A Space of Weight 1 Modular Forms Attached to Totally Real Cubic Number Fields

2015 ◽  
pp. 131-137
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Modular Forms ◽  
Number Fields ◽  
Cubic Number Fields ◽  
Totally Real

scholarly journals Eisenstein series in the Kohnen plus space for Hilbert modular forms

2016 ◽  
Vol 12 (03) ◽  
pp. 691-723 ◽  
Author(s):  
Ren-He Su
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Hilbert Modular Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
The One ◽  
Totally Real

In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, say [Formula: see text], whose [Formula: see text]th Fourier coefficient only occurs when [Formula: see text] is congruent to 0 or 1 modulo 4. The space of modular forms whose Fourier coefficients have the above property is called Kohnen plus space, initially introduced by Kohnen in 1980. Recently, Hiraga and Ikeda generalized the plus space to the spaces for half-integral weight Hilbert modular forms with respect to general totally real number fields. The [Formula: see text]th Fourier coefficients [Formula: see text] of a Hilbert modular form of parallel weight [Formula: see text] lying in the generalized Kohnen plus space does not vanish only if [Formula: see text] is congruent to a square modulo 4. In this paper, we use an adelic way to construct Eisenstein series of parallel half-integral weight belonging to the generalized Kohnen plus spaces and give an explicit form for their Fourier coefficients. These series give a generalization of the one introduced by Cohen. Moreover, we show that the Kohnen plus space is generated by the cusp forms and the Eisenstein series we constructed as a vector space over [Formula: see text].

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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