Fragments of the GL2 Iwasawa theory of elliptic curves without complex multiplication

1999 ◽  
pp. 1-50 ◽  
Author(s):  
John Coates
Keyword(s):  
Elliptic Curves ◽  
Complex Multiplication ◽  
Iwasawa Theory

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 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
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