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

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

class field theory diophantine analysis and the asymptotic fermat s last theorem abstract recent results of freitas kraus engun and siksek give sufficient criteria for the asymptotic fermat s last theorem to hold over a specific number field those works in turn build on many deep theorems in arithmetic geometry in this paper we combine the aforementioned results with techniques from class field theory the theory of p groups and p extensions diophantine approximation and linear forms in logarithms to establish the asymptotic fermat s last theorem for many infinite families of number fields and for thousands of number fields of small degree for example we prove the effective asymptotic fermat s last theorem for the infinite family of fields q 2 r where r 2

differential algebraic systems as differential equations on manifolds based on the theory of differential equations on manifolds existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications both the autonomous and nonautonomous case are considered moreover a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower dimensional manifolds this class includes systems for which the application of ode solvers is known to lead to difficulties finally some solution approach based on continuation techniques is outlined

characterizing diophantine henselian valuation rings and valuation ideals we give a characterization in terms of the residue field of those henselian valuation rings and those henselian valuation ideals that are diophantine this characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature we also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal

the cyclotomic trace and the k theoretic analogue of novikov s conjecture abstract a trace construction the cyclotomic trace is given it associates to algebraic k theory of a group ring or better to waldhausen s a theory equivariant stable homotopy classes of the free loop space of its classifying space the cyclotomic trace detects the borel classes in algebraic k theory of the integers it is used to prove for a wide class of groups that the k theory assembly map is rationally injective this is the k theoretic analogue of novikov s conjecture

asymptotic expansions of the global error of fixed stepsize methods in his fundamental paper on general fixed stepsize methods skeel 6 studied convergence properties but left the existence of asymptotic expansions as an open problem in this paper we give a complete answer to this question for the special cases of one step and linear multistep methods our proof is shorter than the published ones asymptotic expansions are the theoretical base for extrapolation methods

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

tractability of algebraic function fields in one variable of genus zero over global fields if the condition described in definition 1 1 on the simultaneous representation of quaternion algebras holds over a field f then f is called tractable in this paper we show that a field f is tractable if and only if any purely transcendental extension of f is tractable next if f is a global field of characteristic unequal to two we consider tractability of an algebraic function field k in one variable of genus zero over f and determine precisely when k is tractable if f has at most one ordering

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

on pseudo algebraically closed extensions of fields abstract the notion of pseudo algebraically closed pac extensions is a generalization of the classical notion of pac fields in this work we develop a basic machinery to study pac extensions this machinery is based on a generalization of embedding problems to field extensions the main goal is to prove that the galois closure of any proper separable algebraic pac extension is its separable closure as a result we get a classification of all finite pac extensions which in turn proves the bottom conjecture for finitely generated infinite fields the secondary goal of this work is to unify proofs of known results about pac extensions and to establish new basic properties of pac extensions e g transitiveness of pac extensions

numerical solution of nonlinear differential equations with algebraic constraints i convergence results for backward differentiation formulas in this paper we investigate the behavior of numerical ode methods for the solution of systems of differential equations coupled with algebraic constraints systems of this form arise frequently in the modelling of problems from physics and engineering we study some particular examples from electrical networks fluid dynamics and constrained mechanical systems we show that backward differentiation formulas converge with the expected order of accuracy for these systems

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

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

on cyclic fixed points of spectra for a finite p group g and a bounded below g spectrum x of finite type mod p the g equivariant segal conjecture for x asserts that the canonical map x g rightarrow x hg from g fixed points to g homotopy fixed points is a p adic equivalence let c p n be the cyclic group of order p n we show that if the c p equivariant segal conjecture holds for a c p n spectrum x as well as for each of its geometric fixed point spectra varphi c p e x for 0 e n then the c p n equivariant segal conjecture holds for x similar results also hold for weaker forms of the segal conjecture asking only that the canonical map induces an equivalence in sufficiently high degrees on homotopy groups with suitable finite coefficients

places of algebraic function fields in arbitrary characteristic we consider the zariski space of all places of an algebraic function field f k of arbitrary characteristic and investigate its structure by means of its patch topology we show that certain sets of places with nice properties e g prime divisors places of maximal rank zero dimensional discrete places lie dense in this topology further we give several equivalent characterizations of fields that are large in the sense of f pop s annals paper embedding problems over large fields we also study the question whether a field k is existentially closed in an extension field l if l admits a k rational place in the appendix we prove the fact that the zariski space with the zariski topology is quasi compact and that it is a spectral space

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

existentially generated subfields of large fields abstract we study subfields of large fields which are generated by infinite existentially definable subsets we say that such subfields are existentially generated let l be a large field of characteristic exponent p and let e l be an infinite existentially generated subfield we show that e contains l p n the p n th powers in l for some n this generalises a result of fehm from 4 which shows e l under the assumption that l is perfect our method is to first study existentially generated subfields of henselian fields since l is existentially closed in the henselian field l t our result follows

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

on the structure of pythagorean fields abstract in this paper we study in detail the relationship between fans and valuations first uncovered by brocker see 2 our techniques are somewhat different as our valuations are constructed by brute force from the properties of the fields involved this approach has the advantage in that the valuation theory it yields can be directly related to the structure theory for spaces of orderings as developed by murray marshall see 11 14 we also discuss several applications which include results in k theory

descent varieties for algebraic covers this paper is about descent theory for algebraic covers typical questions concern fields of definition models moduli spaces families etc of covers here we construct descent varieties associated to any given cover f they have the property that whether they have rational points on a given field k is the obstruction to descending the field of definition of f to k our constructions have a global version above moduli spaces of covers hurwitz spaces here descent varieties are parameter spaces for hurwitz families with some versal property descent varieties provide a new diophantine viewpoint on descent theory by reducing the questions to that of finding points on varieties there are concrete applications we answer a question raised in deha about totally p adic models of covers we also show that the subset of a given moduli space of covers of p where the field of moduli is a field of definition is zariski dense 2000 msc primary 14e08 14e20 secondary 18d30 14a20 11gxx

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

one step and extrapolation methods for differential algebraic systems the paper analyzes one step methods for differential algebraic equations dae in terms of convergence order in view of extrapolation methods certain perturbed asymptotic expansions are shown to hold for the special dae extrapolation solver based on the semi implicit euler discretization the perturbed order pattern of the extrapolation tableau is derived in detail the theoretical results lead to modifications of the known code the efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion

demuskin fields with valuations we study the arithmetic structure of fields f of characteristic neq p containing a primitive pth root of unity for which the maximal pro p galois group of f is a finitely generated demuskin group

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

a generalization of marshall s equivalence relation for p prime and for a field f containing a root of unity of order p we generalize marshall s equivalence relation on orderings to arbitrary subgroups of f of index p the equivalence classes then correspond to free pro p factors of the maximal pro p galois group of f we generalize to this setting results of jacob on the maximal pro 2 galois group of a pythagorean field

on rational fixed points of finite group actions on the affine space consider a finite l group acting on the affine space of dimension n over a field k whose characteristic differs from l we prove the existence of a fixed point rational over k in the following cases the field k is p special for some prime p different from its characteristic the field k is perfect and fertile and n 3

weak approximation over function fields of curves over large or finite fields let k k c be the function field of a curve over a field k and let x be a smooth projective separably rationally connected k variety with x k neq emptyset under the assumption that x admits a smooth projective model pi mathcal x to c we prove the following weak approximation results 1 if k is a large field then x k is zariski dense 2 if k is an infinite algebraic extension of a finite field then x satisfies weak approximation at places of good reduction 3 if k is a nonarchimedean local field and r equivalence is trivial on one of the fibers mathcal x p over points of good reduction then there is a zariski dense subset w subseteq c k such that x satisfies weak approximation at places in w as applications of the methods we also obtain the following results over a finite field k 4 if k 10 then for a smooth cubic hypersurface x k the specialization map x k longrightarrow prod p in p mathcal x p kappa p at finitely many points of good reduction is surjective 5 if mathrm char k neq 2 3 and k 47 then a smooth cubic surface x over k satisfies weak approximation at any given place of good reduction

the sylow subgroups of the absolute galois group gal q abstract we describe the l sylow subgroups of gal q for an odd prime l by observing and studying their decomposition as f z l where f is a free pro l group and z l are the l adic integers we determine the finite z l quotients of f and more generally show that every split embedding problem of z l groups for f is solvable moreover we analyze the z l action on generators of f

step size adjustment and extrapolation for time stepping schemes in non smooth dynamics in this paper we use step size adjustment and extrapolation methods to improve moreau s time stepping scheme for the numerical integration of non smooth mechanical systems i e systems with impact and friction the scheme yields a system of inclusions which is transformed into a system of projective equations these equations are solved iteratively switching points are time instants for which the structure of the mechanical system changes for example time instants for which a sticking friction element begins to slide we show how switching points can be localized and how these points can be resolved by choosing a minimal step size in order to improve the integration of non smooth systems in the smooth parts we show how the time stepping method can be used as a base integration scheme for extrapolation methods which allow for an increase in the integration order switching points are processed by a small time step while time intervals during which the structure of the system does not change are computed with a larger step size and improved integration order the overall algorithm which consists of a time stepping module an extrapolation module and a step size adjustment module is discussed in detail and some examples are given

an etale approach to the novikov conjecture we show that the rational novikov conjecture for a group of finite homological type follows from the mod 2 acyclicity of the higson compactification of an e we then show that for groups of finite asymptotic dimension the higson compactification is mod p acyclic for all p and deduce the integral novikov conjecture for these groups 2007 wiley periodicals inc

isomorphism conjectures in algebraic k theory duff in this paper we are concerned with the four functors y y3 if and which map from the category of topological spaces x to the category diff of q spectra the functor 3 or if maps the space x to the qspectrum of stable topological or smooth pseudoisotopies of x the spaces diff dif 3i x id9 x i 0 are hatcher s deloopings of 970 x yd x cf 29 the functor j maps a path connected space x to the algebraic ktheoretic s2 spectrum for the integral group ring z7r1x the spaces x x i 0 are the gersten wagoner deloopings of z o x cf 27 52 5 x is also defined on a nonpath connected space x to be x1 where xi i e i are the path components of x and ii xi indicates the direct limit of all finite products of the i xi i e i if x is path connected then the functor maps x to the l surgery classifying spaces for oriented surgery problems with fundamental group r i1x identified by their fourfold periodicity k 90 x k 4j x cf 35 39 44 47 in addition if x is not path connected then we set y77 x tiei5 fj xi where xi i e i are the path components of x results obtained by the authors over the past five years cf 15 25 and in particular 19 16 a 1 a 18 together with many earlier results and conjectures which are reviewed in 1 6 1 1 6 6 suggest that for any connected cwcomplex x and for 5 denoting any of the above functors the q spectrum x should be computable in a simple way from the values taken by 9 on covering spaces of x which have very simple fundamental groups in more detail we let 9 x denote the collection of all subgroups h c ir ix which are either finite or virtually infinite cyclic a group h is virtually infinite cyclic if there is a short exact sequence of groups 1 z h f 1 with f equal to a finite group and z equal to the infinite cyclic group for each such group we let xh x denote the covering space projection corresponding to h then the q spectrum 5 x should be computable in a simple way from the q spectra 5 xh h e 9 x the purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any x

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

linearly implicit extrapolation methods for differential algebraic systems we consider the numerical solution of implicit differential equations in which the solution derivative appears multiplied by a solution dependent singular matrix we study extrapolation methods based on two linearly implicit euler discretizations their error behaviour is explained by perturbed asymptotic expansions

extrapolation at stiff differential equations asymptotic expansions of the global error of numerical methods are well understood if the differential equation is non stiff this paper is concerned with such expansions for the implicit euler method the linearly implicit euler method and the linearly implicit mid point rule when they are applied tostiff differential equations in this case perturbation terms are present whose dominant one is given explicitly this permits us to better understand the behaviour ofextrapolation methods at stiff differential equations numerical examples supporting the theoretical results are included

the drazin inverse in multibody system dynamics the numerical analysis of multibody system dynamics is based on the equations of motion as differential algebraic systems a thorough analysis of the linearized equations and their solution theory leads to an equivalent system of ordinary differential equations which gives deeper insight into the derivation of integration schemes and into the stabilization approaches the main tool is the drazin inverse a generalized matrix inverse which preserves the eigenvalues the results are illustrated by a realistic truck model finally the approach is extended to the nonlinear index 2 formulation

on the topological hochschild homology of bu i the purpose of this paper and its sequel is to determine the homotopy groups of the spectrum thh here p is an odd prime is the adams summand of p local connective k theory see for example 25 and thh is the topological hochschild homology construction introduced by bokstedt in 3 in the present paper we will determine the mod p homotopy groups of thh and also the integral homotopy groups of thh l where l denotes the periodic adams summand in the sequel we will investigate the integral homotopy groups of thh using our present results as a starting point the thh construction appears to be of basic importance in algebraic k theory because it combines two useful properties it can be used to construct good approximations to the algebraic k theory functor and it is very accessible to calculation we shall review what is known about the first property in a moment the second property was demonstrated by bokstedt s calculation in his paper 4 of the homotopy groups of thh hz p and thh hz here hz p and hz denote the evident eilenberg mac lane spectra it is natural to ask about thh r for other popular ring spectra r and our work is a first step in this direction we pay special attention to the connective case because this is the case which is likely to be relevant in applications see subsection 1 4 below the calculation which we present in this paper is a homotopy theoretic one which uses the adams spectral sequence hereafter abbreviated ass this calculation has several interesting features in particular it is a pleasing example of an ass calculation in which although there are infinitely many differentials it is still possible to get the complete answer here is a summary of the contents of the paper in section 2 we review the facts we need to know about ordinary hochschild homology in section 3 we do the same

rationally connected varieties over local fields let x be a proper variety defined over a field k following ma we say that two points x x x k are r equivalent if they can be connected by a chain of rational curves defined over k cf 4 1 in essence two points are r equivalent if they are obviously rationally equivalent several authors have proved finiteness results over local and global fields cubic hypersurfaces ma sd linear algebraic groups ct sa vo1 gi vo2 rational surfaces ct co ct1 ct sk1 quadric bundles and intersections of two quadrics ct sa sd pa su r equivalence is only interesting if there are plenty of rational curves on x at least over k such varieties have been studied in the series of papers komi mo1 ko mi mo3 see also ko1 there are many a priori different ways of defining what plenty of rational curves should mean fortunately many of these turn out to be equivalent and this leads to the notion of rationally connected varieties see ko mi mo2 ko1 iv 3 ko2 4 1 2

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

on the stability of volterra runge kutta methods we investigate the stability of extended runge kutta methods for volterra integral equations of the second kind as test equations we choose linear convolution equations with positive definite l 1 kernels our results are strongly connected with the contractivity of runge kutta methods for nonlinear dissipative differential equations

ode methods for the solution of differential algebraic systems in this paper we study the numerical solution of the differential algebraic systems f t y y 0 many of these systems can be solved conveniently and economically using a range of ode methods others can be solved only by a small subset of ode methods and still others present insurmountable difficulty for all current ode methods we examine the first two groups of problems and indicate which methods we believe to be best for them then we explore the properties of the third group which cause the methods to fail we describe a reduction technique which allows systems to be reduced to ones that can be solved it also provides a tool for the analytical study of the structure of systems

compatible valuations and generalized milnor k theory given a field and a subgroup of there is a minimal group for which there exists an compatible valuation whose units are contained in assuming that has finite index in and contains for prime we describe in computable theoretic terms