Let $n>1$, $e\geq 0$ and a prime number $p\geq 2^{n+2+2e}+3$, such that the index of regularity of $p$ is $\leq e$. We show that there are infinitely many irreducible Galois representations $ρ: Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow {GL}_n(\mathbb{Q}_p)$ unramified at all primes $l\neq p$. Furthermore, these representations are shown to have image containing a fixed finite index subgroup of ${SL}_n(\mathbb{Z}_p)$. Such representations are constructed by lifting suitable residual representations $\barρ$ with image in the diagonal torus in ${GL}_n(\mathbb{F}_p)$, for which the global deformation problem is unobstructed.
We prove that the density of ramified primes in semisimple p-adic representations of Galois groups of number fields is 0. Ravi Ramakrishna has produced examples of such representations that are infinitely ramified.
This paper gives some restrictions on finite groups that can occur as Galois groups of extensions over $\Q$ and over $\F_q(t)$ ramified only at one finite prime. Over $\Q$, we strengthen results of Jensen and Yui about dihedral extensions and rule out some non-solvable groups. Over $\F_q(t)$ restrictions are given for symmetric groups and dihedral groups to occur as tamely ramified extension over $\F_q(t)$ ramified only at one prime.
We show that an infinite family of odd complex 2-dimensional Galois representations ramified at 5 having nonsolvable projective image are modular, thereby verifying Artin's conjecture for a new case of examples. Such a family contains the original example studied by Buhler. In the process, we prove that an infinite family of residually modular Galois representations are modular by studying $Λ$-adic Hecke algebras.
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 specialize various three-point covers to find number fields with Galois group $M_{12}$, $M_{12}.2$, $2.M_{12}$, or $2.M_{12}.2$ and light ramification in various senses. One of our $2.M_{12}.2$ fields has the unusual property that it is ramified only at the single prime $11$.
Bley, Burns and Hahn used relative algebraic $K$-theory methods to formulate a precise conjectural link between the (second Adams-operator twisted) Galois-Gauss sums of weakly ramified Artin characters and the square root of the inverse different of finite, odd degree, Galois extensions of number fields. We provide concrete new evidence for this conjecture in the setting of extensions of odd prime-power degree by using a refined version of a well-known result of Ullom.
We study the number of ramified prime numbers in finite Galois extensions of $\mathbb{Q}$ obtained by specializing a finite Galois extension of $\mathbb{Q}(T)$. Our main result is a central limit theorem for this number. We also give some Galois theoretical applications.
In the local, unramified case the determinantal functions associated to the group-ring of a finite group satisfy Galois descent. This note examines the obstructions to Galois determinantal descent in the ramified case.
In this paper we show how to construct, for most p >= 5, two types of surjective representations ρ:G_Q=Gal(\bar{Q}/Q) -> GL_2(Z_p) that are ramified at an infinite number of primes. The image of inertia at almost all of these primes will be torsion-free. The first construction is unconditional. The catch is that we cannot say whether ρ|_{G_p=Gal(\bar{Q_p}/Q_p) is crystalline or even potentially semistable. The second construction assumes the Generalized Riemann Hypothesis (GRH). With this assumption we can further arrange that ρ|_{G_p} is crystalline at p. We remark that infinitely ramified *r…
Let L/K be a finite Galois extension of complete local fields with finite residue fields and let G=Gal(L/K). Let G_1 and G_2 be the first and second ramification groups. Thus L/K is tamely ramified when G_1 is trivial and we say that L/K is weakly ramified when G_2 is trivial. Let O_L be the valuation ring of L and let P_L be its maximal ideal. We show that if L/K is weakly ramified and n is congruent to 1 mod |G_1| then P_L^n is free over the group ring O_K[G], and we construct an explicit generating element. Under the additional assumption that L/K is wildly ramified, we then show that every…
Let $p>5$ be a prime integer and $K/\mathbb{Q}_p$ a finite ramified extension with ring of integers $\mathcal{O}$ and uniformizer $π$. Let $n>1$ be a positive integer and $ρ_n:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O}/π^n)$ be a continuous Galois representation. In this article we prove that under some technical hypotheses the representation $ρ_n$ can be lifted to a representation $ρ:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O})$. Furthermore, we can pick the lift restriction to inertia at any finite set of primes (at the cost of allowing some extra ramification) and get a deformation problem whose…
Given a totally real number field $F$ and a mod $p$ Galois representation $ρ\colon G_F\to \mathrm{GL}_2(\bar{\mathbf{F}}_p)$, we propose an explicit definition of the set of Serre weights $W(ρ)$ attached to $ρ$. We prove that our explicit definition is equivalent to previous definitions available in the literature. As a consequence we obtain an explicit Serre's modularity conjecture for Hilbert modular forms over totally real number fields. Our work generalises previous work of Dembélé--Diamond--Roberts and Calegari--Emerton--Gee--Mavrides which together give explicit and equivalent sets of we…
Let $ K $ be a number field and let $ L/K $ be a tamely ramified radical extension of prime degree $ p $. If $ K $ contains a primitive $ p^{th} $ root of unity then $ L/K $ is a cyclic Kummer extension; in this case the group algebra $ K[G] $ (with $ G=\mbox{Gal}(L/K) $) gives the unique Hopf-Galois structure on $ L/K $, the ring of algebraic integers $ \mathfrak{O}_L $ is locally free over $ \mathfrak{O}_{K}[G] $ by Noether's theorem, and Gómez Ayala has determined a criterion for $ \mathfrak{O}_L $ to be a free $ \mathfrak{O}_{K}[G] $-module. If $ K $ does not contain a primitive $ p^{th} $…
Given a Galois cover of curves X to Y with Galois group G which is totally ramified at a point x and unramified elsewhere, restriction to the punctured formal neighborhood of x induces a Galois extension of Laurent series rings k((u))/k((t)). If we fix a base curve Y , we can ask when a Galois extension of Laurent series rings comes from a global cover of Y in this way. Harbater proved that over a separably closed field, this local-to-global principle holds for any base curve if G is a p-group, and gave a condition for the uniqueness of such an extension. Using a generalization of Artin-Schrei…
We construct infinitely ramified Galois representations $ρ$ such that the $a_l (ρ)$'s have distributions in contrast to the statements of Sato-Tate, Lang-Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalising a result of Ramakrishna.
In this paper, we show the existence of a non-solvable Galois extension of $\Q$ which is unramified outside 2. The extension $K$ we construct has degree $2251731094732800=2^{19}(3\cdot 5\cdot 17\cdot 257)^2$ and has root discriminant $δ_K <2^{47/8}=58.68...$, and is totally complex.
We compute equivariant Euler characteristics of locally free sheaves on curves, thereby generalizing several results of Kani and Nakajima. For instance, we extend Kani's computation of the Galois module structure of the space of global meromorphic differentials which are logarithmic along the ramification locus from the tamely ramified to the weakly ramified case.
Given a Galois cover of curves over $\mathbb{F}_p$, we relate the $p$-adic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result of Chinburg from the tamely to the weakly ramified case. We furthermore apply Chinburg's result to obtain a `weak' relation in the general case. In the Appendix, we study, in this arbitrarily wildly ramified case, the integrality of $p$-adic valuations of epsilon constants.
We examine whether it is possible to realize finite groups $G$ as Galois groups of minimally tamely ramified extensions of $\mathbb{Q}$ and also specify both the inertia groups and the further decomposition of the ramified primes.
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…
The Steinitz class of a number field extension K/k is an ideal class in the ring of integers O_k of k, which, together with the degree [K:k] of the extension determines the O_k-module structure of O_K. We call rt(k,G) the classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A'-groups inductively, starting by abelian groups and then considering semidirect products of A'-groups with abelian groups of relatively prime order and direc…
Cyclic, ramified extensions $L/K$ of degree $p$ of local fields with residue characteristic $p$ are fairly well understood. Unless $\mbox{char}(K)=0$ and $L=K(\sqrt[p]{π_K})$ for some prime element $π_K\in K$, they are defined by an Artin-Schreier equation. Additionally, through the work of Ferton, Aiba, de Smit and Thomas, and others, much is known about their Galois module structure of ideals, the structure of each ideal $\mathfrak{P}_L^n$ as a module over its associated order $\mathfrak{A}_{K[G]}(n)=\{x\in K[G]:x\mathfrak{P}_L^n\subseteq \mathfrak{P}_L^n\}$ where $G=\mbox{Gal}(L/K)$. This p…
We prove a large finite field version of the Boston--Markin conjecture on counting Galois extensions of the rational function field with a given Galois group and the smallest possible number of ramified primes. Our proof involves a study of structure groups of (direct products of) racks.
Let $G$ be a split reductive group with $\dim Z(G) \leq 1$. We show that for any prime $p$ that is large enough relative to $G$, there is a finitely ramified Galois representation $ρ\colon Γ_{\mathbb Q} \to G(\mathbb Z_p)$ with open image. We also show that for any given integer $e$, if the index of irregularity of $p$ is at most $e$ and if $p$ is large enough relative to $G$ and $e$, then there is a Galois representation $Γ_{\mathbb Q} \to G(\mathbb Z_p)$ ramified only at $p$ with open image, generalizing a theorem of A. Ray. The first type of Galois representation is constructed by lifting a…
We study wildly ramified G-Galois covers $ϕ:Y \to X$ branched at B (defined over an algebraically closed field of characteristic p). We show that curves Y of arbitrarily high genus occur for such covers even when G, X, B and the inertia groups are fixed. The proof relies on a Galois action on covers of germs of curves and formal patching. As a corollary, we prove that for any nontrivial quasi-p group G and for any sufficiently large integer $σ$ with $p \nmid σ$, there exists a G-Galois étale cover of the affine line with conductor $σ$ above the point $\infty$.
Typos in the abstract have been corrected. Let $ρ_n$ be an ordinary weight two representation of absolute Galois group of the rationals to $GL_2(\mathcal O/π^n)$. Here $\mathcal O$ is a ramified DVR with uniformiser $π$. If $ρ_n$ satisfies mild hypotheses we lift it to a characteristic zero $\mathcal O$-valued geometric weight two representation. The earlier methods could handle only the unramified case. We show that the deformation ring of a residual representation arising from a newform can be arranged to be a prescribed DVR provided we choose a suitable auxiliary level. We extend earlier pr…
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…
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.
We use Galois cohomology methods to produce optimal mod $p^d$ level lowering congruences to a $p$-adic Galois representation that we construct as a well chosen lift of a given residual mod $p$ representation. Using our explicit Galois cohomology methods, we construct for a reductive group $G$ and a given residual representation $\barρ: Γ_F \to G(k)$, ramified at a finite set of primes $S$, in favorable conditions that we identify, a finite set of lifts $ρ$, $\{ρ^q\}$ of $\barρ$ to $G(W(k))$ with the following properties: $ρ: Γ_F \to G(W(k))$ is ramified precisely at $S \cup Q$, with $Q$ a fini…
Let $K$ be a local field and let $L/K$ be a totally ramified Galois extension of degree $p^n$. Being semistable and possessing a Galois scaffold are two conditions which facilitate the computation of the additive Galois module structure of $L/K$. In this note we show that $L/K$ is semistable if and only if $L/K$ has a Galois scaffold. We also give sufficient conditions in terms of Galois scaffolds for the extension $L/K$ to be stable.
In this paper we show that two dimensional (mod p) Galois representations satisfying mild hypotheses can be lifted to p-adic Galois representations ramified at infinitely many primes such that the characteristic polynomials of Frobenius at a density one set of unramified primes are defined over the rational integers. In particular we show that often one can "density one compatibly" lift mod p and mod q Galois representations.
We study the possible weights of an irreducible 2-dimensional modular mod p representation of the absolute Galois group of F, where F is a totally real field which is totally ramified at p, and the representation is tamely ramified at the prime above p. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions.
Let F be a totally real field and p an odd prime. If r is a continuous, semisimple, totally odd mod p representation of the absolute Galois group of F which is tamely ramified at all places of F dividing p, then we formulate a conjecture specifying the weights for which r is modular. This extends the conjecture of Diamond, Buzzard, and Jarvis, which supposed that p was unramified in F. We also prove a theorem towards the conjecture and provide some computational evidence.
Let $G$ be a finite nilpotent group and $K$ a number field with torsion relatively prime to the order of $G$. By a sequence of central group extensions with cyclic kernel we obtain an upper bound for the minimum number of prime ideals of $K$ ramified in a Galois extension of $K$ with Galois group isomorphic to $G$. This sharpens and extends results of Geyer and Jarden and of Plans. Also we confirm Boston's conjecture on the minimum number of ramified primes for a family of central extensions by the Schur multiplicator.
Let $G$ be a finite group and let $ram^{t}(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote the minimal non negative integer for which there exists a subset $X$ of $G$ with $d(G)$ elements such that the normal subgroup of $G$ generated by $X$ is all of $G$. It is known that $d(G)\leq ram^{t}(G)$. However, it is unknown whether or not every finite group $G$ can be realized as a Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly $…
The inverse Galois problem is concerned with finding a Galois extension of a field $K$ with given Galois group. In this paper we consider the particular case where the base field is $K=\F_p(t)$. We give a conjectural formula for the minimal number of primes, both finite and infinite, ramified in $G$-extensions of $K$, and give theoretical and computational proofs for many cases of this conjecture.
Given a reducible Galois representation $\overlineρ: G_{\mathbb{Q}} \rightarrow GL_2( \mathbb{F}_q)$ we show there exists an irreducible deformation $ρ: G_{\mathbb{Q}} \rightarrow GL_2 (\mathbb{W} [[T_1, T_2,.., T_r,....,]])$ of $\overlineρ$ ramified at infinitely many primes, where $\mathbb{W}$ denotes the ring of Witt vectors of $\mathbb{F}_q$. This is a modification of Ramakrishna's result for the irreducible case.
Let p be an odd prime. We consider the cyclotomic extension T := Z_(p)[zeta_{p^2}] of S := Z_(p), with galois group G := (Z/p^2)^*. Since this extension is wildly ramified, the SG-module T is not projective. We calculate its cohomology ring H^*(G, T; S), carrying the cup product induced by the ring structure of T. Proceeding in a somewhat greater generality, our results also apply to certain Lubin-Tate extensions.
We give a simple characterization of the totally wild ramified valuations in a Galois extension of fields of characteristic p. This criterion involves the valuations of Artin-Schreier cosets of the F_{p^r}^\times-translation of a single element. We apply the criterion to construct some interesting examples.
We sketch a method to compute mod $\ell$ Galois representations contained in the H2 étale of surfaces. We apply this method to the case of a representation with values in GL(3,9) attached to an eigenform over a congruence subgroup of SL(3). We obtain in particular a polynomial with Galois group isomorphic to the simple group PSU(3,9) and ramified at 2 and 3 only.
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…
For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long standing Q-admissibility conjecture. We extend a theorem of Neukirch on embedding problems with local constraints in order to determine for every number field K, what finite solvable groups G appear as Galois groups of tame maximal subfields of K-division algebras, generalizing Liedahl's theorem for metacyclic G and Sonn's solution of the Q-admissibility conj…
We extend the modularity lifting result of the arXiv:1111.2804 to allow Galois representations with some ramification at p. We also prove modularity mod 2 and 5 of certain Galois representations. We use these results to prove many new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent p-adic Hilbert modular forms of finite slope which substantially generalizes a similar result in arXiv:1111.2804.
Let $K$ be a local field of characteristic $p$ and let $L/K$ be a totally ramified Galois extension such that Gal$(L/K)\cong C_{p^n}$. In this paper we find sufficient conditions for $L/K$ to admit a Galois scaffold. This leads to sufficient conditions for the ring of integers $O_L$ to be free of rank 1 over its associated order $A_0$, and to stricter conditions which imply that $A_0$ is a Hopf order in the group ring $K[C_{p^n}]$.
As an analogue of a link group, we consider the Galois group of the maximal pro-$p$-extension of a number field with restricted ramification which is cyclotomically ramified at $p$, i.e, tamely ramified over the intermediate cyclotomic $\mathbb Z_p$-extension of the number field. In some basic cases, such a pro-$p$ Galois group also has a Koch type presentation described by linking numbers and mod $2$ Milnor numbers (Rédei symbols) of primes. Then the pro-$2$ Fox derivative yields a calculation of Iwasawa polynomials analogous to Alexander polynomials.
We consider continuous representations of the Galois group G of a number field K taking values in the completion C of an algebraic closure A of the field of l-adic numbers. We give a construction of irreducible representations of G in GL(2,C) which cannot be conjugated into GL(2,A) but for which there exist density-one sets of primes with Frobenius traces in A. We also show that any such representation must be ramified at infinitely many primes of K.
Let $L/K$ be a finite Galois extension of local fields. The Hasse-Arf theorem says that if Gal$(L/K)$ is abelian then the upper ramification breaks of $L/K$ must be integers. We prove the following converse to the Hasse-Arf theorem: Let $G$ be a nonabelian group which is isomorphic to the Galois group of some totally ramified extension $E/F$ of local fields with residue characteristic $p>2$. Then there is a totally ramified extension of local fields $L/K$ with residue characteristic $p$ such that Gal$(L/K)\cong G$ and $L/K$ has at least one nonintegral upper ramification break.
For a prime number p, we denote by K the cyclotomic Z_p-extension of a number field k. For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension of K unramified outside S. This paper treats the case where S does not contain p and k is the rational number field or an imaginary quadratic field. In this case, we prove the explicit formulae for the free ranks of the S-ramified Iwasawa modules as abelian pro-p groups, by using Brumer's p-adic version of Baker's theorem on the linear independence of logarithms of a…
