Numerical Evaluation at Negative Integers of the Dedekind Zeta Functions of Totally Real Cubic Number Fields

2004 ◽  
pp. 318-326 ◽  
Author(s):  
Stéphane R. Louboutin
Keyword(s):  
Numerical Evaluation ◽  
Zeta Functions ◽  
Number Fields ◽  
Cubic Number Fields ◽  
Totally Real

scholarly journals p-Adic properties of Siegel modular forms of degree 2

1978 ◽  
Vol 71 ◽  
pp. 43-60 ◽  
Author(s):  
Shōyū Nagaoka
Keyword(s):  
Real Number ◽  
Modular Forms ◽  
Zeta Functions ◽  
Number Fields ◽  
Prime Numbers ◽  
Siegel Modular Forms ◽  
Totally Real

H. P. F. Swinnerton-Dyer determined the structure of the algebra of modular forms mod p for all prime numbers p in elliptic modular case (cf. [10]). Using his result, J.-P. Serre investigated the properties of p-adic modular forms and succeeded to construct the p-adic zeta functions for any totally real number fields (cf. [8]).

scholarly journals Torsion of rational elliptic curves over different types of cubic fields

2020 ◽  
Vol 16 (06) ◽  
pp. 1307-1323
Author(s):  
Daeyeol Jeon ◽  
Andreas Schweizer
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Torsion Group ◽  
Number Fields ◽  
Cubic Field ◽  
Different Types ◽  
Cubic Fields ◽  
Cubic Number Fields ◽  
Totally Real

Let [Formula: see text] be an elliptic curve defined over [Formula: see text], and let [Formula: see text] be the torsion group [Formula: see text] for some cubic field [Formula: see text] which does not occur over [Formula: see text]. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) [Formula: see text] can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves [Formula: see text] together with cubic fields [Formula: see text] so that [Formula: see text].

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