zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

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

technical communique who needs qp for linear mpc anyway conventional mpc uses quadratic programming qp to minimise on line a cost over n linearly constrained control moves however stability constraints often require the use of large n thereby increasing the on line computation rendering the approach impracticable in the case of fast sampling here we explore an alternative that requires a fraction of the computational cost which increases only linearly with n and propose an extension which in all but a small class of models matches to within a fraction of a percent point the performance of the optimal solution obtained through qp the provocative title of the paper is intended to point out that the proposed approach offers a very attractive alternative to qp based mpc

galois module structure of the square root of the inverse different in even degree tame extensions of number fields abstract let g be a finite group and let n e be a tamely ramified g galois extension of number fields whose inverse different c n e is a square let a n e denote the square root of c n e then a n e is a locally free z g module which is in fact free provided n e has odd order as shown by erez using m taylor s theorem we can rephrase this result by saying that when n e has odd degree the classes of a n e and o n the ring of integers of n in cl z g are equal and in fact both trivial we show that the above equality of classes still holds when n e has even order assuming that n e is locally abelian this result is obtained through the study of the frohlich representatives of the classes of some torsion modules which are independently introduced in the setting of cyclotomic number fields jacobi sums together with the hasse davenport formula are involved in this study finally when g is the binary tetrahedral group we use our result in conjunction with taylor s theorem to exhibit a tame g galois extension whose square root of the inverse different has nontrivial class in cl z g

optimizing prediction dynamics for robust mpc a convex formulation is derived for optimizing dynamic feedback laws for constrained linear systems with polytopic uncertainty we show that when it exists the maximal invariant ellipsoidal set for the plant state under a dynamic feedback law incorporating any chosen static feedback gain is equal to the maximal invariant ellipsoidal set under any linear feedback law the dynamic controller and its associated invariant set define a computationally efficient robust model predictive control mpc law with prediction dynamics belonging to a polytopic uncertainty set

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

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

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

tamagawa numbers for motives with noncommutative coefficients ii we provide corroborative evidence for the equivariant tamagawa number conjecture which was formulated in the first part of this article

neron models lie algebras and reduction of curves of genus one let k be a discrete valuation field let ok denote the ring of integers of k and let k be the residue field of ok of characteristic p 0 let s specok let x k be a smooth geometrically connected projective curve of genus 1 over k denote by ek the jacobian of x k let x s and e s be the minimal regular models of x k and ek respectively in this article we investigate the possible relationships between the special fibers xk and ek in doing so we are led to study the geometry of the picard functor pic x s when x s is not necessarily cohomologically flat as an application of this study we are able to prove in full generality a theorem of gordon on the equivalence between the artin tate and birch swinnertondyer conjectures recall that when k is algebraically closed the special fibers of elliptic curves are classified according to their kodaira type which is denoted by a symbol t in i n n z 0 ii ii iii iii iv iv given a type t and a positive integer m we denote by mt the new type obtained from t by multiplying all the multiplicities of t by m when k is algebraically closed the relationships between the type of a curve of genus 1 and the type of its jacobian can be summarized as follows

time optimal model predictive control for linear systems based on ellipsoidal and polytopic sets time optimal model predictive control mpc strategies are investigated in this paper we first derive a series of controllable sets described by ellipsoids or convex polytopes offline then obtain

active walker models tracks and landscapes abstract the track patterns from the active walker models awms are compared with experimental retinal neuron and dielectric breakdown of liquid patterns respectively excellent qualitative and quantitative agreements are obtained the landscapes from the boltzmann awm in 1 1 dimensions form rough surfaces with a first order phase transition as the height of the landscaping function w0 is varied landscapes and statistics of the tracks from the probabilistic awm in 2 1 dimensions are presented

boltzmann active walkers and rough surfaces abstract an active walker model awm was recently proposed by freimuth and lam for the generation of various filamentary patterns in an awm the walker changes the landscape as it walks and its steps are in turn influenced by the changing landscape the landscape so obtained is a rough surface in this paper the properties of such a rough surface with average height conserved generated by a boltzmann active walker in 1 1 dimensions is investigated in detail the scaling properties of the surface thickness t is found to belong to a new class quite different from other types of fractal surfaces for example t is independent of the system size l but is a function of the temperature t soliton propagation is found when t 0

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

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

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

galois structure of zariski cohomology for weakly ramified covers of curves we compute equivariant euler characteristics of locally free sheaves on curves thereby generalizing several results of kani and nakajima for instance we extend kani s computation of the galois module structure of the space of global meromorphic differentials which are logarithmic along the ramification locus from the tamely ramified to the weakly ramified case

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

scaling and universality in animate and inanimate systems we illustrate the general principle that in biophysics econophysics and possibly even city growth the conceptual framework provided by scaling and universality may be of use in making sense of complex statistical data specifically we discuss recent work on dna sequences heartbeat intervals avalanche like lung inflation urban growth and company growth although our main focus is on data we also discuss statistical mechanical models

clustering of active walkers in a two component system this paper investigates the agglomeration of active walkers component a on a two dimensional surface described by a potential u r t that determines the motion of the walkers the walkers are able to change u rt locally by producing a second component b that decreases u r t and which can diffuse and decompose the nonlinear feedback between the spatio temporal density distributions of both components results in a clustering of the walkers the analytic description is based on a set of langevin and fokker planck equations for the active walkers coupled by a reaction diffusion equation for the component b we investigate the stability of homogeneous solutions the selection equation and an effective diffusion coefficient which is negative in the case of the agglomeration process computer simulations demonstrate the time evolution of the surface potential at two different time scales the scale of independent growth and of competition of the attraction regions they also show the time dependent distribution of the active walkers and spatio temporal development of the effective diffusion coefficient

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

ellipsoidal techniques for reachability analysis internal approximation for the reach tube of a linear time varying system with ellipsoidal bounds on the control variable consider the following approximation problem find a tight ellipsoid valued tube inside the reach tube that touches it along any prespecified smooth curve on the boundary we construct this ellipsoidal tube and show that if the given curve is itself a system trajectory the tube can be calculated recursively with minimum computational burden

robust time optimal control of constrained linear systems a version of dynamic programming which computes level sets of the value function rather than the value function set itself is used to design robust non linear controllers for linear discrete time dynamical systems subject to hard constraints on controls and states the controller stabilizes the system and steers all trajectories emanating in a prescribed set to a control invariant set in minimum time for the robust regulator problem the control invariant terminal set is a neighborhood preferably small of the origin for the robust tracking problem the control invariant terminal set is a neighborhood of the invariant set in which the tracking error is zero two non linear controllers which utilize the level sets of the value function are described the first requires the controller to solve on line a modest linear program whose dimension is approximately the same as that of the control variable the second decomposes each level set into a set of simplices a piecewise linear control law affine in each simplex is then constructed

cohomology of g sheaves in positive characteristic let g be a finite group and x a noetherian g scheme defined on an algebraically closed field k whose characteristic divides the order of g we define a refinement of the equivariant k theory of x devoted to give a better account of the information related to modular representation theory the construction relies in an essential way on the work of m auslander in modular representation theory and the use of sheaves of rings with several objects the main applications of this modular k theory are in dimension one where we show how it allows to extend the work of s nakajima

modularity lifting beyond the taylor wiles method we prove new modularity lifting theorems for p adic galois representations in situations where the methods of wiles and taylor wiles do not apply previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology in practice this imposes several restrictions one must be in a shimura variety setting and the automorphic forms must be of regular weight at infinity in this paper we essentially show how to remove these restrictions our most general result is a modularity lifting theorem which on the automorphic side applies to automorphic forms on the group mathrm gl n over a general number field it is contingent on a conjecture which in particular predicts the existence of galois representations associated to torsion classes in the cohomology of the associated locally symmetric space we show that if this conjecture holds then our main theorem implies the following if e is an elliptic curve over an arbitrary number field then e is potentially automorphic and satisfies the sato tate conjecture in addition we also prove some unconditional results for example in the setting of mathrm gl 2 over mathbf q we identify certain minimal global deformation rings with the hecke algebras acting on spaces of p adic katz modular forms of weight 1 such algebras may well contain p torsion moreover we also completely solve the problem for p odd of determining the multiplicity of an irreducible modular representation overline rho in the jacobian j 1 n where n is the minimal level such that overline rho arises in weight two

brill noether loci and the gonality stratification of m g for an irreducible smooth projective complex curve c of genus g the gonality defined as gon c min d z 1 there exists a gd on c is perhaps the second most natural invariant it gives an indication of how far c is from being rational in a way different from what the genus does for g 3 we consider the stratification of the moduli space mg of smooth curves of genus g given by gonality mg 2 m 1 g 3 m 1 g k mg

on the galois structure of equivariant line bundles on curves let it be a finite field and let x be a smooth protective curve over k with structure sheaf o let g be a finite group and write cl 0 g for the reduced grothendieck group of the category of g vector bundles in this paper we describe explicitly the subgroup of cl 0 g which is generated by the classes arising from g stable invertible sheaves on tame galois covers of x which have galois group g introduction let k be a finite field of characteristic p9 and let g be a finite abelian group suppose that y x is a tamely ramified galois covering of smooth projective curves over k with galois group g we shall refer to such coverings as tame g covers of x in this paper we study the structure of g stable line bundles on such curves y in order to be more precise we let cl ox g 9 respectively cl k g 9 denote the reduced grothendieck group of x g vector bundles respectively of finitely generated fc g modules which are cohomologically trivial for g suppose that a is a g stable line bundle on y then 4 is an ox g vector bundle and it gives rise to a class f a g c ox g we shall say that an element of c1 0x g is realizable if it may be obtained in this manner for some choice of y and a we shall combine techniques of c m and bu to obtain an explicit description of the subgroup 9 x g of cl ox g which is generated by the subset of realizable classes in conjunction with a riemann roch theorem these results are sufficient to describe explicitly those elements of c k g which arise as the image of the vector bundles a under the refined euler poincare characteristic

hypersurfaces in projective schemes and a moving lemma let x s be a quasi projective morphism over an affine base we develop in this article a technique for proving the existence of closed subschemes h s of x s with various favorable properties we offer several applications of this technique including the existence of finite quasi sections in certain projective morphisms and the existence of hypersurfaces in x s containing a given closed subscheme c and intersecting properly a closed set f assume now that the base s is the spectrum of a ring r such that for any finite morphism z s pic z is a torsion group this condition is satisfied if r is the ring of integers of a number field or the ring of functions of a smooth affine curve over a finite field we prove in this context a moving lemma pertaining to horizontal 1 cycles on a regular scheme x quasi projective and flat over s we also show the existence of a finite surjective s morphism to psd for any scheme x projective over s when x s has all its fibers of a fixed dimension d

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

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

globally and locally minimal weight spanning tree networks abstract the competition between local and global driving forces is significant in a wide variety of naturally occurring branched networks we have investigated the impact of a global minimization criterion versus a local one on the structure of spanning trees to do so we consider two spanning tree structures the generalized minimal spanning tree gmst defined by dror et al eur j oper res 120 2000 583 and an analogous structure based on the invasion percolation network which we term the generalized invasive spanning tree gist in general these two structures represent extremes of global and local optimality respectively structural characteristics are compared between the gmst and gist for a fixed lattice in addition we demonstrate a method for creating a series of structures which enable one to span the range between these two extremes two structural characterizations the occupied edge density i e the fraction of edges in the graph that are included in the tree and the tortuosity of the arcs in the trees are shown to correlate well with the degree to which an intermediate structure resembles the gmst or gist both characterizations are straightforward to determine from an image and are potentially useful tools in the analysis of the formation of network structures

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

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

on frohlich s conjecture for rings of integers of tame extensions where is the complex conjugate of z the constant w z is called the artin root number of in m l j martinet showed that om is not always a free zf module when n q and when f is a quaternion group of order 8 subsequently in f5 a frtihlich showed that the class of u in this special case is determined by the sign of the artin root number of the irreducible sympletic character of 2 we now return to the general case and let ci tif be the class group of locally free zf modules given a locally free 2 f module x we denote its class by x philippe cassou nogu6s defined a class t w in ci f which is defined solely in terms of the values of artin root numbers of sympletic characters this class will be defined in w after numerous computations by fr6hlich et al cf f6 c t3 for instance fr6hlich conjectured the result given below which we prove in this paper

on diophantine definability and decidability in some infinite totally real extensions of ℚ let m be a number field and w m a set of its non archimedean primes then let o m wm x m ord t x 0 t w m let p 1 p r be a finite set of prime numbers let f inf be the field generated by all the p j i th roots of unity as j and i 1 r let k inf be the largest totally real subfield of f inf then for any e 0 there exist a number field m k inf and a set w m of non archimedean primes of m such that w m has density greater than 1 e and z has a diophantine definition over the integral closure of o m w m in k inf