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By work of Belyĭ [2] , the absolute Galois group G Q = Gal ( Q ‾ / Q ) of the field Q of rational numbers can be embedded into A = Aut ( F ˆ 2 ) , the automorphism group of the free profinite group F ˆ 2 on two generators. The image of G Q lies inside G T ˆ , the Grothendieck-Teichmuller group. While it is known that every abelian representation of G Q can be extended to G T ˆ , Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of G T ˆ . We do this virtually, namely by showing that a rich class of arithmetically defined representations of G Q can be extended to finite index subgroups of G T ˆ . This is achieved, in fact, by extending these representations all the way to finite index subgroups of A = Aut ( F ˆ 2 ) . We do this by developing a profinite version of the work of Grunewald and Lubotzky [7] , which provided a rich collection of representations for the discrete group Aut ( F d ) .

Inspired by the classical setting, Goss defined the $L$-series of Drinfeld $A$-modules corresponding to representations of the absolute Galois group of a rational function field. In this paper, for a given Drinfeld $A$-module $\phi$ of rank 2 defined over the finite field $\mathbb{F}_q$, we give explicit formulas for the values of Goss $L$-series at positive integers $n$ such that $2n+1\leq q$ in terms of polylogarithms and coefficients of the logarithm series of $\phi$.

We prove a potential automorphy theorem for suitable Galois representations $$\Gamma _{F^+} \rightarrow \mathrm {GSpin}_{2n+1}(\overline{\mathbb {F}}_p)$$ and $$\Gamma _{F^+} \rightarrow \mathrm {GSpin}_{2n+1}({\overline{{\mathbb {Q}}}}_p)$$ , where $$\Gamma _{F^+}$$ is the absolute Galois group of a totally real field $$F^+$$ . We also prove results on solvable descent for $$\mathrm {GSp}_{2n}(\mathbb {A}_{F^+})$$ and use these to put representations $$\Gamma _{F^+} \rightarrow \mathrm {GSpin}_{2n+1}({\overline{{\mathbb {Q}}}}_p)$$ into compatible systems of $$\mathrm {GSpin}_{2n+1}({\overline{{\mathbb {Q}}}}_l)$$ -valued representations.

The theory of dessins d’enfants on compact Riemann surfaces, which are bipartite maps on compact orientable surfaces, are combinatorial objects used to study branched covers between compact Riemann surfaces and the absolute Galois group of the field of rational numbers. In this paper, we show how this theory is naturally extended to non-compact orientable surfaces and, in particular, we observe that the Loch Ness monster (LNM; the surface of infinite genus with exactly one end) admits infinitely many regular dessins d’enfants (either chiral or reflexive). In addition, we study different holomorphic structures on the LNM, which come from homology covers of compact Riemann surfaces, and infinite hyperelliptic and infinite superelliptic curves.

Let [Formula: see text] be a Galois extension of [Formula: see text]-adic number fields and let [Formula: see text] be a de Rham representation of the absolute Galois group [Formula: see text] of [Formula: see text]. In the case [Formula: see text], the equivariant local [Formula: see text]-constant conjecture describes the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin [Formula: see text]-functions and it can be formulated as the vanishing of a certain element [Formula: see text] in [Formula: see text]; a similar approach can be followed also in the case of unramified twists [Formula: see text] of [Formula: see text] (see [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383; D. Izychev and O. Venjakob, Equivariant epsilon conjecture for 1-dimensional Lubin–Tate groups, J. Théor. Nr. Bordx. 28(2) (2016) 485–521]). One of the main technical difficulties in the computation of [Formula: see text] arises from the so-called cohomological term [Formula: see text], which requires the construction of a bounded complex [Formula: see text] of cohomologically trivial modules which represents [Formula: see text] for a full [Formula: see text]-stable [Formula: see text]-sublattice [Formula: see text] of [Formula: see text]. In this paper, we generalize the construction of [Formula: see text] in Theorem 2 of [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383] to the case of a higher dimensional [Formula: see text].

Let $S$ be a finite set of primes. We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of $\mathbb{Z}_{S}$-points on integral models of Hilbert modular varieties, extending a result of D.Helm and F.Voloch about modular curves. Let $L$ be a totally real field. Under (a special case of) the absolute Hodge conjecture and a weak Serre's conjecture for mod $\ell$ representations of the absolute Galois group of $L$, we prove that the same holds also for the $\mathcal{O}_{L,S}$-points., Comment: To appear in Canadian Mathematical Bulletin

Let $K$ be a finite field extension of $Q_p$ and let $G_K$ be its absolute Galois group. We construct the universal family of filtered $(\phi,N)$-modules, or (more generally) the universal family of $(\phi,N)$-modules with a Hodge-Pink lattice, and study its geometric properties. Building on this, we construct the universal family of semi-stable $G_K$-representations in $Q_p$-algebras. All these universal families are parametrized by moduli spaces which are Artin stacks in schemes or in adic spaces locally of finite type over $Q_p$ in the sense of Huber. This has conjectural applications to the $p$-adic local Langlands program., Comment: final version, to appear in ANT

Given a field K, a polynomial f ∈ K [ x ] of degree d, and a suitable element t ∈ K , the set of preimages of t under the iterates f ∘ n carries a natural structure of a d-ary tree. We study conditions under which the absolute Galois group of K acts on the tree by the full group of automorphisms. When d ≥ 20 is even and K = Q we exhibit examples of polynomials with maximal Galois action on the preimage tree, partially affirming a conjecture of Odoni. We also study the case of K = F ( t ) and f ∈ F [ x ] in which the corresponding Galois groups are the monodromy groups of the ramified covers f ∘ n : P F 1 → P F 1 .

Let K be a finite extension of ℚ p and G K the absolute Galois group. Then G K acts on the fundamental curve X of p -adic Hodge theory and we may consider the abelian category ℳ ( G K ) of coherent O X -modules equipped with a continuous and semilinear action of G K . An almost ℂ p -representation of G K is a p -adic Banach space V equipped with a linear and continuous action of G K such that there exists d ∈ ℕ , two G K -stable finite dimensional sub- ℚ p -vector spaces U + of V , U − of ℂ p d , and a G K -equivariant isomorphism V ∕ U + → ℂ p d ∕ U − . These representations form an abelian category C ( G K ) . The main purpose of this paper is to prove that C ( G K ) can be recovered from ℳ ( G K ) by a simple construction (and vice-versa) inducing, in particular, an equivalence of triangulated categories D b ( ℳ ( G K ) ) → D b ( C ( G K ) ) .

Given an elliptic curve $E$ over a local field $K$ with residue characteristic $3$, we investigate the action of the absolute Galois group of $K$ in the case of potentially good reduction. In particular the only not completely known case is that of the $\ell$-adic Galois representation attached to an elliptic curve such that the image of inertia is non-cyclic, and isomorphic to $C_3 \rtimes C_4$. In this work we describe such a representation explicitly., 16 pages, final version to appear in Acta Arithmetica

We use tools from combinatorial group theory in order to study actions of three types on groups acting on a curve, namely the automorphism group of a compact Riemann surface, the mapping class group acting on a surface (which now is allowed to have some points removed) and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ in the case of cyclic covers of the projective line., Comment: 23 pages

We prove that the irreducible components of the space of framed deformations of a 2-dimensional mod 2 representation with scalar semi-simplification of the absolute Galois group of Q 2 are in natural bijection with those of its determinant, confirming a conjecture of Bockle–Juschka.

For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory., Comment: 39 pages

Fix an arbitrary prime p. Let F be a field containing a primitive p-th root of unity, with absolute Galois group G F , and let H n denote its mod p cohomology group H n ( G F , Z / p Z ) . The triple Massey product of weight ( n , k , m ) ∈ N 3 is a partially defined, multi-valued function 〈 ⋅ , ⋅ , ⋅ 〉 : H n × H k × H m → H n + k + m − 1 . In this work we prove that for an arbitrary prime p, any defined 3MP of weight ( n , 1 , m ) , where the first and third entries are symbols, contains zero; and that any defined 3MP of weight ( 1 , k , 1 ) , where the middle entry is a symbol, contains zero.

We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck-Teichm\"{u}ller group. Using a result of Drummond-Cole, we deduce that the Grothendieck-Teichm\"{u}ller group acts nontrivially on $\overline{\mathcal{M}}_{0,\bullet+1}$, the operad of stable curves of genus zero. As a second application, we give an alternative proof that the framed little 2-disks operad is formal., Comment: 36 pages

Given two pure representations of the absolute Galois group of an l-adic number field with coefficients in Q ‾ p (with l ≠ p ), we show that the Frobenius-semisimplifications of the associated Weil–Deligne representations are twists of each other by an integral power of a certain unramified character if they have equal normalized traces. This is an analogue of a recent result of Patankar and Rajan in the context of local Galois representations.

For a representation of the absolute Galois group of the rationals over a finite field of characteristic $p$, we study the existence of a lift to characteristic zero that is geometric in the sense of the Fontaine-Mazur conjecture. For two-dimensional representations, Ramakrishna proved that under technical assumptions odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. A key step is generalizing and studying a local deformation condition at $p$ arising from Fontaine-Laffaille theory., This is one piece of the earlier submission which was split for publication

This paper determines the Brauer $p$-dimension Brd$_{p}(K)$ and the absolute Brauer $p$-dimension abrd$_{p}(K)$ of a Henselian valued field $(K, v)$, for a prime $p \neq {\rm char}(\widehat K)$, under restrictions on the residue field $\widehat K$, such as the condition abrd$_{p}(\widehat K) = 0$. It describes the set $\Sigma_{0}$ of sequences ${\rm abrd}_{p}(E), {\rm Brd}_{p}(E)$, $p \in \mathbb P$, where $\mathbb P$ is the set of prime numbers and $E$ runs across the class of Henselian fields with char$(\widehat E) = 0$ and a projective absolute Galois group $\mathcal{G}_{\widehat E}$. Specifically, $\Sigma_{0}$ contains a sequence $a_{p}, b_{p} \in \mathbb N \cup \{0, \infty \}$, $p \in \mathbb P$, whenever $a_{2} \le 2b_{2}$ and $a_{p} \ge b_{p}$, for each $p$. Similar results are obtained in characteristic $q > 0$., Comment: 28 pages, LaTeX. Incorporates referee's suggestions. Accepted for publication in: Journal of Pure and Applied Algebra

Let $p$ be a prime number and let ${K}$ be a field containing a root of 1 of order $p$. If the absolute Galois group $G_{K}$ satisfies $\dim H^1(G_{K},\mathbb{F}_p), Comment: Updated version

We present possible extensions of the quantum statistical mechanical formulation of class field theory to the non-abelian case, based on the action of the absolute Galois group on Grothendieck's dessins d'enfant, the embedding in the Grothendieck-Teichm\"uller group, and the Drinfeld-Ihara involution.