Computing a Lower Bound for the Canonical Height on Elliptic Curves over Totally Real Number Fields

2008 ◽  
pp. 139-152
Author(s):  
Thotsaphon Thongjunthug
Keyword(s):  
Lower Bound ◽  
Real Number ◽  
Elliptic Curves ◽  
Number Fields ◽  
Canonical Height ◽  
Totally Real

scholarly journals On the parity of ranks of Selmer groups IV. With an appendix by Jean-Pierre Wintenberger

2009 ◽  
Vol 145 (6) ◽  
pp. 1351-1359 ◽  
Author(s):  
Jan Nekovář
Keyword(s):  
Real Number ◽  
Elliptic Curves ◽  
Large Class ◽  
Number Fields ◽  
Selmer Groups ◽  
Parity Conjecture ◽  
Totally Real

AbstractWe prove the parity conjecture for the ranks of p-power Selmer groups (p⁄=2) of a large class of elliptic curves defined over totally real number fields.

On linear relations for special L-values over certain totally real number fields

2021 ◽  
pp. 1-18
Author(s):  
Seiji Kuga
Keyword(s):  
Real Number ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Arithmetic Functions ◽  
Hilbert Modular Form ◽  
Linear Relations ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
Totally Real

In this paper, we give linear relations between the Fourier coefficients of a special Hilbert modular form of half integral weight and some arithmetic functions. As a result, we have linear relations for the special [Formula: see text]-values over certain totally real number fields.

scholarly journals Constructions of Dense Lattices of Full Diversity

2020 ◽  
Vol 21 (2) ◽  
pp. 299
Author(s):  
A. A. Andrade ◽  
A. J. Ferrari ◽  
J. C. Interlando ◽  
R. R. Araujo
Keyword(s):  
Real Number ◽  
Fading Channels ◽  
Packing Density ◽  
Signal Transmission ◽  
Cyclotomic Field ◽  
Sphere Packing ◽  
Number Fields ◽  
Trace Form ◽  
Full Diversity ◽  
Totally Real

A lattice construction using Z-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number field can be calculated by the trace form of the field restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subfield of a cyclotomic field. Our focus is on totally real number fields since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

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