Galois Extension Of Number Fields Research Articles

The Ostrowski quotient of an elliptic curve

For [Formula: see text] a finite Galois extension of number fields, the relative Pólya group [Formula: see text] is the subgroup of the ideal class group of [Formula: see text] generated by all the strongly ambiguous ideal classes in [Formula: see text]. The notion of Ostrowski quotient [Formula: see text], as the cokernel of the capitulation map into [Formula: see text], has been recently introduced in [E. Shahoseini, A. Rajaei and A. Maarefparvar, Ostrowski quotients for finite extensions of number fields, Pacific J. Math. 321 (2022) 415–429, doi: 10.2140/pjm.2022.321.415]. In this paper, using some results of [C. D. González-Avilés, Capitulation, ambiguous classes and the cohomology of the units, J. Reine Angew. Math. 613 (2007) 75–97], we find a new approach to define [Formula: see text] and [Formula: see text] which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For [Formula: see text] an elliptic curve defined over [Formula: see text], we define the Ostrowski quotient [Formula: see text] and the coarse Ostrowski quotient [Formula: see text] of [Formula: see text] relative to [Formula: see text], for which in the latter group we do not take into account primes of bad reduction. Our main result is a nontrivial structure theorem for the group [Formula: see text] and we analyze this theorem, in some details, for the class of curves [Formula: see text] over quadratic extensions [Formula: see text].

A supplement to Chebotarev’s density theorem

Let L/K be a Galois extension of number fields with Galois group G. We show that if the density of prime ideals in K that split totally in L tends to 1/∣G∣ with a power saving error term, then the density of prime ideals in K whose Frobenius is a given conjugacy class C ⊂ G tends to ∣C∣/∣G∣ with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros of ζL(s)/ζK(s).

Ostrowski quotients for finite extensions of number fields

For $L/K$ a finite Galois extension of number fields, the relative P\'olya group $\Po(L/K)$ coincides with the group of strongly ambiguous ideal classes in $L/K$. In this paper, using a well known exact sequence related to $\Po(L/K)$, in the works of Brumer-Rosen and Zantema, we find short proofs for some classical results in the literatur. Then we define the ``Ostrowski quotient'' $\Ost(L/K)$ as the cokernel of the capitulation map into $\Po(L/K)$, and generalize some known results for $\Po(L/\mathbb{Q})$ to $\Ost(L/K)$.

Mertens’ theorem for Chebotarev sets

We generalize Mertens’ product theorem to Chebotarev sets of prime ideals in Galois extensions of number fields. Using work of Rosen, we extend an argument of Williams from cyclotomic extensions to this more general case. Additionally, we compute these products for Cheboratev sets in abelian extensions, [Formula: see text] sextic extensions, and sets of primes represented by some quadratic forms.

Campana points and powerful values of norm forms

We give an asymptotic formula for the number of weak Campana points of bounded height on a family of orbifolds associated to norm forms for Galois extensions of number fields. From this formula we derive an asymptotic for the number of elements with m-full norm over a given Galois extension of mathbb {Q}. We also provide an asymptotic for Campana points on these orbifolds which illustrates the vast difference between the two notions, and we compare this to the Manin-type conjecture of Pieropan, Smeets, Tanimoto and Várilly-Alvarado.

Multiplicities in Selmer groups and root numbers of Artin twists

Let K/F be a finite Galois extension of number fields and let σ be an absolutely irreducible, self-dual, complex valued representation of Gal(K/F). Let p be an odd prime and consider two elliptic curves E1,E2 defined over Q with good, ordinary reduction at primes above p and equivalent mod-p Galois representations. In this article, we study the variation of the parity of the multiplicities of σ in the representation space associated to the p∞-Selmer groups of E1 and E2 over K. We also compare the root numbers for the twists of E1 and E2 over F by σ and show that the p-parity conjecture holds for the twist of E1/F by σ if and only if it holds for the twist of E2/F by σ. We also express Mazur-Rubin-Nekovář's arithmetic local constants in terms of certain local Iwasawa invariants.

Constructing totally p-adic numbers of small height

Bombieri and Zannier gave an effective construction of algebraic numbers of small height inside the maximal Galois extension of the rationals which is totally split at a given finite set of prime numbers. They proved, in particular, an explicit upper bound for the lim inf of the height of elements in such fields. We generalize their result in an effective way to maximal Galois extensions of number fields with given local behavior at finitely many places.

A genus formula for the positive étale wild kernel

Let F be a number field and let i≥2 be an integer. In this paper, we study the positive étale wild kernel WK2i−2ét,+F, which is the twisted analogue of the 2-primary part of the narrow class group. If E/F is a Galois extension of number fields with Galois group G, we prove a genus formula relating the order of the groups (WK2i−2ét,+E)G and WK2i−2ét,+F.

On the self-duality of rings of integers in tame and abelian extensions

Let $L/K$ be a tame and Galois extension of number fields with group $G$. It is well-known that any ambiguous ideal in $L$ is locally free over $\mathcal{O}_KG$ (of rank one), and so it defines a class in the locally free class group of $\mathcal{O}_KG$, where $\mathcal{O}_K$ denotes the ring of integers of $K$. In this paper, we shall study the relationship among the classes arising from the ring of integers $\mathcal{O}_L$ of $L$, the inverse different $\mathfrak{D}_{L/K}^{-1}$ of $L/K$, and the square root of the inverse different $A_{L/K}$ of $L/K$ (if it exists), in the case that $G$ is abelian. They are naturally related because $A_{L/K}^2 = \mathfrak{D}_{L/K}^{-1} = \mathcal{O}_L^*$, and $A_{L/K}$ is special because $A_{L/K} = A_{L/K}^*$, where $*$ denotes dual with respect to the trace of $L/K$.

Composantes isotypiques de pro-$p$-extensions de corps de nombres et $p$-rationalite

Soit p un nombre premier et soit K/k une extension galoisienne finie de corps de nombres de groupe de Galois ∆ d’ordre etranger a p. Soit S un ensemble fini de places finies de k contenant l’ensemble S p des places p-adiques et soit KS la pro-p-extension maximale de K non-ramifiee en dehors de S. Soi tG:“GS{Hun quotient de GS:“Gal KS/K) sur lequel ∆ agit trivialement. Posons X:“H{rH, Hs. Dans ce travail,nous etudions la φ-composante X φ de X pour les caracteres Qp-irreductibles φ de ∆ et nous montrons entre autres, sous la conjecture de Leopoldt et pour tout caractere non trivial φ, que X φ est ZpvGw-libre si et seulement si la φ-composante de la Zp-torsion de Gs /{Gs, Gs est triviale. Nous faisons egalement une etude numerique sur la liberte de Xφ dans des extensions cycliques K{Qde degre 3 et de degre 4 (`a partir de familles de polynomes donnees par Balady, Lecacheux et plus recemment par Balady et Washington),et ainsi que dans des extensions diedrales de degre 6 de Q. Les resultats de nos calculs renforcent une recente conjecture de Gras

Annihilating Wild Kernels

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Let $p$ be an odd prime and $r>1$ be an integer. Assuming a conjecture of Schneider, we formulate a conjecture that relates special values of equivariant Artin $L$-series at $s=r$ to the compact support cohomology of the etale $p$-adic sheaf $\mathbb Z_p(r)$. We show that our conjecture is essentially equivalent to the $p$-part of the equivariant Tamagawa number conjecture for the pair $(h^0(\mathrm{Spec}(L))(r), \mathbb Z[G])$. We derive from this explicit constraints on the Galois module structure of Banaszak's $p$-adic wild kernels.

Formulas for Chebotarev densities of Galois extensions of number fields

We generalize the Chebotarev density formulas of Dawsey (Res Number Theory 3:27, 2017) and Alladi (J Number Theory 9:436–451, 1977) to the setting of arbitrary finite Galois extensions of number fields L / K. In particular, if \(C \subset G = {{\mathrm{Gal}}}(L/K)\) is a conjugacy class, then we establish that the Chebotarev density is the following limit of partial sums of ideals of K: $$\begin{aligned} -\lim _{X\rightarrow \infty } \sum _{\begin{array}{c} 2\le N(I)\le X I \in S(L/K; C) \end{array}} \frac{\mu _K(I)}{N(I)} = \frac{|C|}{|G|}, \end{aligned}$$ where \(\mu _K(I)\) denotes the generalized Mobius function and S(L / K; C) is the set of ideals \(I\subset \mathcal {O}_K\) such that I has a unique prime divisor \(\mathfrak {p}\) of minimal norm and the Artin symbol \(\left[ \frac{L/K}{\mathfrak {p}}\right] \) is C. To obtain this formula, we generalize several results from classical analytic number theory, as well as Alladi’s concept of duality for minimal and maximal prime divisors, to the setting of ideals in number fields.

Ultrasoluble coverings of some nilpotent groups by a cyclic group over number fields and related questions

Let be a finite nilpotent group of odd order. For every finite cyclic subgroup of odd order we find necessary and sufficient conditions for a class to determine an ultrasoluble extension (under the additional assumption of minimality of all -Sylow subextensions to the extension with class for all non-Abelian -Sylow subgroups of ), that is, for the existence of a Galois extension of number fields with group such that the corresponding embedding problem is ultrasoluble (it has solutions and all such solutions are fields). We also establish a number of related results.

A note on Galois groups and local degrees

In this paper, we consider infinite Galois extensions of number fields and study the relation between their local degrees and the structure of their Galois groups. It is known that, if K is a number field and L / K is an infinite Galois extension of group G, then the local degrees of L are uniformly bounded at all rational primes if and only if G has finite exponent. In this note we show that the non uniform boundedness of the local degrees is not equivalent to any group theoretical property. More precisely, we exhibit several groups that admit two different realisations over a given number field, one with bounded local degrees at a given set of primes and one with infinite local degrees at the same primes.

Densities of primes and realization of local extensions

In this paper we introduce new densities on the set of primes of a number field. If $K/K_0$ is a Galois extension of number fields, we associate to any element $x \in {\rm Gal}_{K/K_0}$ a density $\delta_{K/K_0,x}$ on primes of $K$. In particular, the density associated to $x = 1$ is the usual Dirichlet density on $K$. After establishing some properties of these densities, we use them to show that the maximal solvable extension of a number field unramified outside an almost Chebotarev set realize the maximal local extension at each prime lying outside this set.

Langlands reciprocity for certain Galois extensions

In this note, we study Artin's conjecture via group theory and derive Langlands reciprocity for certain solvable Galois extensions of number fields, which extends the previous work of Arthur and Clozel. In particular, we show that all nearly nilpotent groups and all groups of order less than 60 are of automorphic type.

A variant of the Bombieri–Vinogradov theorem in short intervals and some questions of Serre

AbstractWe generalise the classical Bombieri–Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extensionL/$\mathbb{Q}$exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modularL-functionL(s, f), the fundamental discriminantsdfor which thed-quadratic twist ofL(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals.

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.

Heilbronn characters

In a seminal paper in 1972 Hans Heilbronn introduced a virtual character associated to representations of Galois extensions of number fields and Artin’s Conjecture on the holomorphy of $L$-series. His construction has evolved in both application and scope, and may now be applied to produce what are called Heilbronn characters of arbitrary finite groups. This article surveys the inception and development of this concept, weaving together its number-theoretic and group-theoretic dimensions, and culminates in a description of the recent classification of unfaithful minimal Heilbronn characters. Connections with other areas of mathematics, variations on these themes, and possible future directions are also explored.

On the existence of large degree Galois representations for fields of small discriminant

Let $L/K$ be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of ${\rm Gal}(L/K)$, the sharper of which is conditional on the Artin Conjecture and the Generalized Riemann Hypothesis. Our bound is nontrivial when $[K : \mathbb{Q}]$ is small and $L$ has small root discriminant, and might be summarized as saying that such fields can't be "too abelian."