We examine the ramification groups of finite Galois extensions over complete discrete valuation fields of equal characteristic $p>0$. Brylinski (1983) calculated the ramification groups in the case where the Galois groups are abelian. We extend the results of Brylinski to some non-abelian cases where the Galois groups are of order $\leq p^{p+1}$ and of maximal nilpotency class.
Ramification groups of Galois extensions over local fields of positive characteristic with Galois group isomorphic to the group of unitriangular matrices
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.
We use techniques of relative algebraic K-theory to develop a common refinement of the existing theories of metrized and hermitian Galois structures in arithmetic. As a first application of this very general approach, we then use it to prove several new results, and to formulate a framework of new conjectures, concerning the detailed arithmetic properties of wildly ramified Galois-Gauss sums.
Let $K$ be a number field and $S$ a set of primes of $K$. We write $K_S/K$ for the maximal extension of $K$ unramified outside $S$ and $G_{K,S}$ for its Galois group. In this paper, we answer the following question under some assumptions: "For $i=1,2$, let $K_i$ be a number field, $S_i$ a (sufficiently large) set of primes of $K_i$ and $σ:G_{K_1,S_1}\to G_{K_2,S_2}$ an isomorphism. Is $σ$ induced by a unique isomorphism between $K_{1,S_1}/K_1$ and $K_{2,S_2}/K_2$?" Here the main assumption is about the Dirichlet density of $S_i$.
This paper focuses on a refinement of the inverse Galois problem. We explore what finite groups appear as the Galois group of an extension of the rational numbers in which only a predetermined set of primes may ramify. After presenting new results regarding extensions in which only a single finite prime ramifies, we move on to studying the more complex situation in which multiple primes from a finite set of arbitrary size may ramify. We then continue by examining a conjecture of Harbater that the minimal number of generators of the Galois group of a tame, Galois extension of the rational numbe…
A generalization of Serre's Conjecture asserts that if $F$ is a totally real field, then certain characteristic $p$ representations of Galois groups over $F$ arise from Hilbert modular forms. Moreover it predicts the set of weights of such forms in terms of the local behavior of the Galois representation at primes over $p$. This characterization of the weights, which is formulated using $p$-adic Hodge theory, is known under mild technical hypotheses if $p > 2$. In this paper we give, under the assumption that $p$ is unramified in $F$, a conjectural alternative description for the set of weight…
We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric and alternating groups in many cases.
We prove the equality of two non-logarithmic ramification filtrations defined by Matsuda and Abbes-Saito for the abelianized absolute Galois group of a complete discrete valuation field in positive characteristic. We also compute the refined Swan conductor and the characteristic form of a character of the fundamental group of a smooth separated scheme over a perfect field of positive characteristic by using sheaves of Witt vectors.
Given a hilbertian field $k$ of characteristic zero and a finite Galois extension $E/k(T)$ with group $G$ such that $E/k$ is regular, we produce some specializations of $E/k(T)$ at points $t_0 \in \mathbb{P}^1(k)$ which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of $\mathbb{Q}$ of various finite groups with specified local behavior - ramified or unramified - at finitely many given primes. Secondly, in the case $k$ is a number field, we pr…
Let $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of an algebraic number field $k$. We denote by $S$ a finite set of prime numbers which does not contain $p$, and $S(k_\infty)$ the set of primes of $k_\infty$ lying above $S$. In the present paper, we will study the structure of the Galois group $\mathcal{X}_S (k_\infty)$ of the maximal pro-$p$ extension unramified outside $S (k_\infty)$ over $k_\infty$. We mainly consider the question whether $\mathcal{X}_S (k_\infty)$ is a non-abelian free pro-$p$ group or not. In the former part, we treat the case when $k$ is an imaginary quadratic f…
The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and $\infty$, is well known including numerical experiments with PARI/GP programs. The case of ``incomplete p-ramification'' (i.e., when the set S of ramified places is a strict subset of the set P of the p-places) is, on the contrary, mostly unknown in a theoretical point of view. We give, in a first part, a way to compute, for any S $\le$ P, the structure of the Galois group of the maximal S-ramified abelian pro-p-extension H\_(K,S) of any field K given by means o…
Andreatta-Iovita-Pilloni constructed eigenvarieties for cuspidal Hilbert modular forms. The eigenvariety has a natural map to the weight space, called the weight map. At a classical point, we compute a lower bound of the dimension of the tangent space of the fiber of the weight map using Galois deformation theory. Along with the classicality theorem due to Tian-Xiao, this enables us to characterize the classical points of the eigenvariety which are ramified over the weight space, in terms of the local splitting behavior of the associated Galois representation.
We classify the upper ramification breaks of totally ramified nonabelian extensions of degree $p^3$ over a local field of characteristic $p>0$. We find that nonintegral upper ramification breaks can occur for each nonabelian Galois group of order $p^3$, except the dihedral group of order $8$.
We give an intrinsic parametrisation of the set of tamely ramified extensions of a local field with finite residue field and bring to the fore the role played by group cohomology. We show that two natural definitions of the cohomology class of a tamely ramified finite galoisian extension coincide, and can be recovered from the parameter. We also give an elementary proof of Serre's mass formula in the tame case and in the simplest wild case, and we classify tame galoisian extensions of degree the cube of a prime.
We prove a local-global principle for the embedding problems of global fields with restricted ramification. By this local-global principle, for a global field $k$, we use only the local information to give a presentation of the maximal pro-$p$ Galois group of $k$ with restricted ramification, when some Galois cohomological conditions are satisfied. For a Galois $p$-extension $K/k$, we use our presentation result for $k$ to study the structure of pro-$p$ Galois groups of $K$. Then for $k=\mathbb{Q}$ and $k=\mathbb{F}_q(t)$ with $p\nmid q$, we give upper and lower bounds for the rank of $p$-tors…
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.
In previous works, we described algorithms to compute the number field cut out by the mod ell representation attached to a modular form of level N=1. In this article, we explain how these algorithms can be generalised to forms of higher level N. As an application, we compute the Galois representations attached to a few forms which are supersingular or admit a companion mod ell with ell=13 (and soon ell=41), and we obtain previously unknown number fields of degree ell+1 whose Galois closure has Galois group PGL(2,ell) and a root discriminant that is so small that it beats records for such numbe…
The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ be the minimal integer $m$ for which there exists a Galois extension $N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infinite one). So, the problem is to compute or to bound $m(G)$. In this paper, we bound the ramification of extensions $N/\mathbb{Q}$ obtained as a specialization of a branched covering $ϕ\colon C\to \mathbb{P}^1_{\mathbb{Q}}$. This leads to novel upper bounds on $m(G)$, for finite groups $G$ that are real…
- Let p be a prime number and K an algebraic number field. What is the arithmetic structure of Galois extensions L/K having p-adic analytic Galois group $Γ$ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecture predicts that such extensions are either deeply ramified (at some prime dividing p) or ramified at an infinite number of primes. In this work, we take up a study (initiated by Boston) of this type of question under the assumption that L is Galois over some subfield k of K such that [K : k] is a prime = p. Letting $σ$ be a generator of Gal(K/k), we study the constraints posed on the …
For a rational prime $p\neq 2$, we compute the sequence of ramification groups of a Galois, radical and finite extension $L/F$ where $F/\mathbb{Q}_p$ is an unramified finite extension. First, we compute it in the case where the exponent of Gal$(L/F)$ is a power of $p$, by a Hecke's theorem and some inductions,. Finally, we deal with the general case. For the case $p=2$, the same method can work, provided that we make an additional hypothesis.
The Poitou-Tate sequence relates Galois cohomology with restricted ramification of a finite Galois module $M$ over a global field to that of the dual module under the assumption that $\#M$ is a unit away from the allowed ramification set. We remove the assumption on $\#M$ by proving a generalization that allows arbitrary "ramification sets" that contain the archimedean places. We also prove that restricted products of local cohomologies that appear in the Poitou-Tate sequence may be identified with derived functor cohomology of an adele ring. In our proof of the generalized sequence we adopt t…
We study irreducible mod p representations, valued in general reductive groups, of the Galois group of a number field. When the number field is totally real, we show that odd representations satisfying local ramification hypotheses and a certain multiplicity-free condition on the adjoint representation admit geometric lifts. For general number fields, we show without any oddness or multiplicity condition that the representation admits a p-adic lift if it does everywhere locally.
In this article, we investigate the shift of Abbes and Saito's ramification filtrations of the absolute Galois group of a complete discrete valuation field of positive characteristic under a purely inseparable extension. We also study a functoriality property for characteristic forms.
Suppose $K$ is a finite extension of $\mathbb{Q}_p$ containing a $p^M$-th primitive root of unity. For $1\leqslant s<p$ denote by $K[s,M]$ the maximal $p$-extension of $K$ with the Galois group of period $p^M$ and nilpotent class $s$. We apply the nilpotent Artin-Schreier theory together with the theory of the field-of-norms functor to give an explicit description of the Galois groups ${Gal}(K[s,M]/K)$. As application we prove that the ramification subgroup $Γ^{(v)}_K$ of the absolute Galois group of $K$ acts trivially on $K[s,M]$ if and only if $v>e_K(M+s/(p-1))-(1-δ_{1s})/p$, where $e_K$ is …
We propose a conjectural characterization of when the dynamical Galois group associated to a polynomial is abelian, and we prove our conjecture in several cases, including the stable quadratic case over ${\mathbb Q}$. In the postcritically infinite case, the proof uses algebraic techniques, including a result concerning ramification in towers of cyclic $p$-extensions. In the postcritically finite case, the proof uses the theory of heights together with results of Amoroso-Zannier and Amoroso-Dvornicich, as well as properties of the Arakelov-Zhang pairing.
We propose an approach for the computation of multi-parameter families of Galois extensions with prescribed ramification type. More precisely, we combine existing deformation and interpolation techniques with recently developed strong tools for the computation of $3$-point covers. To demonstrate the applicability of our method in relatively large degrees, we compute several families of polynomials with symplectic Galois groups, in particular obtaining the first totally real polynomials with Galois group PSp(6,2).
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.
We determine the representation of the group of automorphisms for cyclotomic function fields in characteristic $p > 0$ induced by the natural action on the space of holomorphic differentials via construction of an explicit basis of differentials. This includes those cases which present wild ramification and automorphism groups with non-cyclic $p$-part, which have remained elusive. We also obtain information on the gap sequences of ramified primes. Finally, we extend these results to rank one Drinfel'd modules.
The \(v\)-function appears in the wild McKay correspondence and it is an invariant of a Galois extension over a local field which measures about the ramification of extension. The ramification filtration is also such invariant. Yasuda presented the problem whether the \(v\)-function is determined by ramification filtration. In this paper, we give a formula for the \(v\)-function for 2-dimensional representations of the bicyclic group of order \(p^2\) with \(p\) the characteristic of our base field. Using this formula, we construct an example of \(v\)-function not determined by the ramification…
This paper verifies Abhyankar's Inertia Conjecture for certain sporadic groups $G$ in particular characteristics by showing that all possible inertia groups occur for $G$-Galois covers of the affine line. For a larger set of sporadic groups, all but finitely many possible ramification invariants are shown to occur. In particular, we prove that all but eight of the possible ramification invariants are realizable for $\text{M}_{11}$-Galois covers of the affine line.
