scholarly journals Torsion growth over cubic fields of rational elliptic curves with complex multiplication

2020 ◽  
Vol 97 (1-2) ◽  
pp. 63-76
Author(s):  
Enrique Gonzalez-Jimenez
Keyword(s):  
Elliptic Curves ◽  
Complex Multiplication ◽  
Cubic Fields

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 On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields

1971 ◽  
Vol 43 ◽  
pp. 199-208 ◽  
Author(s):  
Goro Shimura
Keyword(s):  
Elliptic Curves ◽  
Cusp Form ◽  
Mellin Transform ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Function Fields ◽  
Complex Multiplication ◽  
Modular Function ◽  
Imaginary Quadratic Field

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that

scholarly journals Torsion of rational elliptic curves over cubic fields

2016 ◽  
Vol 46 (6) ◽  
pp. 1899-1917 ◽  
Author(s):  
Enrique González-Jiménez ◽  
Filip Najman ◽  
José M. Tornero
Keyword(s):  
Elliptic Curves ◽  
Cubic Fields

scholarly journals Linnik’s theorem for Sato-Tate laws on elliptic curves with complex multiplication

2015 ◽  
Vol 1 (1) ◽  
Author(s):  
Evan Chen ◽  
Peter S. Park ◽  
Ashvin A. Swaminathan
Keyword(s):  
Elliptic Curves ◽  
Complex Multiplication

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.

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.

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