In our previous paper we describe the Galois module structures of $p$th-power class groups $K^\times/{K^{\times p}}$, where $K/F$ is a cyclic extension of degree $p$ over a field $F$ containing a primitive $p$th root of unity. Our description relies upon arithmetic invariants associated with $K/F$. Here we construct field extensions $K/F$ with prescribed arithmetic invariants, thus completing our classification of Galois modules $K^{\times}/K^{\times p}$.
We investigate some Galois groups of linearized polynomials over fields such as $\mathbb{F}_q(t)$. The space of roots of such a polynomial is a module for its Galois group. We present a realization of the symmetric powers of this module, as a subspace of the splitting field of another linearized polynomial.
A Galois scaffold, in a Galois extension of local fields with perfect residue fields, is an adaptation of the normal basis to the valuation of the extension field, and thus can be applied to answer questions of Galois module structure. Here we give a sufficient condition for a Galois scaffold to exist in fully ramified Galois extensions of degree $p^2$ of characteristic $p$ local fields. This condition becomes necessary when we restrict to $p=3$. For extensions $L/K$ of degree $p^2$ that satisfy this condition, we determine the Galois module structure of the ring of integers by finding necessa…
Let $L/K$ be a finite Galois extension of local or global fields in characteristic $0$ or $p$ with nonabelian Galois group $G$, and let ${\mathfrak B}$ be a $G$-stable fractional ideal of $L$. We show that ${\mathfrak B}$ 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.
We say a tame Galois field extension $L/K$ with Galois group $G$ has trivial Galois module structure if the rings of integers have the property that $\Cal{O}_{L}$ is a free $\Cal{O}_{K}[G]$-module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes $l$ so that for each there is a tame Galois field extension of degree $l$ so that $L/K$ has nontrivial Galois module structure. However, the proof does not directly yield specific primes $l$ for a given algebraic number field $K.$ For…
Recently, much work has been done to investigate Galois module structure of local field extensions, particularly through the use of Galois scaffolds. Given a totally ramified $p$-extension of local fields $L/K$, a Galois Scaffold gives us a $K$-basis for $K[G]$ whose effect on the valuation of elements of $L$ is easy to determine. In 2013, N.P. Byott and G.G. Elder gave sufficient conditions for the existence of Galois scaffolds for cyclic extensions of degree $p^2$ in characteristic $p$. We take their work and adapt it to cyclic extensions of degree $p^2$ in characteristic $0$.
We construct various explicit Herr complexes that compute the Galois cohomology of a $p$-adic representation of the absolute Galois group of a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$, using the associated $(φ,τ)$-modules (defined by Xavier Caruso), instead of $(φ,Γ)$-modules. We also give an application to $p$-divisible groups.
The smallest non-abelian p-groups play a fundamental role in the theory of Galois p-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups --- as well as other closely related, larger p-groups --- occur as Galois groups over given base fields. We show further how the appearance of some Galois groups forces the appearance of other Galois groups in an interesting way.
The class-invariant homomorphism allows one to measure the Galois module structure of extensions obtained by dividing points on abelian varieties. In this paper, we consider the case when the abelian variety is the Jacobian of a Fermat curve. We give examples of torsion points whose associated Galois structure is trivial, as well as points of infinite order whose associated Galois structure is non-trivial.
The regular subgroup determining an induced Hopf Galois structure for a Galois extension $L/K$ is obtained as the direct product of the corresponding regular groups of the inducing subextensions. We describe here the associated Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. In order to deal with their associated orders we develop a general method to compute bases and free generators in terms of matrices coming from representation theory of Hopf modules. In the case of an induced Hopf Galois structure it …
This paper justifies an assertion in (Elder, Proc AMS 137 (2009), no 4, 1193--1203) that Galois scaffolds make the questions of Galois module structure tractable. Let $k$ be a perfect field of characteristic $p$ and let $K=k((T))$. For the class of characteristic $p$ elementary abelian $p$-extensions $L/K$ with Galois scaffolds described in mentioned paper, we give a necessary and sufficient condition for the valuation ring $\mathfrak{O}_L$ to be free over its associated order $\mathcal{A}_{L/K}$ in $K[\Gal(L/K)]$. Interestingly, this condition agrees with the condition found by Y. Miyata, con…
Given a quartic Galois extension $L/\mathbb{Q}$ of number fields and a Hopf-Galois structure $H$ on $L/\mathbb{Q}$, we study the freeness of the ring of integers $\mathcal{O}_L$ as module over the associated order $\mathfrak{A}_H$ in $H$. For the classical Galois structure $H_c$, we know by Leopoldt's theorem that $\mathcal{O}_L$ is $\mathfrak{A}_{H_c}$-free. If $L/\mathbb{Q}$ is cyclic, it admits a unique non-classical Hopf-Galois structure, whereas if it is biquadratic, it admits three such Hopf-Galois structures. In both cases, we obtain that freeness depends on the solvability in $\mathbb{…
For fields F of characteristic not p containing a primitive $p$th root of unity, we determine the Galois module structure of the group of $p$th-power classes of K for all cyclic extensions K/F of degree p.
Let $K$ be a number field and $G$ a finite abelian group. We study the asymptotic behaviour of the number of tamely ramified $G$-extensions of $K$ with ring of integers of fixed realisable class as a Galois module.
In the mid-1960s Borevic and Faddeev initiated the study of the Galois module structure of groups of pth-power classes of cyclic extensions K/F of pth-power degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F.
For a $p$-adic local field $K$ with perfect residue field, L. Herr constructed a complex which computes the Galois cohomology of a $p$-torsion representation of the absolute Galois group of $K$ by using the theory of $(Φ,Γ)$-modules. We shall generalize his work to the imperfect residue field (the residue field has a finite $p$-basis) case.
Let F denote an unramified extension of the cyclotomic extension of Q_p by (p^n)th roots of unity, for an odd prime p. We determine the conductors of those Kummer extensions of F of degree dividing p^n which are Galois over the maximal unramified subfield of F. This follows from a comparison of the Galois module structure of the multiplicative group of F with the unit filtration of F.
Let $p$ be a prime, and let $K$ be a finite extension of $\mathbf{Q}_p$, with absolute Galois group $\cal{G}_K$. Let $π$ be a uniformizer of $K$ and let $K_\infty$ be the Kummer extension obtained by adjoining to $K$ a system of compatible $p^n$-th roots of $π$, for all $n$, and let $L$ be the Galois closure of $K_\infty$. Using these extensions, Caruso has constructed étale $(φ,τ)$-modules, which classify $p$-adic Galois representations of $K$. In this paper, we use locally analytic vectors and theories of families of $φ$-modules over Robba rings to prove the overconvergence of $(φ,τ)$-module…
Let $K$ be any finite extension of $Q_{p}$, $L$ any finite Galois extension of $K$ and $E$ any finite large enough coefficient field containing $L$. We classify two-dimensional, F-semistable $E$-representations of $G_{K}$, by listing the isomorphism classes of rank two weakly admissible filtered $(ϕ,N,L/K,E)$-modules.
We prove three theorems concerning the Hopf-Galois module structure of fractional ideals in a finite tamely ramified extension of $ p $-adic fields or number fields which is $ H $-Galois for a commutative Hopf algebra $ H $. Firstly, we show that if $ L/K $ is a tame Galois extension of $ p $-adic fields then each fractional ideal of $ L $ is free over its associated order in $ H $. We also show that this conclusion remains valid if $ L/K $ is merely almost classically Galois. Finally, we show that if $ L/K $ is an abelian extension of number fields then every ambiguous fractional ideal of $ L…
Let p be a prime and F(p) the maximal p-extension of a field F containing a primitive p-th root of unity. We give a new characterization of Demuskin groups among Galois groups Gal(F(p)/F) when p=2, and, assuming the Elementary Type Conjecture, when p>2 as well. This characterization is in terms of the structure, as Galois modules, of the Galois cohomology of index p subgroups of Gal(F(p)/F).
To each Drinfeld module over a finitely generated field with generic characteristic, one can associate a Galois representation arising from the Galois action on its torsion points. Recent work of Pink and Rütsche has described the image of this representation up to commensurability. Their theorem is qualitative, and the objective of this paper is to complement this theory with a worked out example. In particular, we give examples of Drinfeld modules of rank 2 for which the Galois action on their torsion points is as large as possible. We will follow the approach that Serre used to give explici…
We determine the Galois module structure of the parameterizing space of elementary $p$-abelian extensions of a field $K$ when $\text{Gal}(K/F)$ is any finite $p$-group, under the assumption that the maximal pro-$p$ quotient of the absolute Galois group of $F$ is a free, finitely generated pro-$p$ group, and that $F$ contains a primitive $p$th root of unity if $\text{char}(F) \neq p$.
The classification of the local Galois representations using $(φ,Γ)$-modules by Fontaine has been generalized by Kisin and Ren over the Lubin-Tate extensions of local fields using the theory of $(φ_q,Γ_{LT})$-modules. In this paper, we extend the work of (Fontaine) Herr by introducing a complex which allows us to compute cohomology over the Lubin-Tate extensions and compare it with the Galois cohomology groups. We further extend that complex to include certain non-abelian extensions. We then deduce some relations of this cohomology with those arising from $(ψ_q,Γ_{LT})$-modules. We also comput…
In this paper, let $ϕ$ be the Drinfeld module over $\mathbb{F}_{q}(T)$ of prime rank $r$ defined by $$ϕ_T=T+τ^{r-1}+T^{q-1}τ^r.$$ We prove that under certain condition on $\mathbb{F}_q$, the adelic Galois representation $$ρ_ϕ:{\rm{Gal}}(\mathbb{F}_q(T)^{\rm{sep}}/\mathbb{F}_q(T))\longrightarrow \varprojlim_{\mathfrak{a}}{\rm{Aut}}(ϕ[\mathfrak{a}])\cong {\rm{GL_r}}(\widehat{A})$$ is surjective.
Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{Z}/p^m\mathbb{Z}[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^m}$. With at most one exception, each indecomposable summand is cyclic and free over some quotient group of $\text{Gal}(K/F)$. For fixed values of $m$ and $n$, there are only finitely many possible isomorphism classes for the non-free indecomposable summand. These Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation provid…
We compare two approaches to the study of Galois module structures: on the one hand factor equivalence, a technique that has been used by Fröhlich and others to investigate the Galois module structure of rings of integers of number fields and of their unit groups, and on the other hand regulator constants, a set of invariants attached to integral group representations by Dokchitser and Dokchitser, and used by the author, among others, to study Galois module structures. We show that the two approaches are in fact closely related, and interpret results arising from these two approaches in terms …
Frobenius Traces for Rank-2 Drinfeld Modules, Higher-Dimensional Galois Representations, and a Strong Multiplicity One Theorem in Positive Characteristic
We study the groups in the unit filtration of a finite abelian extension K of the field of p-adic numbers. We determine explicit generators of these groups as modules over the pro-p group ring of the Galois group of K over the p-adic numbers. We work in eigenspaces for powers of the Teichmueller character, first at the level of the field of norms for the extension of K by p-power roots of unity and then at the level of K.
For a Galois extension $K/F$ with $\text{char}(K)\neq 2$ and $\text{Gal}(K/F) \simeq \mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$, we determine the $\mathbb{F}_2[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times 2}$. Although there are an infinite number of (pairwise non-isomorphic) indecomposable $\mathbb{F}_2[\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}]$-modules, our decomposition includes at most $9$ indecomposable types. This paper marks the first time that the Galois module structure of power classes of a field has been fully determined when the modular representatio…
Let $p$ be an odd prime. For field extensions $L/\mathbb{Q}_p$ with Galois group isomorphic to the dihedral group $D_{2p}$ of order $2p$, we consider the problem of computing a basis of the associated order in each Hopf Galois structure and the module structure of the ring of integers $\mathcal{O}_L$. We solve the case in which $L/\mathbb{Q}_p$ is not totally ramified and present a practical method which provides a complete answer for the cases $p=3$ and $p=5$. We see that within this family of dihedral extensions, the ring of integers is always free over the associated orders in the different…
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.
Suppose $K$ is unramified over $\mathbb Q _p$ and $Γ_K=\operatorname{Gal}(\bar K/K)$. Let $H$ be a torsion $Γ_K$-equivariant subquotient of crystalline $\mathbb Q _p[Γ_K]$-module with HT weights from $[0,p-2]$. We give a new proof of Fontaine's conjecture about the triviality of action of some ramification subgroups $Γ_K^{(v)}$ on $H$. The earlier author's proof from [1] contains a gap and proves this conjecture only for some subgroups of index $p$ in $Γ_K^{(v)}$.
For given rational prime number $p$ consider the tower of finite extensions of fields $K_0/\mathbb{Q}_p,$ $K/K_0, L/K, M/L$, where $K/K_0$ is unramified and $M/L$ is a Galois extension with Galois group $G$. Suppose one dimensional Honda formal group over the ring $\mathcal{O}_K$, relative to the extension $K/K_0$ and uniformizer $π\in K_0$ is given. The operation $x\underset{F}+y=F(x,y)$ sets a new structure of abelian group on the maximal ideal $\mathfrak{p}_M$ of the ring $\mathcal{O}_M$ which we will denote by $F(\mathfrak{p}_M)$. In this paper the structure of $F(\mathfrak{p}_M)$ as $\mat…
We use new over-convergent p-adic exponential power series, inspired by work of Pulita, to build self-dual normal basis generators for the square root of the inverse different of certain abelian weakly ramified extensions of an unramified extension K of Qp. These extensions, whose set we denote by M, are the degree p subextensions over K of Mp,2, the maximal abelian totally, wildly and weakly ramified extension of K, whose norm group contains p. Our construction follows Pickett's, who dealt with the same set M of extensions of K, but does not depend on the choice of a basis of the residue fiel…
We investigate the relation between p-adic Galois representations and overconvergent (phi,Gamma)-modules in families. Especially we construct a natural open subspace of a family of (phi,Gamma)-modules, over which it is induced by a family of Galois-representations.
Let $F$ be a number field with ring of integers $O_F$ and let $G$ be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group $Cl(O_FG)$ of $O_FG$ that involves applying the work of the second-named author in the context of relative algebraic $K$ theory. When $G$ is of odd order, we show (subject to certain conditions) that the set of realisable classes is a subgroup of $Cl(O_FG)$. This may be viewed as being a partial analogue of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups in the setting of Ga…
Explicit descriptions of local integral Galois module generators in certain extensions of $p$-adic fields due to Pickett have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of number fields. In parallel, Pulita has generalised the theory of Dwork's power series to a set of power series with coefficients in Lubin-Tate extensions of $\Q_p$ to establish a structure theorem for rank one solvable p-adic differential equations. In this paper we first generalise Pulita's power series using the theories of formal group exponent…
Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and $G$ a finite group of odd order. If $K_h$ is a weakly ramified $G$-Galois $K$-algebra, then its square root $A_h$ of the inverse different is a locally free $\mathcal{O}_{K}G$-module and hence determines a class in the locally free class group $\mbox{Cl}(\mathcal{O}_KG)$ of $\mathcal{O}_KG$. We show that for $G$ abelian and under suitable assumptions, the set of all such classes is a subgroup of $\mbox{Cl}(\mathcal{O}_KG)$.
Suppose we are given a Drinfeld Module $ϕ$ over $\mathbb{F}_q(t)$ of rank $r$ and a prime ideal $\mathfrak{l}$ of $\mathbb{F}_q[T]$. In this paper, we prove that the reducibility of mod $\mathfrak{l}$ Galois representation $${\rm{Gal}}(\mathbb{F}_q(T)^{\rm{sep}}/\mathbb{F}_q(T))\rightarrow {\rm{Aut}}(ϕ[\mathfrak{l}])\cong {\rm{GL}}_r(\mathbb{F}_\mathfrak{l})$$ gives a bound on the degree of $\mathfrak{l}$ which depends only on the rank $r$ of Drinfeld module $ϕ$ and the minimal degree of place $\mathcal{P}$ where $ϕ$ has good reduction at $\mathcal{P}$. Then, we apply this reducibility bound t…
In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal $\fl$ of $\F_q[T]$, the question essentially asks whether, up to isogeny, a Drinfeld module $ϕ$ over $\F_q(T)$ contains a rational $\fl$-torsion point if the reduction of $ϕ$ at almost all primes of $\F_q[T]$ contains a rational $\fl$-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is $2$, but negative if the rank…
Let $ϕ$ be a rank $r$ Drinfeld $\BF_q[T]$-module determined by $ϕ_T(X) = TX+g_1X^q+...+g_{r-1}X^{q^{r-1}}+X^{q^r}$, where $g_1,...,g_{r-1}$ are algebraically independent over $\BF_q(T)$. Let $N\in\BF_q[T]$ be a polynomial, and $k/\BF_q$ an algebraic extension. We show that the Galois group of $ϕ_N(X)$ over $k(T,g_1,...,g_{r-1})$ is isomorphic to $\GL_r(\BF_q[T]/N\BF_q[T])$, settling a conjecture of Abhyankar.
The power classes of a field are well-known for their ability to parameterize elementary $p$-abelian Galois extensions. These classical objects have recently been reexamined through the lens of their Galois module structure. Module decompositions have been computed in several cases, providing deep new insight into absolute Galois groups. The surprising result in each case is that there are far fewer isomorphism types of indecomposables than one would expect generically, with summands predominately free over associated quotient rings. Though non-free summands are the exception both in their for…
Let $L/K$ be a finite, totally ramified $p$-extension of complete local fields with residue fields of characteristic $p > 0$, and let $A$ be a $K$-algebra acting on $L$. We define the concept of an $A$-scaffold on $L$, thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where $L/K$ was Galois and $A=K[G]$ for $G=\mathrm{Gal}(L/K)$. When a suitable $A$-scaffold exists, we show how to answer questions generalizing those of classical integral Galois module theory. We give a necessary and sufficient condition, involving only numerical parameters, f…
Let $p$ be prime. Let $L/K$ be a finite, totally ramified, purely inseparable extension of local fields, $\left[ L:K\right] =p^{n},\;n\geq2.$ It is known that $L/K$ is Hopf Galois for numerous Hopf algebras $H,$ each of which can act on the extension in numerous ways. For a certain collection of such $H$ we construct "Hopf Galois scaffolds" which allow us to obtain a Hopf analogue to the Normal Basis Theorem for $L/K.$ The existence of a scaffold structure depends on the chosen action of $H$ on $L.$ We apply the theory of scaffolds to describe when the fractional ideals of $L$ are free over th…
Let $N/F$ be an odd degree Galois extension of number fields with Galois group $G$ and rings of integers ${\mathfrak O}_N$ and ${\mathfrak O}_F=\bo$ respectively. Let $\mathcal{A}$ be the unique fractional ${\mathfrak O}_N$-ideal with square equal to the inverse different of $N/F$. Erez has shown that $\mathcal{A}$ is a locally free ${\mathfrak O}[G]$-module if and only if $N/F$ is a so called weakly ramified extension. There have been a number of results regarding the freeness of $\mathcal{A}$ as a $\Z[G]$-module, however this question remains open. In this paper we prove that $\mathcal{A}$ i…
Pink has given a qualitative answer to the Mumford-Tate conjecture for Drinfeld modules in the 90s. He showed that the image of the v-adic Galois representation is v-adically open in the motivic Galois group for any prime v. In contrast to this result, we provide a family of uniformizable Anderson t-modules for which the Galois representations of their t-adic Tate-modules are "far from" having t-adically open image in their motivic Galois groups. Nevertheless, the image is still Zariski-dense in the motivic Galois group which is in accordance to the Mumford-Tate conjecture. For the proof, we e…
