parametric galois extensions abstract given a field k and a finite group h an h parametric extension over k is a finite galois extension of k t of galois group containing h which is regular over k and has all the galois extensions of k of group h among its specializations we are mainly interested in producing non h parametric extensions which relates to classical questions in inverse galois theory like the beckmann black problem and the existence of one parameter generic polynomials we develop a general approach started in a preceding paper and provide new non parametricity criteria and new examples

henselian implies large in this note we show that the quotient field of a domain which is henselian with respect to a nontrivial ideal is a large field and give some applications of this fact using a specialization theorem for ramified covers of the line over generalized krull fields

frequency domain elastic wave modeling by finite differences a tool for crosshole seismic imaging the migration imaging or inversion of wide aperture cross hole data depends on the ability to model wave propagation in complex media for multiple source positions computational costs can be considerably reduced in frequency domain imaging by modeling the frequency domain steady state equations rather than the time domain equations of motion i develop a frequency domain approach in this note that is competitive with time domain modeling when solutions for multiple sources are required or when only a limited number of frequency components of the solution are required

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

a new approach to analyticity of dirichlet neumann operators this paper outlines the theoretical background of a new approach towards an accurate and well conditioned perturbative calculation of dirichlet neumann operators dnos on domains that are perturbations of simple geometries previous work on the analyticity of dnos has produced formulae that as we have found are very ill conditioned we show how a simple change of variables can lead to recursions that satisfy analyticity estimates without relying on subtle cancellation properties at the heart of previous formulae

limits of galois representations in fundamental groups along maximal degeneration of marked curves i we study gal inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i actions on the fundamental groups arising in deformation of maximally degenerate marked curves and give explicit descriptions of them in terms of the principal parameters of the grothendieck teichmuller group inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i as a special application we present galois theoretic connections between the moduli spaces m 0 4 m 0 5 m 1 1 and m 1 2 which lead to a newtype equation satisfied by the image of gal inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i

three dimensional acoustic scattering by layered media a novel surface formulation with operator expansions implementation the scattering of acoustic waves by irregular structures plays an important role in a wide range of problems of scientific and technological interest this contribution focuses on the rapid and highly accurate numerical approximation of solutions of helmholtz equations coupled across irregular periodic interfaces meant to model acoustic waves incident upon a multi layered medium we describe not only a novel surface formulation for the problem in terms of boundary integral operators dirichlet neumann operators but also a boundary perturbation methodology the method of operator expansions for its numerical simulation the method requires only the discretization of the layer interfaces so that the number of unknowns is an order of magnitude smaller than volumetric approaches while it avoids not only the need for specialized quadrature rules but also the dense linear systems characteristic of boundary integral element methods the approach is a generalization to multiple layers of malcolm nicholls operator expansions algorithm for dielectric structures with two layers as with this precursor this approach is efficient and spectrally accurate

fokas integral equations for three dimensional layered media scattering abstract the scattering of acoustic waves by periodic structures is of central importance in a wide range of problems of scientific and technological interest this paper describes a rapid high order numerical algorithm for simulating solutions of helmholtz equations coupled across irregular non trivial interfaces meant to model acoustic waves incident upon a multiply layered medium building upon an interfacial formulation from previous work we describe an integral equation strategy inspired by recent developments of fokas and collaborators for its numerical approximation the method requires only the discretization of the layer interfaces so that the number of unknowns is an order of magnitude smaller than volumetric approaches while it requires neither specialized quadrature rules nor periodized fundamental solutions characteristic of many popular boundary integral element methods as with previous contributions by the authors on this formulation this approach is efficient and spectrally accurate for smooth interfaces

subfields of ample fields rational maps and definability abstract pop proved that a smooth curve c over an ample field k with c k has k many rational points we strengthen this result by showing that there are k many rational points that do not lie in a given proper subfield even after applying a rational map this has several consequences for example we gain insight into the structure of existentially definable i e diophantine subsets of ample fields

large normal extension of hilbertian fields the goal of this note is to consider a certain natural family of closed normal subgroups of g q and to prove that each group in this family is free more generally consider a countable separably hilbertian field k denote the absolute galois group of k by g k then for almost all g k e the field ks is pac and e free fj2 thms 16 13 and 16 18 here ks is the separable closure of k and ks is the fixed field of in ks being pac means that every nonvoid absolutely irreducible variety defined over ks has a ks rational point we say that ks is e free if g ks i e the closed subgroup 1 e of g k generated by 1 e is free on e generators denote the largest galois extension of k which is contained in ks by ks it is the intersection of all k conjugates of ks and also the fixed field of the smallest closed normal subgroup of g k which contains 1 e if char k 0 then for almost all g k e the field ks is pac fj2 thm 16 47 lemma 1 2 below generalizes this result to arbitrary characteristic if we knew that ks is separably hilbertian then a theorem of fried volklein and pop would imply that ks is free that is g ks is isomorphic to the free profinite group fw on countably many generators unfortunately it is not clear how to prove the hilbertianity of almost all ks directly so we use instead a forerunner to the above mentioned theorem of fried volklein pop and a recent theorem of neumann neu to prove directly that g ks f for almost all g k e a theorem of roquette then implies that ks is also separably hilbertian let k be the algebraic closure of k denote the maximal purely inseparable extension of ks by k then for almost all g k e the field k

split embedding problems over complete domains we prove that every finite split embedding problem is solvable over the field k x1 x n of formal power series in n 2 variables over an arbitrary field k as well as over the field quot a x 1 x n of formal power series in n 1 variables over a noetherian integrally closed domain a this generalizes a theorem of harbater and stevenson who settled the case k x1 x 2

the irreducibility of certain pure cycle hurwitz spaces we study pure cycle hurwitz spaces parametrizing covers of the projective line having only one ramified point over each branch point we start with the case of genus 0 covers using a combination of limit linear series theory and group theory to show that these spaces are always irreducible in the case of four branch points we also compute the associated hurwitz numbers finally we give a conditional result in the higher genus case requiring at least 3g simply branched points these results have equivalent formulations in group theory and in this setting complement results of conway fried parker v olklein

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

projective pairs of profinite groups abstract 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 fundamental group of affine curves in positive characteristic abstract it is shown that the commutator subgroup of the fundamental group of a smooth irreducible affine curve over an uncountable algebraically closed field k of positive characteristic is a profinite free group of rank equal to the cardinality of k

hurwitz spaces of galois coverings of p1 whose galois groups are weyl groups abstract we prove the irreducibility of the hurwitz spaces which parametrize equivalence classes of galois coverings of p 1 whose galois group is an arbitrary weyl group and the local monodromies are reflections this generalizes a classical theorem due to luroth clebsch and hurwitz

the embedding problem over a hilbertian pac field we show that the absolute galois group of a countable hilbertian p seudo a lgebraically c losed field of characteristic 0 is a free profinite group of countably infinite rank theorem a as a consequence g q q is the extension of groups with a fairly simple structure e g 2 sn by a countably free group in addition we characterize those pac fields over which every finite group is a galois group as those with the rg hilbertian property

a hybrid analytical numerical technique for elliptic pdes recent work has given rise to a novel and simple numerical technique for solving elliptic boundary value problems formulated in convex polygons in two dimensions the method based on the unified t

embedding problems over large fields in this paper we study galois theoretic properties of a large class of fields a class which includes all fields satisfying a universal local global principle for the existence of rational points on varieties as applications we give a positive answer to a long standing conjecture which originates in an unpublished note of roquette and asserts that the absolute galois group of a countable hilbertian pac field is profinite free see f j problem 24 41 secondly we give new evidence for the conjecture of shafarevich which asserts that the absolute galois group of qab is profinite free finally one of the most interesting applications of the theory we develop here is the insight in the galois structure of the totally eg adic numbers

stability of high order perturbative methods for the computation of dirichlet neumann operators abstract in this paper we present results on the stability of perturbation methods for the evaluation of dirichlet neumann operators dno defined on domains that are viewed as complex deformations of a basic simple geometry in such cases geometric perturbation methods based on variations of the spatial domains of definition have long been recognized to constitute efficient and accurate means for the approximation of dno and in fact several numerical implementations have been previously proposed inspired by our recent analytical work here we demonstrate that the convergence of these algorithms is quite generally limited by numerical instability indeed we show that these standard perturbative methods for the calculation of dno suffer from significant ill conditioning which is manifest even for very smooth boundaries and whose severity increases with boundary roughness moreover and again motivated by our previous work we introduce an alternative perturbative approach that we show to be numerically stable this approach can be interpreted as a reformulation of classical perturbative algorithms in suitable independent variables and thus it allows for both direct comparison and the possibility of analytic continuation of the perturbation series it can also be related to classical preconditioned spectral approaches and as such it retains in finite arithmetic the spectral convergence properties of classical perturbative methods albeit at a higher computational cost as it does not take advantage of possible dimensional reductions still as we show an alternative approach such as the one we propose may be mandated in cases where substantial information is contained in high order harmonics and or perturbation coefficients of the solution

patching and thickening problems in recent years much progress has been made on the structure of fundamental groups of algebraic curves by means of patching techniques in formal and rigid geometry a number of these results have concerned curves over algebraically closed fields of characteristic p e g the proofs of the abhyankar conjecture ra ha5 and of the geometric case of the shafarevich conjecture po1 ha6 and the realization of galois groups over projective curves sa1 st1 while the rigid approach to patching is often regarded as more intuitive than the formal approach its foundations are less well established but constructions involving the formal approach have tended to be technically more cumbersome the purpose of the current paper is to build on previous formal patching results in order to create a framework in which such constructions are facilitated in the process we prove a result asserting that singular curves over a field k can be thickened to curves over k t with prescribed behavior in a formal neighborhood of the singular locus and similarly for covers of curves afterwards we obtain applications to fundamental groups of curves over large fields the structure of the paper is as follows section 1 concerns patching problems for projective curves x over a power series ring r k t1 tn it is shown theorem 1 that giving a coherent projective module over x is equivalent to giving such modules compatibly on a formal neighborhood of each singular point of the closed fibre x and on the formal thickening along the complement of the singular locus s of x section 2 applies this to thickening problems in both the absolute and relative senses namely it shows theorem 3 that such an x can be constructed from its closed fibre x and from complete local thickenings near s such that x is a trivial deformation away from s moreover theorem 2 given a morphism x z and a thickening z of z such that the local thickenings near s are compatible with that of z there is a unique such thickening x of x that is compatible with the given data in section 3 these results are applied to the problem of thickening and deforming covers theorem 4 there combines the results of sections 1 and 2 to show that covers of reducible k curves can be thickened to covers of curves over k t and theorem 5 then shows how this can be used over large fields to

tchebotarev theorems for function fields abstract the central theme of the paper is the specialization of algebraic function field extensions our main results are tchebotarev type theorems for galois function field extensions finite or infinite over various base fields under some conditions we extend the classical finite field case to number fields p adic fields pac fields function fields x etc we also compare the tchebotarev conclusion existence of unramified local specializations with galois group any cyclic subgroup of the generic galois group up to conjugation to the hilbert specialization property for a function field extension with the tchebotarev property the exponent of the galois group is bounded by the l c m of the local specialization degrees local global questions arise for which we provide answers examples and counter examples

galois groups of unramified covers of projective curves in characteristicp abstract we show several results concerning the finite groups that occur as galois groups of unramified covers of projective curves in characteristic p in particular we prove that every finite group with g generators occurs over some curve of genus g this implies for example that every finite simple group occurs in genus 2 by similar methods we obtain several other families of groups which occur in genus 2 in addition we show that if a group g occurs over some curve of genus g then it must occur over almost all curves of genus g or greater the results are obtained using formal patching

the index of nilpotent lie poset algebras abstract we establish combinatorial formulas for the index of a class of matrix lie algebras whose matrix forms are encoded by strict partial orderings our methods are linear algebraic

computation of a cover of shimura curves using a hurwitz space using a hurwitz space computation we determine the canonical model of the cover of shimura curves x 0 2 to x 1 associated to the quaternion algebra over the cubic field of discriminant 13 2 which is ramified at exactly two real places and unramified at finite places then we list the coordinates of some rational cm points on x 1

homological stability for hurwitz spaces and the cohen lenstra conjecture over function fields we prove a homological stabilization theorem for hurwitz spaces moduli spaces of branched covers of the complex projective line this has the following arithmetic consequence let 2 be prime and a a nite abelian group then there exists q q a such that for q greater than q a positive fraction of quadratic extensions of fq t have the part of their class group isomorphic to a

galois theory over rings of arithmetic power series let r be a domain complete with respect to a norm which denes a non discrete topology on r we prove that the quotient eld of r is ample generalizing a theorem of pop we then consider the case where r is a ring of arithmetic power series which are holomorphic on the closed disc of radius 0 r 1 around the origin and apply the above result to prove that the absolute galois group of the quotient eld of r is semi free this strengthens a theorem of harbater who solved the inverse galois problem over these elds

differential embedding problems over laurent series fields abstract we solve the inverse differential galois problem over the fraction field of k t x and use this to solve split differential embedding problems over k t x that are induced from k x the proofs use patching as well as prior results on inverse problems and embedding problems

study and computation of a hurwitz space and totally real psl2 f8 extensions of q abstract developing on works by fried volklein matzat malle debes wewers we give a method for computing a hurwitz space and illustrate it on some example of number theoretic interest we study and compute a family of degree 9 covers of p c 1 with monodromy group psl 2 f 8 and having four branch points we deduce explicit regular psl 2 f 8 extensions of the rational function field q with totally real fibers this gives rise to totally real polynomials over q with galois group psl 2 f 8

the deligne simpson problem a survey abstract the deligne simpson problem dsp respectively the weak dsp is formulated like this give necessary and sufficient conditions for the choice of the conjugacy classes c j gl n c or c j gl n c so that there exist irreducible respectively with trivial centralizer p 1 tuples of matrices m j c j or a j c j satisfying the equality m 1 m p 1 i or a 1 a p 1 0 the matrices m j and a j are interpreted as monodromy operators of regular linear systems and as matrices residua of fuchsian ones on riemann s sphere the present paper offers a survey of the results known up to now concerning the dsp

fake congruence modular curves and subgroups of the modular group abstract we construct a special class of noncongruence modular subgroups and curves analogous in some ways to the usual congruence ones the subgroups are obtained via the burau representation and the associated quotient curves have a natural moduli space interpretation in fact they are reduced hurwitz spaces corresponding to covers with 4 branch points and monodromy group equal to semi direct products of a cyclic and an abelian group furthermore they form a modular tower in the sense of fried we study representations on the cohomology of these fake congruence modular curves and also calculate the genera of certain quotient curves

harbater mumford subvarieties of moduli spaces of covers we define harbater mumford subvarieties which are special kinds of closed subvarieties of hurwitz moduli spaces obtained by fixing some of the branch points we show that for many finite groups finding geometrically irreducible hm subvarieties defined over open image in new window is always possible this provides information on the arithmetic of hurwitz spaces and applies in particular to the regular inverse galois problem with almost all fixed branch points profinite versions of our results can also be stated providing new tools to study the geometry of modular towers and the regular inverse galois problem for profinite groups

split embedding problems over function fields over complete domains let r be a krull domain complete with respect to a nonzero ideal let k be the quotient field of r we prove that every finite split embedding problem is solvable over every function field in one variable over k if dim r 1 then every finite split embedding problem over k is solvable

co homology of lie algebras via algebraic morse theory abstract the fundamental theorem of cancellation amt 4 and 11 which is the algebraic generalization of discrete morse theory 2 for simplicial complexes and smooth morse theory 10 for differentiable manifolds is discussed in the context of general chain complexes of free modules the chevalley co homology table of a lie algebra is often a tremendous beast using amt we compute the homology of the lie algebra of all triangular matrices sol n over q or z p for large enough primes p we determine the column and row in the table of h k sol n z where the p torsion first appears module h k sol n z p is expressed by the homology of a chain subcomplex for the lie algebra of all strictly triangular matrices nil n using the kunneth formula all conclusions are accompanied by computer experiments then we generalize some results to lie algebras of strictly triangular matrices gl n and gl n with respect to any partial ordering on n we determine the multiplicative structure of h gl n w r t the cup product over fields of zero or sufficiently large characteristic the result being the exterior algebra matchings used here can be analogously defined for other lie algebra families and in other co homology theories we collectively call them normalization matchings they are useful for theoretical as well as computational purposes

an improved formalism for wave scattering from rough surfaces the scattered field and its normal gradient obey a mutual linear relation at the scattering surface that is distinct from the physical boundary condition connecting either quantity to the incident radiation on a moderately rough surface this relation can be represented by an operator whose series expansion in surface slope converges nicely even for large values of the rayleigh parameter this allows the helmholtz integral for scattering amplitude to be written as a series of readily computable terms one or two of which provide good approximations for surfaces too irregular for the usual bragg expansion this formulation reproduces the bragg series for small surface elevations and in the limit of low roughness wavenumber gives the kirchhoff approximation with an explicit correction term in surface curvature the usual results for a composite surface also emerge naturally on several test profiles the method produces better overall accuracy than other multiscale approximations at comparable efficiency

a field expansions method for scattering by periodic multilayered media the interaction of acoustic and electromagnetic waves with periodic structures plays an important role in a wide range of problems of scientific and technological interest this contribution focuses upon the robust and high order numerical simulation of a model for the interaction of pressure waves generated within the earth incident upon layers of sediment near the surface herein described is a boundary perturbation method for the numerical simulation of scattering returns from irregularly shaped periodic layered media the method requires only the discretization of the layer interfaces so that the number of unknowns is an order of magnitude smaller than finite difference and finite element simulations while it avoids not only the need for specialized quadrature rules but also the dense linear systems characteristic of boundary integral element methods the approach is a generalization to multiple layers of bruno and reitich s method of field expansions for dielectric structures with two layers by simply considering the entire structure simultaneously rather than solving in individual layers separately the full field can be recovered in time proportional to the number of interfaces as with the original field expansions method this approach is extremely efficient and spectrally accurate

weierstrass preparation and algebraic invariants we prove a form of the weierstrass preparation theorem for normal algebraic curves over complete discrete valuation rings while the more traditional algebraic form of weierstrass preparation applies just to the projective line over a base our version allows more general curves this result is then used to obtain applications concerning the values of u invariants and on the period index problem for division algebras over fraction fields of complete two dimensional rings our approach uses patching methods and matrix factorization results that can be viewed as analogs of cartan s lemma

local galois theory in dimension two this paper proves a generalization of shafarevich s conjecture for fields of laurent series in two variables over an arbitrary field this result says that the absolute galois group gk of such a field k is quasi free of rank equal to the cardinality of k i e every non trivial finite split embedding problem for gk has exactly cardk proper solutions we also strengthen a result of pop and haran jarden on the existence of proper regular solutions to split embedding problems for curves over large fields our strengthening concerns integral models of curves which are two dimensional

connected components of affine primitive permutation groups abstract for a finite group g the hurwitz space h r g i n g is the space of genus g covers of the riemann sphere with r branch points and the monodromy group g in this paper we give a complete list of primitive genus one systems of affine type that is we assume that g is a primitive group of affine type under this assumption we determine the braid orbits on the suitable nielsen classes which is equivalent to finding connected components in h r 1 i n g furthermore we give a new algorithm for computing large braid orbits on nielsen classes this algorithm utilizes a correspondence between the components of h r 1 i n g and h r 1 i n m where m is the point stabilizer in g

analytic continuation of dirichlet neumann operators the analytic dependence of dirichlet neumann operators dno with respect to variations of their domain of definition has been successfully used to devise diverse computational strategies for their estimation these strategies have historically proven very competitive when dealing with small deviations from exactly solvable geometries as in this case the perturbation series of the dno can be easily and recursively evaluated in this paper we introduce a scheme for the enhancement of the domain of applicability of these approaches that is based on techniques of analytic continuation we show that in fact dno depend analytically on variations of arbitrary smooth domains in particular this implies that they generally remain analytic beyond the disk of convergence of their power series representations about a canonical separable geometry and this in turn guarantees that alternative summation mechanisms such as pade approximation can be effectively used to numerically access this extended domain of analyticity our method of proof is motivated by our recent development of stable recursions for the coefficients of the perturbation series here we again utilize this recursion as we compare and contrast the performance of our new algorithms with that of previously advanced perturbative methods the numerical results clearly demonstrate the beneficial effect of incorporating analytic continuation procedures into boundary perturbation methods moreover the results also establish the superior accuracy and applicability of our new approach which as we show allows for precise calculations corresponding to very large perturbations of a basic geometry

normal subgroups of the algebraic fundamental group of affine curves in positive characteristic let 1 c be the algebraic fundamental group of a smooth connected affine curve defined over an algebraically closed field of characteristic p 0 of countable cardinality let n be a normal respectively characteristic subgroup of 1 c under the hypothesis that the quotient 1 c n admits an infinitely generated sylow p subgroup we prove that n is indeed isomorphic to a normal respectively characteristic subgroup of a free profinite group of countable cardinality as a consequence every proper open subgroup of n is a free profinite group of countable cardinality

fundamental group in positive characteristic abstract it is shown that the commutator subgroup of the fundamental group of a smooth irreducible affine curve over a countable algebraically closed field of nonzero characteristic is profinite free of countable rank a description of the abelianizations of the fundamental groups of affine curves over an algebraically closed field of nonzero characteristic is also given

the inverse galois problem and rational points on moduli spaces we reduce the regular version of the inverse galois problem for any finite group g to finding one rational point on an infinite sequence of algebraic varieties as a consequence any finite group g is the galois group of an extension l p x with l regular over any pac field p of characteristic zero a special case of this implies that g is a galois group over fp x for almost all primes p

morse theory from an algebraic viewpoint forman s discrete morse theory is studied from an algebraic viewpoint and we show how this theory can be extended to chain complexes of modules over arbitrary rings as applications we compute the homologies of a certain family of nilpotent lie algebras and show how the algebraic morse theory can be used to derive the classical anick resolution as well as a new two sided anick resolution