on the descending central sequence of absolute galois groups let p be an odd prime number and f a field containing a primitive p th root of unity we prove a new restriction on the group theoretic structure of the absolute galois group g f of f namely the third subgroup g f 3 in the descending p central sequence of g f is the intersection of all open normal subgroups n such that g f n is 1 bbb z p 2 or the extra special group m p 3 of order p 3 and exponent p 2

a mackey imprimitivity theory for algebraic groups let g be an affine algebraic group over an algebraically closed field k and let h be a closed subgroup of g if v is a rational h module a comodule for the coordinate ring of h there is a now well known notion of an induced module vi g for g defined as the space morph g v of all h equivariant morphisms from g to a finite dimensional subspace of v with obvious g action the question arises given a rational g module m how can one recognize m as an induced module vig for a finite group g the answer is the mackey imprimitivity theorem the module m is induced if and only if it is a direct sum of subspaces permuted transitively by g with h the stabilizer of one of these subspaces called v one uses this result for example in proving the famous mackey decomposition theorem which describes the restriction of any induced module to a second subgroup l as a direct sum of suitable induced modules given the imprimitivity theorem the proof is just a matter of grouping the summands permuted by g into their orbits under l in the case of algebraic groups the situation is quite different for some subgroups h all g modules are induced this occurs for example if g is connected and k g h k e g if h is parabolic also if g is a connected unipotent group then a rational g module m is induced from some proper subgroup if and only if its endomorphism ring contains a two dimensional submodule e for the conjugation action of g with e k 1 and such that k l is precisely the subspace annihilated by the action of the lie algebra of g on e 27 cf also 5 5 below the latter is an application of a general criterion in case g h is affine a rational g module m is induced if and only if there is an action of the coordinate ring a of g h compatible with the action of g on both a and m note that this generalizes the imprimitivity theorem in the case of finite groups since then a has a k basis of g h orthogonal idempotents permuted transitively by g

cubic structures equivariant euler characteristics and lattices of modular forms we use the theory of cubic structures to give a fixed point riemann roch formula for the equivariant euler characteristics of coherent sheaves on projective at schemes over z with a tame action of a finite abelian group this formula supports a conjecture concerning the extent to which such equivariant euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus our results are applied to study the module structure of modular forms having fourier coeficients in a ring of algebraic integers as well as the action of diamond hecke operators on the mordell weil groups and tate shafarevich groups of jacobians of modular curves

witt groups of curves and surfaces we study witt groups of smooth curves and surfaces over algebraically closed fields of characteristic not two in both dimensions we determine both the classical witt group and balmer s shifted witt groups in the case of curves the results are supplemented with a complete description of the shifted grothendieck witt groups in a second step we analyse the relationship of witt groups of smooth complex curves and surfaces with their real topological k groups they turn out to be surprisingly close for all curves and for all projective surfaces of geometric genus zero the witt groups may be identified with the quotients of their even ko groups by the images of their complex topological k groups under realification

instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings this paper reports estimates of the effects of jtpa training programs on the distribution of earnings the estimation uses a new instrumental variable iv method that measures program impacts on quantiles the quantile treatment effects qte estimator reduces to quantile regression when selection for treatment is exogenously determined qte can be computed as the solution to a convex linear programming problem although this requires first step estimation of a nuisance function we develop distribution theory for the case where the first step is estimated nonparametrically for women the empirical results show that the jtpa program had the largest proportional impact at low quantiles perhaps surprisingly however jtpa training raised the quantiles of earnings for men only in the upper half of the trainee earnings distribution

higher chern classes in iwasawa theory we begin a study of mth chern classes and mth characteristic symbols for iwasawa modules which are supported in codimension at least m this extends the classical theory of characteristic ideals and their generators for iwasawa modules which are torsion i e supported in codimension at least 1 we apply this to an iwasawa module constructed from an inverse limit of p parts of ideal class groups of abelian extensions of an imaginary quadratic field when this module is pseudo null which is conjecturally always the case we determine its second chern class and show that it has a characteristic symbol given by the steinberg symbol of two katz p adic l functions

homological mirror symmetry for singularities of type d we prove homological mirror symmetry for lefschetz fibrations obtained as sebastiani thom sums of polynomials of types a or d the proof is based on the behavior of the fukaya category under sebastiani thom summation of a polynomial of type d

triangular witt groups part ii from usual to derived we establish that the derived witt group is isomorphic to the usual witt group when 2 is invertible this key result opens the ali baba s cave of triangular witt groups linking the abstract results of part i to classical questions for the usual witt group for commercial purposes we survey the future applications of triangular witt groups in the introduction we also establish a connection between odd indexed witt groups and formations finally we prove that over a commutative local ring in which 2 is a unit the shifted derived witt groups are all zero but the usual one

on the structure of iv estimands abstract when the overidentifying restrictions of the constant effect linear instrumental variables model fail common iv estimators converge to different probability limits i characterize the estimands of two stage least squares two step gmm and limited information maximum likelihood as functions of the single instrument estimands from the just identified iv regressions which consider each instrument separately the limited information maximum likelihood estimand is found to be discontinuous on a set of dimension equal to the number of instruments minus one and to equal the full parameter space on a set of dimension equal to the number of instruments minus two

using instrumental variables for inference about policy relevant treatment effects we propose a method for using instrumental variables iv to draw inference about causal effects for individuals other than those affected by the instrument at hand policy relevance and external validity turn on the ability to do this reliably our method exploits the insight that both the iv estimand and many treatment parameters can be expressed as weighted averages of the same underlying marginal treatment effects since the weights are identified knowledge of the iv estimand generally places some restrictions on the unknown marginal treatment effects and hence on the values of the treatment parameters of interest we show how to extract information about the treatment parameter of interest from the iv estimand and more generally from a class of iv like estimands that includes the two stage least squares and ordinary least squares estimands among others our method has several applications first it can be used to construct nonparametric bounds on the average causal effect of a hypothetical policy change second our method allows the researcher to flexibly incorporate shape restrictions and parametric assumptions thereby enabling extrapolation of the average effects for compliers to the average effects for different or larger populations third our method can be used to test model specification and hypotheses about behavior such as no selection bias and or no selection on gain

a characterization of c fields via galois groups abstract it is known that those fields f which are of class c in the sense of r ware can be characterized by their witt ring structure as well as by the behavior of their binary quadratic forms and the structure of their galois 2 extensions in this paper we show that c fields which have at least 8 square classes are characterized by the absence of a certain group dc of order 16 as a galois group over them moreover we show that the level of such a field is determined by the presence or absence of the cyclic group of order 4 and the dihedral group of order 8 as galois groups over it these results provide an interesting continuation of the work of ware in the appendix we give an explicit construction of field extensions with galois group dc

stiefel whitney classes of real representations of finite groups abstract we prove the existence and uniqueness of stiefel whitney classes of real representations subject to the hirzebruch axioms within the category of finite groups we also give a method of constructing and computing with these classes using multiplicative transfer and induced representations and apply our formulae in certain simple cases

on the classification of quasihomogeneous functions we give a criterion for the existence of a non degenerate quasihomogeneous polynomial in a configuration i e in the space of polynomials with a fixed set of weights and clarify the relation of this criterion to the necessary condition derived from the formula for the poincare polynomial we further prove finiteness of the number of configurations for a given value of the singularity index for the value 3 of this index which is of particular interest in string theory a constructive version of this proof implies an algorithm for the calculation of all non degenerate configurations

on arithmetic class invariants let f be a number field with ring of integers of and let s be a finite set of places of f assume that s contains the set s of archimedean places of f and write sf for the set of finite places contained in s let os or o when there is no danger of confusion denote the ring of sf integers of f write f for an algebraic closure of f let y be any scheme over spec o suppose thatg is a finite flat commutative group scheme over y of exponent n and let g denote the cartier dual of g let x y be ag torsor and write 0 g y for the trivialg torsor thenox is anog comodule and so it is also anogd module see 12 as an ogd module ox is locally free of rank one and it therefore gives a line bundle m over g set l m m 1 0 then the map h y g pic g l

triangulated categories of matrix factorizations for regular systems of weights with e 1 abstract we construct a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a nondegenerate regular system of weights whose smallest exponents are equal to 1 in the associated grothendieck group the strongly exceptional collection defines a root basis of a generalized root system of sign l 0 2 and a coxeter element of finite order whose primitive eigenvector is a regular element in the expanded symmetric domain of type iv with respect to the weyl group

twists of symmetric bundles we establish comparison results between the hasse witt invariants w t e of a symmetric bundle e over a scheme and the invariants of one of its twists e alpha for general twists we describe the difference between w t e and w t e alpha up to terms of degree 3 next we consider a special kind of twist which has been studied by a frohlich this arises from twisting by a cocycle obtained from an orthogonal representation a simple important example of this twisting procedure is the bilinear trace form of an etale algebra which is obtained by twisting the standard sum of squares form by the orthogonal representation attached to the algebra we show how to explicitly describe the twist for representations arising from very general tame actions this involves the square root of the inverse different which serre esnault kahn viehweg and ourselves had studied before for torsors we show that in our geometric set up jardine s generalization of frohlich s formula holds namely let x g be a torsor with quotient y let e be a symmetric bundle over y let rho g rightarrow bf o e be an orthogonal representation and let e rho x be the corresponding twist of e then we verify up to degree 3 that the formula w t e rho x sp t rho w t e w t rho holds here sp t rho and w t rho are respectively the spinor invariant and the stiefel whitney class of rho the case of genuinely tamely ramified actions is geometrically more involved and leads us to introduce an invariant of ramification which in a sense gives a decomposition in terms of representations of the inertia groups of the invariant introduced by serre for curves the comparison result in the tamely ramified case proceeds by reduction to the case of a torsor the reduction is carried out by means of a partial normalisation procedure which we had introduced in a previous paper an important lemma of esnault kahn and viehweg allows us to express the difference between the invariants of bundles before and after the normalisation procedure in terms of chern classes of certain sub bundles as noted elsewhere this result can be best understood in the context of symmetric complexes and their invariants our results are new even for bundles over curves and they allow us to weaken the regularity assumptions that we had to impose in previous work of ours

galois structure of de rham cohomology of tame covers of schemes a class on in the grothendieck group ko z g of all finitely generated projective z g modules the class group cl z g of z g is defined to be the quotient of ko z g by the subgroup generated by the class of z g let on stab be the image of on in cl z g in 34 taylor proved fr6hlich s conjecture that on stab is equal to another invariant wn k in cl z g which cassou nogues had defined by means of the root numbers of symplectic representations of the galois group g the root number of a representation v of g is the constant which appears in the functional equation of the artin lfunction of v the connection between galois structure invariants and artin l functions has been a basic theme in research on galois structure for further discussion see 12 and 4

galois module structure of unramified covers we use the theory of n cubic structures to study the galois module structure of the coherent cohomology groups of unramified galois covers of varieties over the integers assuming that all the sylow subgroups of the covering group are abelian we show that the invariant that measures the obstruction to the existence of a virtual normal integral basis is annihilated by a product of certain bernoulli numbers with orders of even k groups of z we also show that the existence of such a basis is closely connected to the truth of the kummer vandiver conjecture for the prime divisors of the degree of the cover

the arason invariant and mod 2 algebraic cycles introduction 73 1 review of the arason invariant 75 2 the special clifford group 76 3 k cohomology of split reductive algebraic groups 78 4 k cohomology of bg 86 5 gl n and cliff n n 92 6 two invariants for clifford bundles 95 7 snaking a bloch ogus differential 100 8 proof of theorem 1 101 9 application to quadratic forms 102 appendix a toral descent 104 appendix b the rost invariant 108 appendix c an amusing example 113 acknowledgements 116 references 116

on the computation of the trace form of some galois extensions abstract we investigate the trace form trl k l k x trl kx2of a finite galois extensionl k in particular we study 2 extensions of degree 16 using some reduction theorems these results yield a classification of nearly all trace forms of galois extensions of degree 31 finally we study the trace form of a cyclotomic extension and of its maximal real subfield

extended canonical algebras and fuchsian singularities the authors introduce a new class of finite dimensional algebras called extended canonical and determine the shape of their derived categories extended canonical algebras arise from a canonical algebra by onepoint extension or coextension by an indecomposable projective module our main results concern the case of negative euler characteristic of the corresponding weighted projective line mathbb x more specifically we establish for a base field of arbitrary characteristic a link to the fuchsian singularity r of mathbb x which for the base field of complex numbers is isomorphic to an algebra of automorphic forms by means of a recent result of orlov we show that the triangulated category of the graded singularities of r in the sense of buchweitz and orlov admits a tilting object whose endomorphism ring is the corresponding extended canonical algebra of particular interest are those cases where the attached coxeter transformation has spectral radius one a k theoretic analysis then shows that this happens exactly for 38 cases including arnold s 14 exceptional unimodal singularities the paper is related to recent independent work by kajiura saito and takahashi

on different notions of tameness in arithmetic geometry the notion of a tamely ramified covering is canonical only for curves several notions of tameness for coverings of higher dimensional schemes have been used in the literature we show that all these definitions are essentially equivalent furthermore we prove finiteness theorems for the tame fundamental groups of arithmetic schemes

induced orthogonal representations of galois groups abstract we prove a result showing the connection between induced orthogonal representations and corestrictions of group extensions derived from their clifford groups by the means of the corestriction map we then obtain new obstructions to the p embedding problems given by the group extensions of the modular p group and to one 2 embedding problem given by a group extension of the dihedral group

construction de p extensions galoisiennes d un corps de caracteristique differente de p abstract let p be a prime number let k be a field with characteristic different from p containing the pth roots of unity and let e k be a galois extension of degree pd d n o with an abelian galois group gal e k g of exponent p the aim of this article is to state explicit formulas by which one can construct any field n of degree p over e such that n k is a galois extension soient p un nombre premier k un corps de caracteristique differente de p contenant les racines p iemes de l unite et e k une extension galoisienne de degre pd d n o de groupe de galois gal e k g abelien d exposant p le but de cet article est d etablir des formules explicites permettant de construire tous les corps n de degre p sur e tels que l extension n k soit galoisienne

the classifying topos of a group scheme and invariants of symmetric bundles let y be a scheme in which 2 is invertible and let v be a rank n vector bundle on y endowed with a non degenerate symmetric bilinear form q the orthogonal group bf o q of the form q is a group scheme over y whose cohomology ring h b bf o q bf z 2 bf z simeq a y hw 1 q hw n q is a polynomial algebra over the etale cohomology ring a y h y et bf z 2 bf z of the scheme y here the hw i q s are jardine s universal hasse witt invariants and b bf o q is the classifying topos of bf o q as defined by grothendieck and giraud the cohomology ring h b bf o q bf z 2 bf z contains canonical classes mathrm det q and c q of degree 1 and 2 respectively which are obtained from the determinant map and the clifford group of q the classical hasse witt invariants w i q live in the ring a y our main theorem provides a computation of det q and c q as polynomials in hw 1 q and hw 2 q with coefficients in a y written in terms of w 1 q w 2 q in a y this result is the source of numerous standard comparison formulas for classical hasses witt invariants of quadratic forms our proof is based on computations with abelian and non abelian cech cocycles in the topos b bf o q this requires a general study of the cohomology of the classifying topos of a group scheme which we carry out in the first part of this paper

the sign of the functional equation of the l function of an orthogonal motive we prove that the sign of the functional equation of the l function of an orthogonal motive of even weight is always positive if we admit the conjecture of the product formula for the epsilon factor of deligne d3 5 2 more precisely we define the sign independently of any conjecture by using the product formula as the definition and by working with one chosen good prime f and we prove its positivity the positivity has been established for the artin motives fr q d2 for the symmetric square of a modular elliptic curve c sc and in positive characteristic by serre fr q as in the proof in d2 we consider the second stiefel whitney class in zbr k of an orthogonal representation of the absolute galois group g of an algebraic number field k the positivity of the sign is then a consequence of the reciprocity law of the square residue symbol fixing a good prime f we work with an f adic representation and compare the local epsilon factor and the local stiefel whitney class for a prime vxr it is a theorem of deligne d2 for a prime vie we compute the stiefel whitney class assuming the representation is cristalline theorem 2 using a generalization of a theorem of fr6hlich fr to f adic representation of hodge tate type theorem 1 by combining them with an elementary computation for the hodge structures at r iot we conclude that the sign is positive theorem 3 the crucial step in the proof of theorem 2 given in section 1 is the determination by fontaine and lafaille fo l of the action of the tame inertia on the semisimplification of reduction modulo p of a cristalline representation the theory of orthogonal representations of galois group including a proof of fr6hlich s theorem is reviewed in a preliminary section 0

tameness and local normal bases for objects of finite hopf algebras let r be a commutative ring s an r algebra h a hopf ralgebra both finitely generated and projective as r modules and suppose s is an h object so that h homr h r acts on s via a measuring let i be the space of left integrals of h we say s has normal basis if s h as h modules and s has local normal bases if sp hp as hp modules for all prime ideals p of r when r is a perfect field h is commutative and cocommutative and certain obvious necessary conditions on s hold then s has normal basis if and only if is r sh if r is a domain with quotient field k h is cocommutative and l s or k has normal basis as h 0 k module then s has local normal bases if and only if is r sh suppose k is a number field with ring of integers r l is a finite galois extension of k with galois group g and s is the integral closure of r in l then g acts as a group of automorphisms of s relative galois module theory seeks to understand s as an rg module via this action the most basic question is to inquire whether s is locally isomorphic to rg that is for all primes p of r sp has a local normal basis as a free rp module this question was answered by emmy noether who showed that s has a local normal basis at every prime p of r if and only if l k is tamely ramified here tamely ramified means that at each prime ideal p of r the ramification index of any prime p of s lying over p is relatively prime to the residue field characteristic it is well known that this latter condition is equivalent to the surjectivity of the trace map tr s r tr s ezacg s the purpose of this paper is to formulate and prove an extension of noether s theorem to objects of finite cocommutative hopf algebras assume now only that r is a commutative ring h is a hopf r algebra which is a finitely generated projective r module and h homr h r the dual hopf algebra a commutative r algebra s is an h object if s is a right h comodule via a map a s s 0 h which is an r algebra homomorphism if s is an h object then a induces a measuring a h s s in the sense of sweedler 21 galois module theory in this setting is the study of s as an h module via the measuring a the trivial example is s h itself with a a the comultiplication on h so we say that the h object has normal basis if s h as h modules and s has local normal bases if for all primes p of r sp hp as h modules since rgf rg as rg modules for g any finite group this notion of normal basis when specialized to h rg is equivalent to the classical notion received by the editors may 7 1985 and in revised form january 7 1986 1980 mathematics subject classification 1985 revision primary 13b05 secondary 16a24 1986 american mathematical society 0002 9947 86 1 00 25 per page

galois structure of homogeneous coordinate rings suppose g is a finite group acting on a projective scheme x over a commutative noetherian ring r we study the rg modules h 0 x f l n when n 0 and f and are coherent g sheaves on x such that l is an ample line bundle we show that the classes of these modules in the grothendieck group g 0 rg of all finitely generated rg modules lie in a finitely generated subgroup under various hypotheses we show that there is a finite set of indecomposable rg modules such that each h 0 x f l n is a direct sum of these indecomposables with multiplicities given by generalized hilbert polynomials for n 0

maximally graded matrix factorizations for an invertible polynomial of chain type abstract in 1977 orlik randell construct a nice integral basis of the middle homology group of the milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles the purpose of this paper is to prove the algebraic counterpart of the orlik randell conjecture under the homological mirror symmetry we may expect that the triangulated category of maximally graded matrix factorizations for the berglund hubsch transposed polynomial admits a full exceptional collection with a nice numerical property indeed we show that the category admits a lefschetz decomposition with respect to a polarization in the sense of kuznetsov smirnov whose euler matrix are calculated in terms of the zeta function of the inverse of the polarization as a corollary it turns out that the homological mirror symmetry holds at the level of the grothendieck group in the following sense the grothendieck group of the category with the euler form is isomorphic to the middle homology group with the quasi seifert form with a suitable sign

on the e constants of a variety over a finite field let x denote an irreducible smooth projective variety over a finite field which supports an action by a finite group g for each complex representation v of g we let e x g v denote the constant in the functional equation of the associated l function in this paper we study the constants e x g v when the character of the representation v is real valued our first main result shows that e x g v is positive either when the dimension d say of x is odd and v is orthogonal or when d is even and v is symplectic this then generalizes serre s theorem for the e constants of orthogonal representations for curves our second main result concerns the situation where g acts tamely on x and where the repre sentation v has real valued character is of even degree and has trivial determinant we than show that the constants e x g v are determined by certain hermitian euler characteristics associated to a stratification of x this result is particularly striking when x is a curve for then our result shows that the symplectic e constants are all determined by the virtual fp g module h ox hl ox 1 introduction let x be a regular projective variety of dimension d over a finite field k of characteristic p with q elements let k denote an algebraic closure of k and put x x x k suppose further that x carries an action by a finite group g which is tamely ramified over a divisor with strictly normal

matrix factorizations and representations of quivers ii type ade case abstract we study a triangulated category of graded matrix factorizations for a polynomial of type ade we show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding dynkin quiver also we discuss a special stability condition for the triangulated category in the sense of t bridgeland which is naturally defined by the grading