Related papers: The Sylow subgroups of the absolute Galois group G…

For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint case and for any sufficiently large regular prime p, we also construct…

Let K/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable Fp[Gal(K/F)]-modules. For these…

Suppose $p$ is a prime and $S$ is a Sylow $p$-subgroup of a finite group $G$. If $S$ is normal in $G$, then $Z(S)$ is the direct product of $S \cap Z(G)$ with $[Z(S), G]$. We prove an analogous result for all groups except in some cases…

We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first…

We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first…

Let $p$ be a prime number. In this article we present a theorem, suggested by Peter Scholze, which states that the absolute Galois group of $\mathbf{Q}_p$ is the \'etale fundamental group of a certain object $Z$ which is defined over an…

Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G_F$ its absolute Galois group. We determine a new canonical quotient $\mathrm{Gal}(F_{(3)}/F)$ of $G_F$ which encodes the full mod-$q$ cohomology ring…

We show that the simple group PSL_2(F_p) occurs as the Galois group of an extension of the rationals for all primes p>3. We obtain our Galois extensions by studying the Galois action on the second etale cohomology groups of a specific…

In this paper, we will study the pseudo-nullity of the fine Selmer group and its related question. Namely, we investigate certain situations, where one can deduce the pseudo-nullity of the dual fine Selmer group of a general Galois module…

Let $p$ be an odd prime number and $F$ a field containing a primitive $p$th root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group $G_F$ of $F$. Namely, the third subgroup $G_F^{(3)}$ in the…

We study the action of the Galois group $G$ of a finite extension $K/k$ of number fields on the points on an elliptic curve $E$. For an odd prime $p$, we aim to determine the structure of the $p$-adic completion of the Mordell-Weil group…

We describe the structure of Sylow {\ell}-subgroups of a finite reduc-tive group G(Fq) when q $\not\equiv$ 0 (mod {\ell}) that we find governed by a complex reflection group attached to G and {\ell}, which depends on {\ell} only through the…

For an odd prime $p$ satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group $G_{\mathbb{Q}(\zeta_p)}$ on the homology of the degree $p$ Fermat curve, building on work of Anderson. Further, we…

Let p be a prime number. It is not known if every finite p-group of rank n>1 can be realized as a Galois group over Q with no more than n ramified primes. We prove that this can be done for the family of finite p-groups which contains all…

Let $n\geq 1$ be an integer, $p$, $q$ be distinct odd primes. Let ${G}$, $N$ be two groups of order $p^nq$ with their Sylow-$p$-subgroups being cyclic. We enumerate the Hopf-Galois structures on a Galois ${G}$-extension, with type $N$. This…

Let l be an odd prime and K/k a Galois extension of totally real number fields with Galois group G such that K/k_\infty and k/Q are finite. We give a full description of the algebraic structure of the semisimple algebra QG=Quot(\Lambda G)…

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p; then G contains a simple group as a subquotient which exhibits the same property. In addition…

Let $p$ be a prime, $k$ a finite extension of $\mathbf{F}_p$ of cardinal $q$, $l$ a finite extension of $k$ of group $\Sigma=\mathrm{Gal}(l|k)$, and $T$ a subgroup of $l^\times$. Using the method of "little groups", we classify irreducible…

In this paper we introduce a new method for finding Galois groups by computer. This is particularly effective in the case of Galois groups of p-extensions ramified at finitely many primes but unramified at the primes above p. Such Galois…

Let k be a global field, p an odd prime number different from char(k) and S, T disjoint, finite sets of primes of k. Let G_S^T(k)(p)=Gal(k_S^T(p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and…