Monte Carlo algorithm, quasilinear in $n$. We have applied our algorithms to the non-CM elliptic curves in Cremona’s tables and the Stein–Watkins database, some 140 million curves of conductor up to $10^{10}$, thereby obtaining a conjecturally complete list of 63 exceptional Galois images $G_{E}(\ell )$ that arise for $E/\mathbf{Q}$ without CM. Under this conjecture, we determine a complete list of 160 exceptional Galois images $G_{E}(\ell )$ that arise for non-CM elliptic curves over quadratic fields with rational $j$-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the $j$-invariant is irrational.

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