MODULAR LATTICES OVER CM FIELDS

2009 ◽  
Vol 05 (05) ◽  
pp. 859-869
Author(s):  
IVAN SUAREZ
Keyword(s):  
Modular Lattice ◽  
Number Fields ◽  
Class Numbers ◽  
Modular Lattices ◽  
Fixed Level ◽  
Ideal Lattices ◽  
Totally Real

We study some properties of Arakelov-modular lattices, which are particular modular ideal lattices over CM fields. There are two main results in this paper. The first one is the determination of the number of Arakelov-modular lattices of fixed level over a given CM field provided that an Arakelov-modular lattice is already known. This number depends on the class numbers of the CM field and its maximal totally real subfield. The first part gives also a way to compute all these Arakelov-modular lattices. In the second part, we describe the levels that can occur for some multiquadratic CM number fields.

scholarly journals Construction of Arakelov-modular lattices over totally definite quaternion algebras

2017 ◽  
Vol 13 (07) ◽  
pp. 1855-1880
Author(s):  
Xiaolu Hou
Keyword(s):  
Real Number ◽  
Quaternion Algebra ◽  
Number Fields ◽  
Prime Integer ◽  
Quaternion Algebras ◽  
Modular Lattices ◽  
Ideal Lattices ◽  
Definition Of ◽  
Totally Real ◽  
Real Number Field

We study ideal lattices constructed from totally definite quaternion algebras over totally real number fields, and generalize the definition of Arakelov-modular lattices over number fields. In particular, we prove for the case where the totally real number field is [Formula: see text] that for [Formula: see text] a prime integer, there always exists a totally definite quaternion algebra over [Formula: see text] from which an Arakelov-modular lattice of level [Formula: see text] can be constructed.

scholarly journals Ideal lattices over totally real number fields and Euclidean minima

2006 ◽  
Vol 86 (3) ◽  
pp. 217-225 ◽  
Author(s):  
Eva Bayer-Fluckiger ◽  
Ivan Suarez
Keyword(s):  
Real Number ◽  
Number Fields ◽  
Ideal Lattices ◽  
Totally Real

The determination of the class-numbers of totally real cubic fields

1961 ◽  
Vol 57 (4) ◽  
pp. 728-730 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Class Number ◽  
Sum Of Squares ◽  
Class Numbers ◽  
Cubic Field ◽  
Cubic Fields ◽  
Totally Real

In a previous paper (2) it was shown how the work of finding the units of a totally real cubic field could be facilitated by consideration of the sum of squares of differences between a number and its conjugates. In the present paper it is shown that the same ideas can be helpful in the calculation of class-numbers, and a list of the fields with class-number greater than unity and discriminant less than 20,000 is given.

scholarly journals Computation of L(0, χ) and of relative class numbers of CM-fields

2001 ◽  
Vol 161 ◽  
pp. 171-191 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Real Number ◽  
Number Field ◽  
Class Group ◽  
Number Fields ◽  
Class Numbers ◽  
Finite Part ◽  
Relative Class ◽  
Relative Class Number ◽  
Totally Real ◽  
Real Number Field

Let χ be a nontrivial Hecke character on a (strict) ray class group of a totally real number field L of discriminant dL. Then, L(0, χ) is an algebraic number of some cyclotomic number field. We develop an efficient technique for computing the exact values at s = 0 of such abelian Hecke L-functions over totally real number fields L. Let fχ denote the norm of the finite part of the conductor of χ. Then, roughly speaking, we can compute L(0, χ) in O((dLfx)0.5+∊) elementary operations. We then explain how the computation of relative class numbers of CM-fields boils down to the computation of exact values at s = 0 of such abelian Hecke L-functions over totally real number fields L. Finally, we give examples of relative class number computations for CM-fields of large degrees based on computations of L(0, χ) over totally real number fields of degree 2 and 6.

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