Norme relative de l'unité fondamentale de certains corps biquadratiques et parité des oongueurs de cycles d'idéaux réduits

RésuméSoit k = Q(√d) (d > 0 libre de carrés) un corps quadratique réel de discriminant D > 0 ayant t facteurs premiers distincts. Il est bien connu que le sous groupe du groupe des classes engendré par les t idéaux premiers ramifiés dans k/Q est d'ordre 2t−1 lorque l'unité fondamentale est de norme −1, et d'ordre 2t−2 lorsqu'elle est de norme + 1. Puisque (1) et (√d) sont principaux, il n'existe pas d'autre relation de principalité entre ces idéaux premiers ramifiés lorsque l'unité fondamentale est de norme −1, et il en existe précisément deux autres non triviales (duales l'une de l'autre) lorsqu'elle est de norme + 1. Si ω0 désigne le générateur habituel de l'anneau des entiers de k, alors l'unité fondamentale est de norme −1 si le développement en fractions continues de ω0 est de longueur de période primitive impaire, et de norme + 1 si il est de longueur de période primitive paire. De plus, dans ce dernier cas les deux relations de principalité non triviales entre les idéaux premiers ramifiés sont bien déterminées par le terme médiant de ce développement (voir [5], [6]). Nous prolongeons ces résultats dans le cas de certains corps biquadratiques totalement imaginaires (donc de rang du groupe d'unités égal à 1) contenant un sous corps quadratique (de sorte que l'on puisse parler de la norme relative de cette unité fondamentale, sa norme absolue valant elle toujours + 1), corps quadratique supposé imaginaire principal. Nous nous demandons si la parité des longueurs de cycles d'idéaux réduits (notion prolongeant celle de fractions continues puisque coïncidant avec elle dans le cas des corps quadratiques réels) donne la norme relative de l'unité fondamentale de ces extensions. Nous verrons qu'une obstruction se présente, et indiquerons quelles propriétés devrait satisfaire une nouvelle notion de cycle d'idéaux réduits pour lever cette obstruction.AbstractLet k/k be a quadratic extension of a principal imaginary quadratic field k. We show that the theory of cycles of reduced ideals as developed by H. Amara provides us with an algorithm to test the relative norm Nk/k(ηk) of the fundamental unit of K. We then show that the well known result connecting the norm of the fundamental unit of a real quadratic field with the parity of the period length of the continued fractional expansion of √D does not generalize

Dedekind sums, mean square value of L-functions at s = 1 and upper bounds on relative class numbers. A survey and open problems

International audienceExplicit formulas for the quadratic mean value at s = 1 of the Dirichlet L-functions associated with the set X − f of the φ(f)/2 odd Dirichlet characters modulo f are known. They have been used to obtain explicit upper bounds for relative class numbers of cyclotomic number fields. Here we present a generalization of these results: we show that explicit formulas for quadratic mean values at s = 1 of Dirichlet L-functions associated with subsets of X − f can be obtained. As an application we obtain explicit upper bounds for relative class numbers of imaginary subfields of cyclotomic number fields

Twisted quadratic moments for Dirichlet L-functions at s = 2

International audienceLet c, n be given positive integers. Let q > 2 be coprime with c. Let X q be the multiplicative group of order φ(q) of the Dirichlet characters modulo q. Set M (q, c, n) := 2 φ(q) χ∈Xq χ(−1)=(−1) n χ(c)|L(n, χ)| 2. The goal of this paper is to explain how one can compute explicit formulas for M (q, c, n) for a given small integers n and c. As an example, we give explicit formulas for M (q, c, 2) for c ∈ {1, 2, 3, 4, 6} and for M (p, 5, 2) for p a prime integer. As a consequence, we show that a previously published formula for M (p, 3, 2) is false. 0 2010 Mathematics Subject Classification. Primary. 11M06

Localization of the complex zeros of parametrized families of polynomials

AbstractLet Pn(x)=xm+pm−1(n)xm−1+⋯+p1(n)x+pm(n) be a parametrized family of polynomials of a given degree with complex coefficients pk(n) depending on a parameter n∈Z≥0. We use Rouché’s theorem to obtain approximations to the complex roots of Pn(x). As an example, we obtain approximations to the complex roots of the quintic polynomials Pn(x)=x5+nx4−(2n+1)x3+(n+2)x2−2x+1 studied by A. M. Schöpp

Second moment of Dirichlet LL-functions, character sums over subgroups and upper bounds on relative class numbers

We prove an asymptotic formula for the mean-square average of $L$- functions associated to subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\cal A}(p,d)$ recently introduced by E. Elma. We obtain an asymptotic formula for ${\cal A}(p,d)$ which holds true for any divisor $d$ of $p-1$ removing previous restrictions on the size of $d$. This anwers a question raised in Elma's paper. Our proof relies both on estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application we deduce the following bound $h_{p,d}^- \leq 2\left (\frac{(1+o(1))p}{24}\right )^{m/4}$ on the relative class numbers of the imaginary number fields of conductor $p\equiv 1\mod d$ and degree $m=(p-1)/d$

Blood-Brain Barrier Abnormalities Caused by HIV-1 gp120: Mechanistic and Therapeutic Implications

The blood-brain barrier (BBB) is compromised in many systemic and CNS diseases, including HIV-1 infection of the brain. We studied BBB disruption caused by HIV-1 envelope glycoprotein 120 (gp120) as a model. Exposure to gp120, whether acute [by direct intra-caudate-putamen (CP) injection] or chronic [using SV(gp120), an experimental model of ongoing production of gp120] disrupted the BBB, and led to leakage of vascular contents. Gp120 was directly toxic to brain endothelial cells. Abnormalities of the BBB reflect the activity of matrix metalloproteinases (MMPs). These target laminin and attack the tight junctions between endothelial cells and BBB basal laminae. MMP-2 and MMP-9 were upregulated following gp120-injection. Gp120 reduced laminin and tight junction proteins. Reactive oxygen species (ROS) activate MMPs. Injecting gp120 induced lipid peroxidation. Gene transfer of antioxidant enzymes protected against gp120-induced BBB abnormalities. NMDA upregulates the proform of MMP-9. Using the NMDA receptor (NMDAR-1) inhibitor, memantine, we observed partial protection from gp120-induced BBB injury. Thus, (1) HIV-envelope gp120 disrupts the BBB; (2) this occurs via lesions in brain microvessels, MMP activation and degradation of vascular basement membrane and vascular tight junctions; (3) NMDAR-1 activation plays a role in this BBB injury; and (4) antioxidant gene delivery as well as NMDAR-1 antagonists may protect the BBB

Devito: Towards a generic Finite Difference DSL using Symbolic Python

Domain specific languages (DSL) have been used in a variety of fields to express complex scientific problems in a concise manner and provide automated performance optimization for a range of computational architectures. As such DSLs provide a powerful mechanism to speed up scientific Python computation that goes beyond traditional vectorization and pre-compilation approaches, while allowing domain scientists to build applications within the comforts of the Python software ecosystem. In this paper we present Devito, a new finite difference DSL that provides optimized stencil computation from high-level problem specifications based on symbolic Python expressions. We demonstrate Devito's symbolic API and performance advantages over traditional Python acceleration methods before highlighting its use in the scientific context of seismic inversion problems.Comment: pyHPC 2016 conference submissio