Abstract:
We show that a finite-dimensional tame division algebra D over a Henselian field F has a maximal subfield Galois over F if and only if its residue division algebra has a maximal subfield Galois over the residue field of F. This generalizes the mechanism behind several known noncrossed product constructions to a crossed product criterion for all tame division algebras, and in particular for all division algebras if the residue characteristic is 0. If the residue field is a global field, the criterion leads to a description of the location of noncrossed products among tame division algebras, and their discovery in new parts of the Brauer group.

Abstract:
We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this context. We assume that $\K$ is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include: (1) Galois-stability above the Hanf number implies that \kappa(K) is less than the Hanf number. Where \kappa(K) is the parallel of \kapppa(T) for f.o. T. (2) We use (1) to construct Morley sequences (for non-splitting) improving previous results of Shelah (from Sh394) and Grossberg & Lessmann. (3) We obtain a partial stability-spectrum theorem for classes categorical above the Hanf number.

Abstract:
We establish automatic realizations of Galois groups among groups M\rtimes G, where G is a cyclic group of order p^n for a prime p and M is a quotient of the group ring Fp[G].

Abstract:
Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally polarized abelian variety A of dimension n over F such that the resulting \ell-torsion representation \rho_{A,\ell} from G_F to GSp(A[\ell](\bar{F})) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(\mathbb{F}_\ell) as the Galois group of a finite tame extension of number fields F'/F such that F is unramified above \ell.

Abstract:
Given a finite group G and a number field k, a well-known conjecture asserts that the set R_t(k,G) of Steinitz classes of tame G-Galois extensions of k is a subgroup of the ideal class group of k. In this paper we investigate an explicit candidate for R_t(k,G), when G is of odd order. More precisely, we define a subgroup W(k,G) of the class group of k and we prove that R_t(k,G) is contained in W(k,G). We show that equality holds for all groups of odd order for which a description of R_t(k,G) is known so far. Furthermore, by refining techniques introduced in arXiv:0910.5080v1, we use the Shafarevich-Weil Theorem in cohomological class field theory, to construct some tame Galois extensions with given Steinitz class. In particular, this allows us to prove the equality R_t(k,G)=W(k,G) when G is a group of order dividing l^4, where l is an odd prime.

Abstract:
We study the middle convolution of local systems on the punctured affine line in the setting of singular cohomology and in the setting of \'etale cohomology. We derive a formula to compute the topological monodromy of the middle convolution in the general case and use it to deduce some irreducibility criteria. Then we give a geometric interpretation of the middle convolution in the \'etale setting. This geometric approach to the convolution and the theory of Hecke characters yields information on the occurring arithmetic determinants. We employ these methods to realize special linear groups regularly as Galois groups over ${\bf Q}(t).$

Abstract:
Let $F$ be a number field with ring of integers $O_F$ and let $G$ be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group $Cl(O_FG)$ of $O_FG$ that involves applying the work of the second-named author in the context of relative algebraic $K$ theory. When $G$ is nilpotent, we show (subject to certain conditions) that the set of realisable classes is a subgroup of $Cl(O_FG)$. This may be viewed as being an analogue of a classical theorem of Scholz and Reichardt on the inverse Galois problem for nilpotent groups in the setting of Galois module theory.

Abstract:
We show that for any tame regular discrete series parameter of GSp_4 or its inner form GU_2(D), the L-packet attached by the local Langlands conjecture agrees with the L-packet of depth zero supercuspidal representations constructed by DeBacker and Reeder.

Abstract:
In this paper we investigate the (classical) weights of mod $p$ Siegel modular forms of degree 2 toward Serre's conjecture for $GSp_4$. To carry out it, we first construct various theta operators on the space of such forms a la Katz and define the theta cycles for specific theta operators. This enable us to obtain a kind of weight reduction theorem for mod $p$ Siegel modular forms without increasing the level. We also give a conjecture of Serre's weight conjecture a la Serre in the tame ordinary case.

Abstract:
Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q_p for all p, or equivalently, with roots mod n for all n. It is known that f(x) cannot be irreducible but can be a product of two or more irreducible polynomials, and that if f(x) is a product of m>1 irreducible polynomials, then its Galois group must be "m-coverable", i.e. a union of conjugates of m proper subgroups, whose total intersection is trivial. We are thus led to a variant of the inverse Galois problem: given an m-coverable finite group G, find a Galois realization of G over the rationals Q by a polynomial f(x) in Z[x] which is a product of m nonlinear irreducible factors (in Q[x]) such that f(x) has a root in Q_p for all p. The minimal value m=2 is of special interest. It is known that the symmetric group S_n is 2-coverable if and only if 2