Elliptic curves over totally real quartic fields not containing √5 are modular

2021 ◽  
Author(s):  
Josha Box
Keyword(s):  
Elliptic Curves ◽  
Totally Real

scholarly journals ON MAZUR'S CONJECTURE FOR TWISTED L-FUNCTIONS OF ELLIPTIC CURVES OVER TOTALLY REAL OR CM FIELDS

2010 ◽  
Vol 53 (1) ◽  
pp. 207-210
Author(s):  
CRISTIAN VIRDOL
Keyword(s):  
Functional Equation ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Order ◽  
Number Field ◽  
Meromorphic Continuation ◽  
Minimal Order ◽  
Finite Set ◽  
Entire Complex ◽  
Totally Real

Let E be an elliptic curve defined over a number field F, and let Σ be a finite set of finite places of F. Let L(s, E, ψ) be the L-function of E twisted by a finite-order Hecke character ψ of F. It is conjectured that L(s, E, ψ) has a meromorphic continuation to the entire complex plane and satisfies a functional equation s ↔ 2 − s. Then one can define the so called minimal order of vanishing ats = 1 of L(s, E, ψ), denoted by m(E, ψ) (see Section 2 for the definition).

scholarly journals Elliptic curves over totally real cubic fields are modular

2020 ◽  
Vol 14 (7) ◽  
pp. 1791-1800
Author(s):  
Maarten Derickx ◽  
Filip Najman ◽  
Samir Siksek
Keyword(s):  
Elliptic Curves ◽  
Cubic Fields ◽  
Totally Real
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