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.

On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

2005 ◽  
Vol 48 (1) ◽  
pp. 16-31 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Ernst Kani
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Upper Bounds

AbstractLet E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer l, let ϕl be the Galois representation associated to the l-division points of E. From a celebrated 1972 result of Serre we know that ϕl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which ϕl is not surjective.

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.

scholarly journals Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields

2021 ◽  
Vol 56 (1) ◽  
pp. 47-61
Author(s):  
Enrique González-Jiménez ◽  
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Complex Multiplication ◽  
Number Fields ◽  
Quadratic Fields ◽  
Fixed Degree ◽  
Explicit Characterization

In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve.

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

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 Constructing elliptic curves from Galois representations

2018 ◽  
Vol 154 (10) ◽  
pp. 2045-2054
Author(s):  
Andrew Snowden ◽  
Jacob Tsimerman
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Good Reduction ◽  
Arithmetic Surface

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius elements, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.

scholarly journals p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

2015 ◽  
Vol 219 ◽  
pp. 269-302
Author(s):  
Kenichi Bannai ◽  
Hidekazu Furusho ◽  
Shinichi Kobayashi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Theta Function ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Good Reduction ◽  
Ring Of Integers

AbstractConsider an elliptic curve defined over an imaginary quadratic fieldKwith good reduction at the primes abovep≥ 5 and with complex multiplication by the full ring of integersof K. In this paper, we constructp-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then provep-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

scholarly journals No Singular Modulus Is a Unit

2018 ◽  
Vol 2020 (24) ◽  
pp. 10005-10041 ◽  
Author(s):  
Yuri Bilu ◽  
Philipp Habegger ◽  
Lars Kühne
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Endomorphism Ring ◽  
Complex Multiplication ◽  
Computer Assisted ◽  
Algebraic Unit ◽  
Special Points ◽  
Effective Proof ◽  
Rule Out

Abstract A result of the 2nd-named author states that there are only finitely many complex multiplication (CM)-elliptic curves over $\mathbb{C}$ whose $j$-invariant is an algebraic unit. His proof depends on Duke’s equidistribution theorem and is hence noneffective. In this article, we give a completely effective proof of this result. To be precise, we show that every singular modulus that is an algebraic unit is associated with a CM-elliptic curve whose endomorphism ring has discriminant less than $10^{15}$. Through further refinements and computer-assisted arguments, we eventually rule out all remaining cases, showing that no singular modulus is an algebraic unit. This allows us to exhibit classes of subvarieties in ${\mathbb{C}}^n$ not containing any special points.

scholarly journals INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS

2012 ◽  
Vol 92 (1) ◽  
pp. 45-60 ◽  
Author(s):  
AARON EKSTROM ◽  
CARL POMERANCE ◽  
DINESH S. THAKUR
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Residue Class ◽  
Complex Multiplication ◽  
Weak Form ◽  
Arithmetic Progressions ◽  
Carmichael Numbers ◽  
The Third ◽  
Fermat’S Little Theorem

AbstractIn 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

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