DETERMINING ELEMENTS OF MINIMAL INDEX IN AN INFINITE FAMILY OF TOTALLY REAL BICYCLIC BIQUADRATIC NUMBER FIELDS

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.

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Power Integral Bases in Composits of Number Fields

Canadian Mathematical Bulletin ◽

10.4153/cmb-1998-025-3 ◽

1998 ◽

Vol 41 (2) ◽

pp. 158-165 ◽

Cited By ~ 9

Author(s):

István Gaál

Keyword(s):

Number Fields ◽

Parametric Family ◽

Prime Degree ◽

Index Form ◽

Form Equation ◽

Integral Bases ◽

Quadratic Extensions ◽

Totally Real

AbstractIn the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree 10 of this type.

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Number fields with large minimal index containing quadratic subfields

International Journal of Number Theory ◽

10.1142/s1793042118501427 ◽

2018 ◽

Vol 14 (09) ◽

pp. 2333-2342 ◽

Cited By ~ 1

Author(s):

Henry H. Kim ◽

Zack Wolske

Keyword(s):

Number Field ◽

Number Fields ◽

Upper Bounds ◽

Minimal Index

In this paper, we consider number fields containing quadratic subfields with minimal index that is large relative to the discriminant of the number field. We give new upper bounds on the minimal index, and construct families with the largest possible minimal index.

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Constructions of Dense Lattices of Full Diversity

TEMA (São Carlos) ◽

10.5540/tema.2020.021.02.299 ◽

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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Nearly ordinary Galois deformations over arbitrary number fields

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748008000327 ◽

2008 ◽

Vol 8 (1) ◽

pp. 99-177 ◽

Cited By ~ 11

Author(s):

Frank Calegari ◽

Barry Mazur

Keyword(s):

Arbitrary Number ◽

Quadratic Field ◽

Galois Representations ◽

Galois Representation ◽

Number Fields ◽

Base Change ◽

Deformation Spaces ◽

Totally Real ◽

Set Of Points ◽

Dense Set

AbstractLet K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

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