the integral monodromy of hyperelliptic and trielliptic curves we compute the mathbb z ell and mathbb z ell monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves in particular we provide a proof that the mathbb z ell monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group rm s p2g mathbb z ell we prove that the mathbb z ell monodromy of the moduli space of trielliptic curves with signature r s is the special unitary group rm su r s mathbb z ell otimes mathbb z zeta 3

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

complexes de chaines etales et courbes de deligne lusztig abstract in a first part we lift the usual constructions of functors between derived categories of etale sheaves over schemes with a sheaf of algebras to pure derived categories for varieties with finite group actions we recover in a more functorial way rickard s construction we apply this to the case of deligne lusztig varieties and show that broue s conjecture holds for curves the additional ingredient we need is obtained from an easy property of the cohomology of etale covers of the affine line

galois groups of some vectorial polynomials previously nice vectorial equations were constructed having various finite classical groups as galois groups here such equations are constructed for the remaining classical groups the previous equations were genus zero equations the present equations are strong genus zero

local systems and finite unitary and symplectic groups abstract for powers q of any odd prime p and any integer n 2 we exhibit explicit local systems on the affine line a 1 in characteristic p 0 if 2 n and on the affine plane a 2 if 2 n whose geometric monodromy groups are the finite symplectic groups sp 2 n q when n 3 is odd we show that the explicit rigid local systems on the affine line in characteristic p 0 constructed in 11 do have the special unitary groups su n q as their geometric monodromy groups as conjectured therein and also prove another conjecture of 11 that predicted their arithmetic monodromy groups

higher adeles and non abelian riemann roch abstract we show a riemann roch theorem for group ring bundles over an arithmetic surface this is expressed using the higher adeles of beilinson parshin and the tame symbol via a theory of adelic equivariant chow groups and chern classes the theorem is obtained by combining a group ring coefficient version of the local riemann roch formula as in kapranov vasserot with results on k groups of group rings and an explicit description of group ring bundles over p 1 our set up provides an extension of several aspects of the classical frohlich theory of the galois module structure of rings of integers of number fields to arithmetic surfaces

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

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

on some difference equations with eventually periodic solutions abstract we study the boundedness and the periodic character of the solutions of the equation formula where x 1 x 0 z and 1 1

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

riemann hurwitz formula for finite morphisms of p adic curves given a finite morphism varphi y rightarrow x of quasi smooth berkovich curves over a complete non archimedean nontrivially valued algebraically closed field k of characteristic 0 we prove a riemann hurwitz formula relating their euler poincare characteristics calculated using de rham cohomology of their overconvergent structure sheaf the main tools are p adic runge s theorem together with valuation polygons of analytic functions using the results obtained we provide another point of view on riemann hurwitz formula for finite morphisms of curves over algebraically closed fields of positive characteristic

subharmonic solutions of systems of difference equations with periodic perturbations part i existence we consider the existence of subharmonic solutions of systems of difference equations with periodic perturbations the theory of coincidence degree coupled with some detailed a priori estimates is applied to show that the perturbed system admits subharmonic solutions near a hyperbolic periodic orbit of the unperturbed system if the perturbation is sufficiently small

schur products and matrix completions we prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain schur product map defined on a certain subspace of matrices is a positive map by analyzing the positive elements of this subspace we obtain new proofs of the results of h dym and i gohberg and grone johnson sa and wolkowitz linear algebra appl 58 1984 109 124 linear algebra appl 36 1981 1 24 we also obtain a new proof of a result of u haagerup decomposition of completely bounded maps on operation algebras preprint characterizing the norm of schur product maps and a new hahn banach type extension theorem for these maps finally we obtain generalizations of many of these results to matrices of operators which we apply to the study of representations of certain subalgebras of the n n matrices

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

algebraic curves with many automorphisms abstract let x be a projective geometrically irreducible non singular algebraic curve of genus g 2 defined over an algebraically closed field k of odd characteristic p let aut x be the group of all automorphisms of x which fix k element wise it is known that if aut x 8 g 3 then the p rank equivalently the hasse witt invariant of x is zero this raises the problem of determining the minimum value function f g such that whenever aut x f g then x has zero p rank for even g we prove that f g 900 g 2 the odd genus case appears to be much more difficult although for any genus g 2 if aut x has a solvable subgroup g such that g 252 g 2 then x has zero p rank and g fixes a point of x our proofs use the hurwitz genus formula and the deuring shafarevich formula together with a few deep results from group theory characterizing finite simple groups whose sylow 2 subgroups have a cyclic subgroup of index 2 we also point out some connections with the abhyankar conjecture and the katz gabber covers

translation and dilation invariant subspaces of l2 r the closed subspaces of the hilbert space l2 r which are invariant under multiplication by hy r functions and the dilation operators f x f sx 1 s y are determined as the two parameter family of subspaces l2 a b 0ea bey which are reducing for multiplication operators together with a four parameter family of nonreducing subspaces the lattice and topological structure are determined and using operator algebra methods the corresponding family of orthogonal projections with the weak operator topology is identified as a compact connected 4 manifold

on colmez s product formula for periods of cm abelian varieties colmez conjectured a product formula for periods of abelian varieties with complex multiplication by a field k analogous to the standard product formula in algebraic number theory he proved this conjecture up to a rational power of 2 for k mathbb q abelian in this paper we complete the proof of colmez for k mathbb q abelian by eliminating this power of 2 our proof relies on analyzing the galois action on the de rham cohomology of fermat curves in mixed characteristic 0 2 which in turn relies on understanding the stable reduction of mathbb z 2 n covers of the projective line branched at three points

resolution of singularities and modular galois theory i shall sketch a brief history of the desingularization problem from riemann thru zariski to hironaka including the part i played in it and the work on galois theory which this led me to and how that caused me to search out many group theory gurus i shall also formulate several conjectures and suggest numerous thesis problems

polynomials with psl 2 monodromy let k be a field of characteristic p 0 let q be a power of p and let u be transcendental over k we determine all polynomials f k x k x p of degree q q 1 2 for which the galois group of f x u over k u has a transitive normal subgroup isomorphic to psl 2 q subject to a certain ramification hypothesis as a consequence we describe all polynomials f k x such that deg f is not a power of p and f is functionally indecomposable over k but f decomposes over an extension of k moreover except for one ramification configuration which is handled in a companion paper with rosenberg we describe all indecomposable polynomials f k x such that deg f is not a power of p and f is exceptional in the sense that x y is the only absolutely irreducible factor of f x f y which lies in k x y it is known that when k is finite a polynomial f is exceptional if and only if it induces a bijection on infinitely many finite extensions of k

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

cyclic quotients of transitive groups abstract let a be a transitive subgroup of s n we show that the largest cyclic quotient of a has order at most n this can be interpreted as an equivalent result about extensions of constants in the galois closure of a covering of curves over a finite field we also prove that the point stabilizer in a finite primitive permutation group always has a faithful orbit

on subharmonic solutions of systems of difference equations with periodic perturbations part ii multiplicity and stability we consider the multiplicity and stability of subharmonic solutions of discrete dynamic systems with periodic perturbations whose existence was established in the first part of this series under the hypotheses that the perturbation is locally lipschitz continuous we determine the precise number of subharmonic solutions and we also show that these subharmonic solutions are stable unstable provide the periodic orbit of the unperturbed system is stable unstable

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

families of wildly ramified covers of curves in this paper i investigate wildly ramified g galois covers of curves inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i branched at exactly one point over an algebraically closed field k of characteristic p i answer a question of m raynaud by showing that proper families of such covers of a twisted projective line are isotrivial the method is to construct an affine moduli space for covers whose inertia group is of the form inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i there are two other applications of this space in the case that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i the first uses formal patching to compute the dimension of the space of nonisotrivial deformations of in terms of the lower jump of the filtration of higher inertia groups the second gives necessary criteria for good reduction of families of such covers these results will be used in a future paper to prove the existence of such covers with specified ramification data

wild ramification and a vanishing cycles formula in duke math j 55 1987 629 659 k kato proved using techniques from k theory a formula which compares the dimensions of the spaces of vanishing cycles in a finite morphism between formal germs of curves over a complete discrete valuation ring to the best of my knowledge kato s formula is explicit only in the case where this morphism is generically separable on the level of special fibres in this note we prove using formal patching techniques a la harbater an analogous explicit formula in the case of a galois cover of degree p between formal germs of curves over a complete discrete valuation ring of unequal characteristic 0 p which includes the case where we have inseparability on the level of special fibres the results of this paper play a key role in math ag 0106249 where is studied the semi stable reduction of galois covers of degree p of curves over a complete discrete valuation ring of unequal characteristics 0 p as well as the galois action on these covers

wild ramification and k pi 1 spaces we prove that every connected affine scheme of positive characteristic is a k pi 1 space for the etale topology the main ingredient is the special case of the affine space mathbf a k n over a field k this is dealt with by induction on n using a key bertini type statement regarding the wild ramification of ell adic local systems on affine spaces which might be of independent interest its proof uses in an essential way recent advances in higher ramification theory due to t saito we also give rigid analytic and mixed characteristic versions of the main result

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

two classes of piecewise linear difference equations with eventual periodicity three in this paper we will study two classes of difference equations which are piecewise linear and of similar forms we will show that all nontrivial solutions of one equation are eventually periodic with prime period three we will show this result for one case of the second equation

teichmuller curves triangle groups and lyapunov exponents we construct a teichmuller curve uniformized by a fuchsian triangle group commensurable to m n for every m n in most cases for example when m 6 n and m or n is odd the uniformizing group is equal to the triangle group m n our construction includes the teichmuller curves constructed by veech and ward as special cases the construction essentially relies on properties of hypergeometric differential operators for small m we find billiard tables that generate these teich muller curves we interpret some of the so called lyapunov exponents of the kontsevich zorich cocycle as normalized degrees of a natural line bundle on a teichmuller curve we determine the lyapunov exponents for the teichmuller curves we construct

rigidity reduction and ramification in this paper we consider wildly ramified g galois covers of curves f y 1k branched at exactly one point over an algebraically closed field k of characteristic p for g equal to ap or psl2 p we prove abhyankar s inertia conjecture that all possible inertia groups occur over infinity for such covers f in addition we prove that the set of conductors that can be realized depends on the group the method we use is to compute the reduction of galois covers of mathbb p 1 overline mathbb q branched at 3 points we observe that the existence of covers with given inertia in characteristic p is closely related to the arithmetic of covers in characteristic zero

connectedness of the moduli space of artin schreier curves of fixed genus abstract we study the moduli space as g of artin schreier curves of genus g over an algebraically closed field k of positive characteristic p the moduli space is partitioned into strata which are irreducible each stratum parameterizes artin schreier curves whose ramification divisors have the same coefficients we construct deformations of these curves to study the relations between those strata as an application when p 3 we prove that as g is connected for all g when p 3 it turns out that as g is connected for a sufficiently large value of g in the course of our work we answer a question of pries and zhu about how a combinatorial graph determines the geometry of as g

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

construction de revetements de courbes pointees resume si k est un corps fertile large field chez f pop g un k schema en groupes fini etale et x un g torseur on construit un g revetement ramifie c p 1k ou c est lisse et geometriquement connexe sur k induisant en 0 p 1 k le torseur x lorsque k est de caracteristique nulle c est un resultat recent de colliot thelene on termine l article par des commentaires sur l utilisation du langage des champs algebriques dans ce type de probleme copyright 2001 academic press if k is a large field in the sense of f pop g a finite etale k group scheme and x a g torsor we construct a ramified g covering c p 1k where c is smooth and geometrically connected over k inducing at 0 p 1 k the given x when k has characteristic zero this was proved recently by colliot thelene we end with some comments on how the language of algebraic stacks can be used for such problems

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

on the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t with periodic parameters we investigate the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t where for each value of i the sequence ain n 0 of positive numbers is periodic with period pi we give both sufficient conditions on the pi s for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded this work essentially complements the work by bidwell and franke who showed that as long as every positive solution of our equation is bounded then every positive solution is eventually periodic thereby leaving open the question as to when solutions are bounded

periodicity of a second order switched difference system over integers abstract in this paper a second order switched difference system which consists of two linear difference equations with a switching rule is proposed to study specifically the periodicity of a particular case is addressed deriving the appropriate rational values for parameter r which possess periodic integer solutions by the transformation method the particular second order difference system is transformed into a first order switched system and we prove that 1 this system possesses periodic integer solutions of prime period two if and only if r 1 2 2 any rational r except for the integers arises periodic integer solutions of prime period three 3 periodic integer solutions of prime period four exist if and only if r 1 2 4 this system possesses no periodic solutions of prime period five we also prove that if r 0 and the system has periodic integer solutions of prime period k 6 then the only possible values of r are reciprocals of integers

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

positive periodic solutions of nonlinear functional difference equations depending on a parameter in this paper we use the upper and lower solutions method to show that there existsa l such that the nonlinear functional difference equation of the form g n 1 a n g n lh n g n t n n e z has at least one positive t periodic solutions for l e 0 l and does not have any positive t periodic solutions for l l where a n h n and t n are t periodic functions

effective methods for vanishing cycles of p cyclic covers of the p adic line abstract this paper studies the stable reduction of p cyclic covers x p k 1 of the projective line over p adic fields so far an algorithm to effectively determine the stable reduction of such covers is only known under additional hypothesis on the branch locus of the cover here rather than restricting the type of cover we consider the general case and obtain results on the structure of the special fiber xk of the stable reduction of x special attention is payed to making all constructions effective the central result is a formula computing the number of vanishing cycles on xk in particular we give criteria for the special fiber of the stable reduction to be tree like and for when x is a mumford curve refining the analysis of vanishing cycles we describe an algorithm that computes all the components of positive p rank in the stable model

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

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

formal and rigid geometry in 1974 raynaud proposed a program raynaud in mem soc math fr 39 40 319 327 1974 where he introduced groundbreaking ideas to rigid geometry by interpreting a rigid analytic space as the generic fiber of a formal schemes over operatorname spf r here operatorname spf r is always the formal spectrum of a complete valuation ring r of height 1 where its topology is given by an ideal pi for some element pi in r with 0 pi 1 due to results on flat modules raynaud and gruson in invent math 13 1 89 1971 his approach also works in the non noetherian case of formal schemes of topologically finite presentation over operatorname spf r

semi discreteness and dilation theory for nest algebras abstract in this paper it is shown that a contractive weakly continuous hilbert space representation of a nest algebra admits a weakly continuous dilation to the containing algebra of all operators this is accomplished by first establishing the complete contractivity of contractive representations through a semi discreteness property for nest algebras relative to finite dimensional nest algebras theorem 2 1 the semi discreteness property is obtained by an examination of the order type spectral type and multiplicity of the nest and by the construction of subalgebras that are completely isometric copies of finite dimensional nest algebras with good approximation properties with complete contractivity at hand the desired dilation follows from arveson s dilation theorem and auxiliary arguments

almost locally free fields abstract using the positive solution of the general abhyankar s conjecture we prove that the fundamental group 1 c of the smooth connected affine curve c is almost free that is for each positive integer e and for almost all 1 e 1 c e in the sense of the haar measure the closed subgroup of 1 c generated by 1 e is profinite free on e generators this implies a theorem of harbater stevenson proved by other means that every finite embedding problem for 1 c is solvable if we restrict the problem to a suitable open subgroups

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

convergent arithmetic power series this paper considers rings of power series on discs of radius r 0 with coefficients lying in a subring a of the rational numbers one such ring denoted here by ar t is an arithmetic version of the disc algebra studied in analysis a slightly smaller ring ar t consists of the holomorphic functions on a neighborhood of the closed disc of radius r with coefficients lying in a the rings z1 t and z1 t are in fact equal to z t and so the families zr ti 0 r i and zr t i 0 r 1 can be regarded as interpolating between z t and z t these convergent power series rings may thus be of use in a deformation theoretic context the maximal spectra of ar t and ar t are described in section 1 and the latter ring is shown to be a noetherian regular unique factorization domain the ring zr t meanwhile is shown to be complete under the uniform norm section 2 shows that every finitely generated projective module over ar t is free the proof uses a patching lemma for free modules this section also contains a more general module patching result which can be viewed as an arithmetic analog of grothendieck s existence theorem from this point of view z is analogous to the polynomial ring k xl over a field k z t and z t correspond to k x t and k x t and zr t corresponds to k t x whose spectrum is a tubular neighborhood of the affine k line section 3 considers the subring ar kitih of ar t consisting of algebraic power series every such power series is shown to lie in an etale neighborhood of the disc it is also shown that the ring z t h of algebraic power series over z on the open unit disc consists entirely of rational functions given an algebraic structure over z t h obtained for example by