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.

scholarly journals On Modified Circular Units and Annihilation of real Classes

2005 ◽  
Vol 177 ◽  
pp. 77-115 ◽  
Author(s):  
Jean-Robert Belliard ◽  
Thống Nguyễn-Quang-Ðỗ
Keyword(s):  
Real Number ◽  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Euler Systems ◽  
Totally Real ◽  
Circular Units ◽  
Functorial Approach ◽  
Real Number Field

For an abelian totally real number field F and an odd prime number p which splits totally in F, we present a functorial approach to special “p-units” previously built by D. Solomon using “wild” Euler systems. This allows us to prove a conjecture of Solomon on the annihilation of the p-class group of F (in the particular context here), as well as related annihilation results and index formulae.

CONOYAUX DE CAPITULATION ET MODULES D'IWASAWA

2011 ◽  
Vol 07 (06) ◽  
pp. 1503-1517
Author(s):  
FRÉDÉRIC PITOUN
Keyword(s):  
Real Number ◽  
Galois Group ◽  
Number Field ◽  
Class Group ◽  
Primary Roots ◽  
Totally Real ◽  
Cyclotomic Extension ◽  
Real Number Field

Soit F un corps de nombres totalement réel et p un premier impair, on note K0 = F(ζp). Pour n ∈ ℕ, Kn désigne le n-ième étage de la ℤp-extension cyclotomique K∞/K0, An est la p-partie du groupe des classes de Kn, [Formula: see text] et N∞ est l'extension de K∞ obtenue en extrayant des racines p-primaires d'unités. Le but de cet article est de montrer que le dual de Pontryagin de la partie plus des conoyaux de capitulation [Formula: see text], sur laquelle l'action de Γ a été tordue une fois par le caractère cyclotomique et la partie moins de la ℤp-torsion du groupe de Galois Gal (N∞ ∩ L∞/K∞) sont isomorphes. Let F be a totally real number field and p an odd prime, we note K0 = F(ζp). For an integer n, Kn is the nth floor of the ℤp-cyclotomic extension K∞/K0, An is the p-part of the class group of Kn, [Formula: see text] and N∞ is the extension of K∞ generated by p-primary roots of units. In this article, we prove that the plus part of the capitulation's cokernel [Formula: see text], on which Γ-action was twisted on time by the cyclotomic character, and the minus part of the ℤp-torsion of the Galois group Gal (N∞ ∩ L∞/K∞) is isomorphic.

scholarly journals Lower bounds for Maass forms on semisimple groups

2020 ◽  
Vol 156 (5) ◽  
pp. 959-1003
Author(s):  
Farrell Brumley ◽  
Simon Marshall
Keyword(s):  
Real Number ◽  
Number Field ◽  
Lower Bounds ◽  
Positive Curvature ◽  
Semisimple Group ◽  
Maass Forms ◽  
Semisimple Groups ◽  
Cartan Involution ◽  
Totally Real ◽  
Real Number Field

Let $G$ be an anisotropic semisimple group over a totally real number field $F$. Suppose that $G$ is compact at all but one infinite place $v_{0}$. In addition, suppose that $G_{v_{0}}$ is $\mathbb{R}$-almost simple, not split, and has a Cartan involution defined over $F$. If $Y$ is a congruence arithmetic manifold of non-positive curvature associated with $G$, we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.

scholarly journals ON NUMBER FIELDS WITH EQUIVALENT INTEGRAL TRACE FORMS

2012 ◽  
Vol 08 (07) ◽  
pp. 1569-1580 ◽  
Author(s):  
GUILLERMO MANTILLA-SOLER
Keyword(s):  
Quadratic Form ◽  
Real Number ◽  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Integral Quadratic Form ◽  
Trace Forms ◽  
Fundamental Discriminant ◽  
Totally Real

Let K be a number field. The integral trace form is the integral quadratic form given by tr k/ℚ(x2)|OK. In this article we study the existence of non-conjugated number fields with equivalent integral trace forms. As a corollary of one of the main results of this paper, we show that any two non-totally real number fields with the same signature and same prime discriminant have equivalent integral trace forms. Additionally, based on previous results obtained by the author and the evidence presented here, we conjecture that any two totally real quartic fields of fundamental discriminant have equivalent trace zero forms if and only if they are conjugated.

A Hypergraph with Commuting Partial Laplacians

2001 ◽  
Vol 44 (4) ◽  
pp. 385-397 ◽  
Author(s):  
Cristina M. Ballantine
Keyword(s):  
Prime Ideal ◽  
Real Number ◽  
Number Field ◽  
Linear Group ◽  
General Linear Group ◽  
Commuting Operators ◽  
Affine Building ◽  
Finite Quotient ◽  
Totally Real ◽  
Real Number Field

AbstractLetFbe a totally real number field and let GLnbe the general linear group of rank n overF. Let р be a prime ideal ofFand Fрthe completion ofFwith respect to the valuation induced by р. We will consider a finite quotient of the affine building of the group GLnover the field Fр. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

scholarly journals Weak Arithmetic Equivalence

2015 ◽  
Vol 58 (1) ◽  
pp. 115-127 ◽  
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Real Number ◽  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Root Numbers ◽  
Local Root ◽  
Arithmetic Equivalence ◽  
Totally Real ◽  
Local Root Numbers

AbstractInspired by the invariant of a number field given by its zeta function, we define the notion of weak arithmetic equivalence and show that under certain ramification hypotheses this equivalence determines the local root numbers of the number field. This is analogous to a result of Rohrlich on the local root numbers of a rational elliptic curve. Additionally, we prove that for tame non-totally real number fields, the integral trace form is invariant under arithmetic equivalence

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