Finite Galois Extension Research Articles

Galois structure ofS-units

Let K/k be a finite Galois extension of number fields with Galois group G, S a large set of primes of K, and E the G-module of S-units of K. Previous work has determined the data which is necessary to determine the stable isomorphism class of E. This paper explains how to build a G-module M from this data which has the property that M is stably isomorphic to the direct sum of E and some free G-module F.

Cyclic Extensions Are Radical

We show that finite Galois extensions with cyclic Galois group are radical.

Canonical Nonclassical Hopf–Galois Module Structure of Nonabelian Galois Extensions

Let L/K be a finite Galois extension of local or global fields in any characteristic with nonabelian Galois group G, and let 𝔅 be an ambiguous ideal of L. We show that 𝔅 is free over its associated order in K[G] if and only if it is free over its associated order in the Hopf algebra giving the canonical nonclassical Hopf-Galois structure on the extension.

The relative Brauer group and generalized cyclic crossed products for a ramified covering

Let T/A be an integral extension of noetherian integrally closed integral domains whose quotient field extension is a finite cyclic Galois extension. Let S/R be a localization of this extension which is unramified. Using a generalized cyclic crossed product construction it is shown that certain reflexive fractional ideals of T with trivial norm give rise to Azumaya R-algebras that are split by S. Sufficient conditions on T/A are derived under which this construction can be reversed and the relative Brauer group of S/R is shown to fit into the exact sequence of Galois cohomology associated to the ramified covering T/A. Many examples of affine algebraic varieties are exhibited for which all of the computations are carried out.

On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions

Abstract Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair ( h 1 ⁢ ( A / F ) ⁢ ( 1 ) , ℤ ⁢ [ Gal ⁢ ( F / k ) ] ) (h^{1}(A_{/F})(1),\mathbb{Z}[{\rm Gal}(F/k)]) . By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell–Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible by p. More generally, our approach leads us to the formulation of certain precise families of conjectural p-adic congruences between the values at s = 1 s=1 of derivatives of the Hasse–Weil L-functions associated to twists of A, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate–Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions.

On the Galois Structure of Selmer Groups

Let A be an abelian variety defined over a number field k and F a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F, we investigate the explicit Galois structure of the p-primary Selmer group of A over F. We also use the results so obtained to derive new bounds on the growth of the Selmer rank of A over extensions of k.

Base change and K-theory for orbits of p-adic GL(n)

Let E=F be a finite Galois extension of a nonarchimedean local field F with zero characteristic and let G=GL(n)=GL(n,F). We investigate base change E=F at the level of K-theory for the spherical C*-algebra $C_r^*(G//K)$ and the Iwahori C*-algebra $C_r^*(G//I)$. Then we proceed to more abstract compact orbits of Langlands parameters and show that, at the level of the K-theory group K1, base change is multiplication by the residue degree f=f(E=F), generalizing previous results.

The growth of the rank of Abelian varieties upon extensions

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties with respect to finite extensions of number fields. First, we show that if \(A\) is an Abelian variety over a number field \(K\) and \(L/K\) is a finite Galois extension such that \({{\mathrm{Gal}}}(L/K)\) does not have an index 2 subgroup, then \({{\mathrm{rk}}}A(L)-{{\mathrm{rk}}}A(K)\) can never be 1. We show that \({{\mathrm{rk}}}A(L)-{{\mathrm{rk}}}A(K)\) is either 0 or \(\ge p-1\), where \(p\) is the smallest prime divisor of \(\# {{\mathrm{Gal}}}(L/K)\), and we obtain more precise results when \({{\mathrm{Gal}}}(L/K)\) is alternating, \({{\mathrm{SL}}}_2(\mathbb {F}_p)\) or \({{\mathrm{PSL}}}_2(\mathbb {F}_p)\) for \(p>2\). This implies a restriction on \({{\mathrm{rk}}}E(K(E[p]))-{{\mathrm{rk}}}E(K(\zeta _p))\) when \(E/K\) is an elliptic curve whose mod \(p\) Galois representation is surjective. We obtain similar results for the growth of the rank over certain non-Galois extensions. Second, we show that for every \(n\ge 2\) there exists an elliptic curve \(E_n\) over a number field \(K_n\) such that \(\mathbb {Q}\otimes {{\mathrm{End}}}_\mathbb {Q}{{\mathrm{Res}}}_{K_n/\mathbb {Q}} E_n\) contains a number field of degree \(2^n\). We ask whether every elliptic curve \(E/K\) has infinite rank over \(K\mathbb {Q}(2)\), where \(\mathbb {Q}(2)\) is the compositum of all quadratic extensions of \(\mathbb {Q}\). We show that if the answer is yes, then for any \(n\ge 2\), there exists an elliptic curve \(E_n\) over a number field \(K_n\) admitting infinitely many quadratic twists whose rank is a positive multiple of \(2^n\).

Base change for elliptic curves over real quadratic fields

Let E be an elliptic curve over a real quadratic field K and F/K a totally real finite Galois extension. We prove that E/F is modular.

Parametric Galois extensions

Given a field k and a finite group H, an H-parametric extension over k is a finite Galois extension of k(T) of Galois group containing H which is regular over k and has all the Galois extensions of k of group H among its specializations. We are mainly interested in producing non-H-parametric extensions, which relates to classical questions in inverse Galois theory like the Beckmann–Black problem and the existence of one parameter generic polynomials. We develop a general approach started in a preceding paper and provide new non-parametricity criteria and new examples.

On K-theoretic invariants of semigroup C*-algebras attached to number fields

We show that semigroup C*-algebras attached to ax+b-semigroups over rings of integers determine number fields up to arithmetic equivalence, under the assumption that the number fields have the same number of roots of unity. For finite Galois extensions, this means that the semigroup C*-algebras are isomorphic if and only if the number fields are isomorphic.

On the Characterization of Galois Extensions

We present a shortcut to the familiar characterizations of finite Galois extensions, based on an idea from an earlier Monthly note by Sonn and Zassenhaus.

Annihilation of Selmer groups for monomial CM-extensions

Let F/k be a finite Galois extension of number fields with Galois group G, and A an abelian variety over k. We fix an odd prime p. When G is isomorphic to the dihedral group of order 4p, assuming the Birch and Swinnerton–Dyer conjecture and some technical assumptions, we prove that the Pontryagin dual of the (classical) Selmer group Selp(AF)∨ is annihilated by some element in the center of the group ring Zp[G] which is constructed from equivariant L-values. More generally, we study the relation between the annihilation of Selmer groups for non-abelian extensions and that for abelian extensions.

Towards a Galoisian lnterpretation of Heisenberg lndeterminacy Principle

We revisit Heisenberg indeterminacy principle in the light of the Galois–Grothendieck theory for the case of finite abelian Galois extensions. In this restricted framework, the Galois–Grothendieck duality between finite K-algebras split by a Galois extension \(L\) and finite \(Gal(L{:}K)\)-sets can be reformulated as a Pontryagin duality between two abelian groups. We define a Galoisian quantum model in which the Heisenberg indeterminacy principle (formulated in terms of the notion of entropic indeterminacy) can be understood as a manifestation of a Galoisian duality: the larger the group of automorphisms \(H\subseteq G\) of the states in a G-set \({\mathcal {O}}\simeq G/H\), the smaller the “conjugate” algebra of observables that can be consistently evaluated on such states. Finally, we argue that states endowed with a group of automorphisms \(H\) can be interpreted as squeezed coherent states, i.e. as states that minimize the Heisenberg indeterminacy relations.

BASES NORMALES AUTODUALES ET GROUPES UNITAIRES EN CARACT ÉRISTIQUE 2

Let k be a field of characteristic 2, and let L/k be a finite Galois extension, with Galois group G. We show the equivalence of the following two properties:(∗) The group G is generated by elements of order 2 and by elements of odd order.(∗∗) There exists x ∈ L such that Tr(x) = 1 and Tr(x.g(x)) = 0 for every g ∈ G, g = 1.

A Quillen model for classical Morita theory and a tensor categorification of the Brauer group

Let K be a commutative ring. In this article we construct a well-behaved symmetric monoidal Quillen model structure on the category of small K-categories which enhances classical Morita theory. Making use of it, we then obtain the usual categorification of the Brauer group and of its functoriality. Finally, we prove that the (contravariant) corestriction map for finite Galois extensions also lifts to this categorification.

Weak crossed-product orders over valuation rings

Let F be a field, let V be a valuation ring of F of arbitrary Krull dimension (rank), let K be a finite Galois extension of F with group G, and let S be the integral closure of V in K. Let f:G×G↦K∖{0} be a normalized two-cocycle such that f(G×G)⊆S∖{0}, but we do not require that f should take values in the group of multiplicative units of S. One can construct a crossed-product V-order Af=∑σ∈GSxσ with multiplication given by xσsxτ=σ(s)f(σ,τ)xστ for s∈S, σ,τ∈G. We characterize semihereditary and Dubrovin crossed-product orders, under mild valuation-theoretic assumptions placed on the nature of the extension K/F.

Carmichael Meets Chebotarev

Abstract.For any finite Galois extension K of ℚ and any conjugacy class C in Gal(K/ℚ), we show that there exist infinitely many Carmichael numbers composed solely of primes for which the associated class of Frobenius automorphisms is C. This result implies that for every natural number n there are infinitely many Carmichael numbers of the form a2 + nb2 with a, b ∊ 핫.

Change of root numbers of elliptic curves under extension of scalars

In this paper we study how the root number attached to an elliptic curve E over a finite field extension K of Q 3 changes when E is considered as an elliptic curve over a finite Galois extension F of K via extension of scalars. The main result is a formula relating the root number W ( E / F ) attached to E × K F to the root number W ( E / K ) attached to E .

Congruences of models of elliptic curves

Let O_K be a discrete valuation ring with field of fractions K and perfect residue field. Let E be an elliptic curve over K, let L/K be a finite Galois extension and let O_L be the integral closure of O_K in L. Denote by X' the minimal regular model of E_L over O_L. We show that the special fibers of the minimal Weierstrass model and the minimal regular model of E over O_K are determined by the infinitesimal fiber X'_m together with the action of Gal(L/K), when m is big enough (depending on the minimal discriminant of E and the different of L/K).