Absolute Galois Group Research Articles

Some bounds for ramification of [formula omitted]-torsion semi-stable representations

Let p be an odd prime, K a finite extension of Q p , G K = Gal ( K ¯ / K ) its absolute Galois group and e = e ( K / Q p ) its absolute ramification index. Suppose that T is a p n -torsion representation of G K that is isomorphic to a quotient of G K -stable Z p -lattices in a semi-stable representation with Hodge–Tate weights { 0 , … , r } . We prove that there exists a constant μ depending only on n, e and r such that the upper numbering ramification group G K ( μ ) acts on T trivially.

Filtered modules corresponding to potentially semi-stable representations

We classify the filtered modules with coefficients corresponding to two-dimensional potentially semi-stable p-adic representations of the absolute Galois groups of p-adic fields under the assumptions that p is odd and the coefficients are large enough.

M-Bigness in compatible systems

Taylor–Wiles type lifting theorems allow one to deduce that if ρ is a “sufficiently nice” l-adic representation of the absolute Galois group of a number field whose semi-simplified reduction modulo l, denoted ρ ¯ , comes from an automorphic representation then so does ρ. The recent lifting theorems of Barnet-Lamb–Gee–Geraghty–Taylor impose a technical condition, called m-big, upon the residual representation ρ ¯ . Snowden–Wiles proved that for a sufficiently irreducible compatible system of Galois representations, the residual images are big at a set of places of Dirichlet density 1. We demonstrate the analogous result in the m-big setting using a mild generalization of their argument.

On Serre's conjecture for mod ℓ Galois representations over totally real fields

In 1987 Serre conjectured that any mod ℓ 2-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalization of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where ℓ is unramified. The hard work is in formulating an analogue of the weight part of Serre's conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a “mod ℓ Langlands philosophy.” Using ideas of Emerton and Vignéras, we formulate a mod ℓ local-global principle for the group D*, where D is a quaternion algebra over a totally real field, split above ℓ and at 0 or 1 infinite places, and we show how it implies the conjecture.

Patching and admissibility over two-dimensional complete local domains

We develop a patching machinery over the field E = K((X, Y )) of formal power series in two variables over an infinite field K. We apply this machinery to prove that if K is separably closed and G is a finite group of order not divisible by char(E), then there exists a G-crossed product algebra with center E if and only if the Sylow subgroups of G are abelian of rank at most 2. MR Classification: 12E30, 16S35 Introduction Complete local domains play an important role in commutative algebra and algebraic geometry, and their algebraic properties were already described in 1946 by Cohen’s The second author was partially supported by the Israel Science Foundation (grant No. 343/07), and by an ERC grant. structure theorem. The Galois theoretic properties of their quotient fields were extensively studied over the past two decades. The pioneering work in this line of research is due to Harbater [Ha87], who introduced the method of patching to prove that if R is a complete local domain with quotient field K, then every finite group occurs as a Galois group over K(x). This result was strengthened by Pop [Po96], and in a different language, by Haran-Jarden [HaJ98], who showed that if moreover R is of dimension 1, then every finite split embedding problem over K(x) is solvable. The first step towards higher dimension was made by Harbater-Stevenson [HaS05], who essentially showed that if R is a complete local domain of dimension 2, then every finite split embedding problem over Quot(R) has |R| independent solutions. That is, the absolute Galois group of Quot(R) is semi-free of rank |R| (see [BHH10] for details on this notion). This result was later generalized by Pop [Po10] and by the second author [Pa10], who showed that if K is the quotient field of a complete local domain of dimension exceeding 1, then Gal(K) is semi-free. Despite the major progress made in the study of Galois theory over these fields, little is known about the structure of division algebras over them. A step in that direction was recently made by Harbater-Hartmann-Krashen [HHK10]. In that work, the authors consider a question relating Galois theory and Brauer theory over a field E – which groups are admissible over E ? That is, which finite groups occur as a Galois group of an adequate Galois extension F/E (recall that an extension F/E is called adequate if F is a maximal subfield in an E-central division algebra). Equivalently, for which groups G there is a G-crossed product division algebra with center E. Note that for E as above and a finite extension F/E, the above maximality requirement can be omitted since any F which is a subfield of an E-division algebra is also a maximal subfield of some E-division algebra (see Remark 3.9). This question was first considered by Schacher over global fields. In [Sch68], Schacher proved that any Q-admissible group has metacyclic Sylow subgroups and conjectured the converse. Admissibility was studied extensively over global fields (for example: [Ste82], [Son83], [SS92], [Lid96], [Fei04], [Nef10]), function fields and fields of Laurent series (for example: [FSS92], [FS95a], [FS95b]).

Projective pairs of profinite groups

We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish a connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic extension of a PAC field K. Then M / K is PAC if and only if the corresponding pair of absolute Galois groups ( Gal ( M ) , Gal ( K ) ) is projective. Moreover any projective pair can be realized as absolute Galois groups of a PAC extension of a PAC field. Using this characterization we construct new examples of PAC extensions of relatively small fields, e.g. unbounded abelian extensions of the rational numbers.

The location of noncrossed products in Brauer groups of Laurent series fields over global fields

Since Amitsur’s discovery of noncrossed product division algebras in 1972, their existence over more familiar fields has been an object of investigation. Brussel’s work was a culmination of this effort, exhibiting noncrossed products over the rational function field k(t) and the Laurent series field k((t)) over any global field k—the smallest possible centers of noncrossed products. Witt’s theorem gives a transparent description of the Brauer group of k((t)) as the direct sum of the Brauer group of k and the character group of the absolute Galois group of k. We classify the Brauer classes over k((t)) containing noncrossed products by analyzing the fiber over χ for each character χ in Witt’s theorem. In this way, a picture of the partition of the Brauer group into crossed products/noncrossed products is obtained, which is in principle ruled by a relation between index and number of roots of unity. As a side consequence of the result there are crossed products that have a noncrossed product primary component.

A generalization of Sen–Brinon’s theory

Let K be a complete discrete valuation field of mixed characteristic and k be its residue field of prime characteristic p > 0. We assume that [k : k p ] = p h < ∞. Let G K be the absolute Galois group of K and $${\mathcal{R}}$$ be a Banach algebra over $${\mathbb{C}_p:=\widehat{\overline{K}}}$$ with a continuous action of G K . When k is perfect (i.e. h = 0), Sen studied the Galois cohomology $${{\rm H}^1(G_K, \mathcal{R}^\ast)}$$ and Sen’s operator associated to each class (Sen Ann Math 127:647–661, 1988). In this paper we generalize Sen’s theory to the case h ≥ 0 by using Brinon’s theory (Brinon Math Ann 327:793–813, 2003). We also give another formulation of Brinon’s theorem (a la Colmez).

Lifting Galois representations over arbitrary number fields

It is proved that every two-dimensional residual Galois representation of the absolute Galois group of an arbitrary number field lifts to a characteristic zero p-adic representation, if local lifting problems at places above p are unobstructed.

Admissible unitary completions of locally ℚ_{𝕡}-rational representations of 𝔾𝕃₂(𝔽)

Let F F be a finite extension of Q p \mathbb {Q}_p , p &gt; 2 p&gt;2 . We construct admissible unitary completions of certain representations of G L 2 ( F ) \mathrm {GL}_2(F) on L L -vector spaces, where L L is a finite extension of F F . When F = Q p F=\mathbb {Q}_p using the results of Berger, Breuil and Colmez we obtain some results about lifting 2 2 -dimensional mod p p representations of the absolute Galois group of Q p \mathbb {Q}_p to crystabelline representations with given Hodge-Tate weights.

Serre’s conjecture and its consequences

After some generalities about the absolute Galois group of $$\mathbb Q$$ , we present the historical context in which Serre made his modularity conjecture. This was recently proved by Wintenberger and the author ([22], [23]), with an input of Kisin ([24]). The focus of these notes is on the applications of the conjecture. Some of the applications are based on the methods used in the proof. We stress some applications to base change and descent for classical modular forms. For base change we have no new results, aside from a fragment for odd A 5-extensions of $$\mathbb Q$$ . Descent for Hilbert modular forms to $$\mathbb Q$$ on the other hand is a simple corollary of Serre’s conjecture.

The semisimplicity conjecture for A-motives

AbstractWe prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

Représentations p-adiques de GL2(L) et Catégories Dérivées

We construct some locally ℚ p -analytic representations of GL2(L), L a finite extension of ℚ p , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel’d’s upper half space has a structure of rank 2 admissible filtered (φ, N)-module. Finally, we prove that this filtered module is associated, via Fontaine’s theory, to the initial Galois representation.

Permanence criteria for semi-free profinite groups

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups.

Non-abelian local reciprocity law

Following our work on the generalized Fesenko reciprocity map, we construct the non-abelian local reciprocity map \({\pmb{\Phi}_K^{(\varphi)}}\) of a local field K as a certain isomorphism from the absolute Galois group GK of K onto a topological group \({\nabla_{K,Y}^{(\varphi)}}\) whose definition involves Fontaine–Wintenberger theory of field of norms, and build the non-abelian local class field theory over K in the sense of Fesenko and Koch.

Galois theory over complete local domains

Complete local domains play an important role in commutative algebra and algebraic geometry, and their algebraic properties were already described by Cohen’s structure theorem in 1946. However, the Galois theoretic properties of their quotient fields only recently began to unfold. In 2005 Harbater and Stevenson considered the two dimensional case. They proved that the absolute Galois group of the field K((X, Y)) (where K is an arbitrary field) is semi-free. In this work we settle the general case, and prove that if R is a complete local domain of dimension exceeding 1, then the quotient field of R has a semi-free absolute Galois group.

On reductions of families of crystalline Galois representations

Let K_f be the finite unramified extension of {Q}_p of degree f and E any finite large enough coefficient field containing K_f. We construct analytic families of étale (\varphi ,\Gamma ) -modules which give rise to families of crystalline E -representations of the absolute Galois group G_{K_f} of K_f. For any irreducible effective two-dimensional crystalline E -representation of G_{K_f} with labeled Hodge-Tate weights {0,-k_i}_{\tau _i} induced from a crystalline character of G_{K_{2f}}, we construct an infinite family of crystalline E -representations of G_{K_f} of the same Hodge-Tate type which contains it. As an application, we compute the semisimplified mod p reductions of the members of each such family.

On Ihara’s lemma for Hilbert modular varieties

AbstractLetρbe a two-dimensional moduloprepresentation of the absolute Galois group of a totally real number field. Under the assumptions thatρhas a large image and admits a low-weight crystalline modular deformation we show that any low-weight crystalline deformation ofρunramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara-type lemma for the local component atρof the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraicp-part of the value at one of the adjointL-function associated with a Hilbert modular newform to the cardinality of the corresponding Selmer group.

GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS

AbstractWe consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponentn– their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graphKn,n, wherenis an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphsKn,n[7] and generalises the analogous results for maps obtained in [9], partly using different methods.

Profinite and Asymptotic Group Theory

This meeting was focused on asymptotic aspects of group theory. The resulting problems lead, in particular, to the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients. Also, properties of infinite families of finite groups are at the center of interest in the field. We had talks on the most important recent developments in the field of asymptotic and profinite group theory. We single out some of these new results. The classification of finite simple groups has provided an impetus for the detailed investigation of properties of families of groups of Lie type and of alternating groups. This has produced many results of asymptotic or probabilistic nature, describing in a quantitative way properties that hold for all sufficiently large simple groups, or else that hold ‘almost surely’ as the orders of the groups are tending to infinity. Here we had a talk of Aner Shalev on his recent solution (with coauthors) of Ore's conjecture. Recently there has been much activity in the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients, or their finite-index subgroups. The theory of subgroup growth is now well established; the theory of the associated zeta functions is making good progress. More subtle variations of subgroup growth are also developing into significant theories in their own right: there are important new results on representation growth – of arithmetic groups, nilpotent groups and pro-p groups; and on maximal subgroup growth, with applications to probabilistic generation properties. Profinite groups are ‘asymptotic limits’ of finite groups. The nature of verbal subgroups and verbal mappings, in both finite and profinite groups, has been the focus of much recent work. This has led among other things to the solution of Serre's problem on subgroups of finite index in finitely generated profinite groups; other significant results include the characterisation of closed verbal subgroups in pro-p groups and new characterisations of solubility in finite groups. Branch groups, both discrete and profinite, are certain groups of automorphisms of rooted trees; introduced by Grigorchuk and others over 20 years ago, they have provided the solution to several outstanding problems, and their study continues to provide new results. Some of them were reported on in our meeting. We also had talks from important fields of mathematics in which profinite and asymptotic group theory finds application. There is, for instance, the study of the absolute Galois group of the rationals and in particular its closed pro-p subgroups. Methods from pro-p group theory have successfully been applied in order to understand fundamental groups of (hyperbolic) 3-manifolds. The interplay between the topological fundamental group of a complex algebraic surface and its profinite completion which is the etale fundamental group of the surface is the subject of recent research. We believe that the meeting was exciting and successful. The high quality lectures covered a lot of very recent results in the field. There were many intense discussions between mathematicians of different fields. Altogether there were over 50 participants from all over the world. The percentage of young researchers was very high. The following extended abstracts were collected and edited by Dr. Evija Ribnere.