Variation of the analytic λ‐invariant over a solvable extension

2019 ◽  
Vol 120 (6) ◽  
pp. 918-960
Author(s):  
Daniel Delbourgo
Keyword(s):  
Solvable Extension

scholarly journals Exactly solvable spin chain models corresponding to BDI class of topological superconductors

2016 ◽  
Vol 6 (1) ◽  
Author(s):  
S. A. Jafari ◽  
Farhad Shahbazi
Keyword(s):  
Time Reversal ◽  
Solvable Models ◽  
Topological Superconductors ◽  
Spin Interactions ◽  
Detailed Classification ◽  
Exactly Solvable ◽  
Solvable Extension ◽  
Xy Chain ◽  
Topological Phase Transitions

Abstract We present an exactly solvable extension of the quantum XY chain with longer range multi-spin interactions. Topological phase transitions of the model are classified in terms of the number of Majorana zero modes, n M which are in turn related to an integer winding number, n W . The present class of exactly solvable models belong to the BDI class in the Altland-Zirnbauer classification of topological superconductors. We show that time reversal symmetry of the spin variables translates into a sliding particle-hole (PH) transformation in the language of Jordan-Wigner fermions – a PH transformation followed by a π shift in the wave vector which we call it the πPH. Presence of πPH symmetry restricts the n W (n M ) of time-reversal symmetric extensions of XY to odd (even) integers. The πPH operator may serve in further detailed classification of topological superconductors in higher dimensions as well.

Kummer's and Iwasawa's Version of Leopoldt's Conjecture

1988 ◽  
Vol 31 (3) ◽  
pp. 338-346 ◽  
Author(s):  
Jonathan W. Sands
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Class Numbers ◽  
Root Of Unity ◽  
Leopoldt's Conjecture ◽  
Solvable Extension ◽  
Cyclotomic Units

AbstractWe present a refinement of Iwasawa's approach to Leopoldt's conjecture on the non-vanishing of the p-adic regulator of an algebraic number field K. As an application, the conjecture for K implies the conjecture for a solvable extension L of degree g over K if g is relatively prime to p — 1 and p does not divide g, the discriminant of K, and the quotient of class numbers where is a primitive pth root of unity. This can be viewed as generalizing a theorem of Kummer on cyclotomic units.

Variation of the algebraic λ-invariant over a solvable extension

2019 ◽  
pp. 1-24
Author(s):  
DANIEL DELBOURGO
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Galois Representation ◽  
Exact Formula ◽  
Abelian Extension ◽  
Hida Family ◽  
Solvable Extension ◽  
Image Position

Abstract Fix an odd prime p. Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $ where g > 0 and $0 \lt n'\leq n$ . Given a modular Galois representation $\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$ which is p-ordinary and also p-distinguished, we shall write $\mathcal{H}(\overline{\rho})$ for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant \begin{equation} \lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of } \text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big) \end{equation} at all $f\in\mathcal{H}(\overline{\rho})$ , under the assumption $\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$ for at least one form f0. We can then easily deduce that $\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$ is constant along branches of $\mathcal{H}(\overline{\rho})$ , generalising a theorem of Emerton, Pollack and Weston for $\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$ . For example, if $\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$ has the structure of a p-adic Lie extension then our formulae include the cases where: either (i) $\mathcal{D}_{\infty}/K$ is a g-fold false Tate tower, or (ii) $\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$ has dimension ≤ 3 and is a pro-p-group.

scholarly journals On the Existence of Unramified Separable Infinite Solvable Extensions of Function Fields over Finite Fields

1958 ◽  
Vol 13 ◽  
pp. 95-100
Author(s):  
Hisasi Morikawa
Keyword(s):  
Finite Field ◽  
Existence Theorem ◽  
Finite Fields ◽  
Present Note ◽  
Function Fields ◽  
Regular Extension ◽  
Solvable Extension ◽  
Dimension One

In the present note, using the results in the previous paper, we shall prove the following existence theorem:THEOREM. Let k be a finite field with q elements and K/k be a regular extension of dimension one over k. Then, if q ≧ 11 and the genus gK of K/k is greater than one, there exists an unramified separable infinite solvable extension of K ivhich is regular over k.

scholarly journals A non-solvable extension of ℚ unramified outside 7

2012 ◽  
Vol 148 (3) ◽  
pp. 669-674 ◽  
Author(s):  
Luis V. Dieulefait
Keyword(s):  
Galois Group ◽  
Cusp Form ◽  
Fourier Coefficients ◽  
Galois Representation ◽  
Maximal Subgroups ◽  
Genus 2 ◽  
Solvable Extension ◽  
Level 1 ◽  
Siegel Cusp Form

AbstractWe consider a mod 7 Galois representation attached to a genus 2 Siegel cusp form of level 1 and weight 28 and using some of its Fourier coefficients and eigenvalues computed by N. Skoruppa and the classification of maximal subgroups of PGSp(4,p) we show that its image is as large as possible. This gives a realization of PGSp(4,7) as a Galois group over ℚ and the corresponding number field provides a non-solvable extension of ℚ which ramifies only at 7.

Solvable Points on Projective Algebraic Curves

2004 ◽  
Vol 56 (3) ◽  
pp. 612-637 ◽  
Author(s):  
Ambrus Pál
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Algebraic Curves ◽  
Residue Field ◽  
Rational Points ◽  
Projective Curve ◽  
Arbitrary Characteristic ◽  
Projective Curves ◽  
Solvable Extension ◽  
Solvable Points

AbstractWe examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus g, when g is at least 40, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus 0, 2, 3 or 4 defined over any field has a solvable point. Finally we prove that every genus 1 curve defined over a local field of characteristic zero with residue field of characteristic p has a divisor of degree prime to 6p defined over a solvable extension.

scholarly journals Hypergroups associated to harmonic NA groups

2002 ◽  
Vol 72 (2) ◽  
pp. 209-216 ◽  
Author(s):  
Bianca Di Blasio
Keyword(s):  
Compact Space ◽  
Lie Group ◽  
Probability Measures ◽  
Nilpotent Lie Group ◽  
Locally Compact ◽  
Structure Preserving ◽  
Borel Measures ◽  
Heisenberg Type ◽  
Convolution Structure ◽  
Solvable Extension

AbstractA harmonic NA group is a suitable solvable extension of a two-step nilpotent Lie group N of Heisenberg type by R+, which acts on N by anisotropic dilations. A hypergroup is a locally compact space for which the space of Borel measures has a convolution structure preserving the probability measures and satisfying suitable conditions. We describe a class of hypergroups associated to NA groups.

scholarly journals The Helgason Fourier transform on a class of nonsymmetric harmonic spaces

1997 ◽  
Vol 55 (3) ◽  
pp. 405-424 ◽  
Author(s):  
Francesca Astengo ◽  
Roberto Camporesi ◽  
Bianca Di Blasio
Keyword(s):  
Fourier Transform ◽  
Riemannian Metric ◽  
Smooth Functions ◽  
One Dimensional ◽  
Symmetric Manifold ◽  
Rank One ◽  
Heisenberg Type ◽  
Solvable Extension ◽  
Left Invariant Riemannian Metric ◽  
The Fourier Transform

Given a group N of Heisenberg type, we consider a one-dimensional solvable extension NA of N, equipped with the natural left-invariant Riemannian metric, which makes NA a harmonic (not necessarily symmetric) manifold. We define a Fourier transform for compactly supported smooth functions on NA, which, when NA is a symmetric space of rank one, reduces to the Helgason Fourier transform. The corresponding inversion formula and Plancherel Theorem are obtained. For radial functions, the Fourier transform reduces to the spherical transform considered by E. Damek and F. Ricci.

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