the neron component series of an abelian variety we introduce the neron component series of an abelian variety a over a complete discretely valued field this is a power series in mathbb z left left t right right which measures the behaviour of the number of components of the neron model of a under tame ramification of the base field if a is tamely ramified then we prove that the neron component series is rational it has a pole at t 1 whose order equals one plus the potential toric rank of a this result is a crucial ingredient of our proof of the motivic monodromy conjecture for abelian varieties we expect that it extends to the wildly ramified case we prove this if a is an elliptic curve and if a has potential purely multiplicative reduction

ramification of local fields with imperfect residue fields we define two decreasing filtrations by ramification groups on the absolute galois group of a complete discrete valuation field whose residue field may not be perfect in the classical case where the residue field is perfect we recover the classical upper numbering filtration the definition uses rigid geometry and log structures we also establish some of their properties

duality and nearby cycles over general bases this paper studies the sliced nearby cycle functor and its commutation with duality over a henselian discrete valuation ring we show that this commutation holds confirming a prediction of deligne as an application we give a new proof of beilinson s theorem that the vanishing cycle functor commutes with duality up to twist over an excellent base scheme we show that the sliced nearby cycle functor commutes with duality up to modification of the base we deduce that duality preserves universal local acyclicity over an excellent regular base we also present gabber s theorem that local acyclicity implies universal local acyclicity over a noetherian base

computing motivic zeta functions on log smooth models we give an explicit formula for the motivic zeta function in terms of a log smooth model it generalizes the classical formulas for snc models but it gives rise to much fewer candidate poles in general this formula plays an essential role in recent work on motivic zeta functions of degenerating calabi yau varieties by the second named author and his collaborators as a further illustration we explain how the formula for newton non degenerate polynomials can be viewed as a special case of our results

tikhonov regularization and the l curve for large discrete ill posed problems discretization of linear inverse problems generally gives rise to very ill conditioned linear systems of algebraic equations typically the linear systems obtained have to be regularized to make the computation of a meaningful approximate solution possible tikhonov regularization is one of the most popular regularization methods a regularization parameter specifies the amount of regularization and in general an appropriate value of this parameter is not known a priori we review available iterative methods and present new ones for the determination of a suitable value of the regularization parameter by the l curve criterion and the solution of regularized systems of algebraic equations

convergence analysis of lsqr for compact operator equations abstract in this paper we analyze the behavior of the lsqr algorithm for the solution of compact operator equations in hilbert spaces we present results concerning existence of krylov solutions and the rate of convergence in terms of an l p sequence where p depends on the summability of the singular values of the operator under stronger regularity requirements we also consider the decay of the error finally we study the approximation of the dominant singular values of the operator attainable with the bidiagonal matrices generated by the lanczos bidiagonalization and the arising low rank approximations some numerical experiments on classical test problems are presented

a regularization parameter in discrete ill posed problems the tikhonov regularization method for discrete ill posed problems is considered for the practical choice of the regularization parameter alpha some authors use a plot of the norm of the regularized solution versus the norm of the residual vector for all alpha considered this paper contains an analysis of the shape of this plot and gives a theoretical justification for choosing the regularization parameter so it is related to the l corner of the plot considered in the logarithmic scale moreover a new criterion for choosing alpha is introduced independent of the shape of the plot which gives a new interpretation of the corner criterion mentioned above the existence of l corner is discussed

on the swan conductor in positive characteristic let x be a proper smooth surface over an algebraically closed field of positive characteristic and u be a complement of a simple normal crossing divisor for a smooth l adic sheaf on u deligne proved a formula calculating the euler characteristic of by local invariants kato gave another formula for the euler characteristic in case where is of rank 1 using class field theory in this paper we show that local terms of these formulas coincide admitting certain conjectures for vanishing cycles

unramified class field theory of arithmetical surfaces let k be an algebraic number field 9k its ring of integers and v a non empty open subscheme of spec ck let x be a projective smooth geometrically irreducible scheme over k and x a regular proper flat scheme over v such that x x vk x the purpose of this paper is to give an explicit description of the abelian fundamental groups ab x and 7t b x using some idele class groups attached to x and x by a k theoretical method its outline is as follows in general for a noetherian scheme z and i 0 or 1 we define the group ski z to be the cokernel of

stable reduction of curves and tame ramification we study stable reduction of curves in the case where a tamely ramified base extension is sufficient if x is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring there is a criterion due to saito that describes precisely in terms of the geometry of the minimal model with strict normal crossings of x when a tamely ramified extension suffices in order for x to obtain stable reduction for such curves we construct an explicit extension that realizes the stable reduction and we furthermore show that this extension is minimal we also obtain a new proof of saito s criterion avoiding the use of l adic cohomology and vanishing cycles

characteristic varieties and vanishing cycles on presente une etude systematique des varietes caracteristiques des systemes holonomes a singularites regulieres ce sont des sous varietes lagrangiennes coniques dans le fibre cotangent t x

on euler characteristics associated to exceptional divisors let k be an algebraically closed field and f e k x1 x fix an embedded resolution h x an i of f i o and denote by ei i e s the irreducible components of h l f l o with multiplicity ni in the 0 0 visor of f o h put also es ei uj1iej and denote by x ei its euler characteristic several conjectures concerning igusa s local zeta function and the topological zeta function of f motivate the study of euler characteristics associated to subsets uie tei of uiese1 which are maximal connected with respect to the property that dini for all i e t here d e n d 1 we prove that if h maps uiet ei to a point then

periods of hodge structures and special values of the gamma function at the end of the 1970s gross and deligne conjectured that periods of geometric hodge structures with multiplication by an abelian number field are products of values of the gamma function at rational arguments with exponents determined by the hodge decomposition we prove an alternating variant of this conjecture for smooth projective varieties acted upon by an automorphism of finite order thus improving previous results of maillot and rossler the proof relies on a product formula for periods of regular singular connections due to saito and terasoma

ramification theory for degree p extensions of arbitrary valuation rings in mixed characteristic 0 p abstract we previously obtained a generalization and refinement of results about the ramification theory of artin schreier extensions of discretely valued fields in characteristic p with perfect residue fields to the case of fields with more general valuations and residue fields as seen in 5 the defect case gives rise to many interesting complications in this paper we present analogous results for degree p extensions of arbitrary valuation rings in mixed characteristic 0 p in a more general setting more specifically the only assumption here is that the base field k is henselian in particular these results are true for defect extensions even if the rank of the valuation is greater than 1 a similar method also works in equal characteristic generalizing the results of 5

vanishing cycles and geometry of curves over a discrete valuation ring 0 introduction geometry of curves over a discrete valuation ring is reflected in the action of the monodromy on the f adic cohomology groups the stable reduction theorem d m gives an equivalent geometrical condition for the action of the inertia group to be unipotent in the global situation where global means that the curve in consideration is proper in this paper we will generalize it in two directions one is to the local situation with the isolated singularity the other is to consider the geometrical condition for the action of the wild ramification group to be trivial we will also give a purely cohomological proof without using picard schemes of the stable reduction theorem and a new proof of the formula of deligne la theorem 5 1 1 based only on the stable reduction theorem here we only treat a strictly local discrete valuation ring with algebraically closed residue field however the results apply to any discrete valuation ring with perfect residue field since they are etale local we introduce some notations and terminologies

some properties of the arnoldi based methods for linear ill posed problems in this paper we study some properties of the classical arnoldi based methods for solving infinite dimensional linear equations involving compact operators these problems are intrinsically ill posed since a compact operator does not admit a bounded inverse we study the convergence properties and the ability of these algorithms to estimate the dominant singular values of the operator

a trace formula for varieties over a discretely valued field we study the motivic serre invariant of a smoothly bounded alge braic or rigid variety x over a complete discretely valued field k with perfect residue field k if k has characteristic zero we extend the definition to arbi trary k varieties using bittner s presentation of the grothendieck ring and a process of neron smoothening of pairs of varieties the motivic serre invariant can be considered as a measure for the set of unramified points on x under certain tameness conditions it admits a cohomological interpretation by means of a trace formula in the curve case we use t saito s geometric criterion for cohomological tameness to obtain more detailed results we discuss some applications to weil chatelet groups chow motives and the structure of the grothendieck ring of varieties msc 2000 11s15 14g05 14g22

on localizations of the characteristic classes of l adic sheaves and conductor formula in characteristic p 0 abbes kato and saito generalize the grothendieck ogg shafarevich formula to an arbitrary dimension kato and saito in ann math 168 33 96 2008 abbes and saito in invent math 168 567 612 2007 in this paper assuming the strong resolution of singularities we prove a localized version of a formula proved using the characteristic class of an l adic sheaf by abbes and saito invent math 168 567 612 2007 we prove a localized version of the lefschetz verdier trace formula proved in grothendieck formule de lefschetz expose iii sga 5 lect notes math vol 589 pp 372 406 exp x springer berlin 1977 theoreme 4 4 as an application we prove a conductor formula in an arbitrary dimension in the equal characteristic case

rational krylov and adi iteration for infinite size quasi toeplitz matrix equations abstract we consider a class of linear matrix equations involving semi infinite matrices which have a quasi toeplitz structure these equations arise in different settings mostly connected with pdes or the study of markov chains such as random walks on bidimensional lattices we present the theory justifying the existence of the solution in an appropriate banach algebra which is computationally treatable and we propose several methods for computing them we show how to adapt the adi iteration to this particular infinite dimensional setting and how to construct rational krylov methods convergence theory is discussed and numerical experiments validate the proposed approaches

conjugate direction methods with variable storage in this paper we study conjugate gradient algorithms for large optimization problems these methods accelerate or precondition the conjugate gradient method by means of quasi newton matrices and are designed to utilize a variable amount of storage depending on how much information is retained in the quasi newton matrices we are concerned with the behaviour of such methods on the underlying quadratic model and in particular with finite termination properties

ramification theory for artin schreier extensions of valuation rings abstract the goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings in the case of artin schreier extensions we define refined versions of invariants of ramification in the classical ramification theory and compare them furthermore we can treat the defect case

qn like variable storage conjugate gradients both conjugate gradient and quasi newton methods are quite successful at minimizing smooth nonlinear functions of several variables and each has its advantages in particular conjugate gradient methods require much less storage to implement than a quasi newton code and therefore find application when storage limitations occur they are however slower so there have recently been attempts to combine cg and qn algorithms so as to obtain an algorithm with good convergence properties and low storage requirements one such method is the code conmin due to shanno and phua it has proven quite successful but it has one limitation it has no middle ground in that it either operates as a quasi newton code using o n 2 storage locations or as a conjugate gradient code using 7n locations but it cannot take advantage of the not unusual situation where more than 7n locations are available but a quasi newton code requires an excessive amount of storage

elliptic curves and hilbert s tenth problem for algebraic function fields over real and p adic fields let k be a field of characteristic zero v a smooth positive dimensional quasiprojective variety over k and d a nonempty effective divisor on v let k be the function field of v and a the semilocal ring of d in k in this paper we prove the diophantine undecidability of 1 a in all cases 2 k when k is formally real and v has a real point 3 k when k is a subfield of a p adic field for some odd prime p to achieve this we use denef s method from an elliptic curve e over q without complex multiplication one constructs a quadratic twist e of e over q t which has mordell weil rank one most of the paper is devoted to proving using a theorem of r noot that one can choose f in k vanishing at d such that the group e k deduced from the field extension k q f q t is equal to e q t then we mimic the arguments of denef for the real case and of kim and roush for the p adic case

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

motivic integration on smooth rigid varieties and invariants of degenerations we develop a theory of motivic integration for smooth rigid varieties as an application we obtain a motivic analogue for rigid varieties of serre s invariant for p adic varieties our construction provides new geometric birational invariants of degenerations of algebraic varieties for degenerations of calabi yau varieties our results take a stronger form

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

cleanliness and log characteristic cycles for vector bundles with flat connections let x be a proper smooth algebraic variety over a field k of characteristic zero and let d be a divisor with simple normal crossings let m be a vector bundle over x d equipped with a flat connection with possible irregular singularities along d we define a cleanliness condition which roughly says that the singularities of the connection are controlled by the singularities at the generic points of d when this condition is satisfied we compute explicitly the associated log characteristic cycle and relate it to the so called refined irregularities as a corollary of a log variant of kashiwara dubson formula we obtain the euler characteristic of the de rham cohomology of the vector bundle under a mild technical hypothesis on m

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

de rham factors we construct a theory of de rham epsilon factors given a smooth projective variety x over a field of characteristic 0 and a open image in new window module open image in new window we may consider the determinant of de rham cohomology open image in new window as a graded super line let s be the singular support of open image in new window u an open in x y x u and a 1 form on u x such that u s then we show the existence of a canonically defined graded super line open image in new window canonically isomorphic to open image in new window the key property is the local nature of the epsilon factor it only depends on the restriction of open image in new window and to any open neighborhood of y restricting to the case of curves gives a theory of epsilon factors for curves previously defined by beilinson bloch deligne and esnault

a trace formula for rigid varieties and motivic weil generating series for formal schemes we establish a trace formula for rigid varieties x over a complete discretely valued field which relates the set of unramified points on x to the galois action on its etale cohomology next we show that the analytic milnor fiber of a morphism f at a point x completely determines the formal germ of f at x we develop a theory of motivic integration for formal schemes of pseudo finite type over a complete discrete valuation ring r and we introduce the weil generating series of a regular formal r scheme mathfrak x of pseudo finite type via the construction of a gelfand leray form on its generic fiber when mathfrak x is the formal completion of a morphism f from a smooth irreducible variety to the affine line then its weil generating series coincides modulo normalization with the motivic zeta function of f when mathfrak x is the formal completion of f at a closed point x of the special fiber f 1 0 we obtain the local motivic zeta function of f at x

a note on the superlinear convergence of gmres in the paper it is shown how the rate of convergence of the gmres method for solving a linear operator equation lambda i k u f in a hilbert space is related to the degree of compactness of k measured by the products of its singular values

the characteristic class and ramification of an l adic etale sheaf we introduce the characteristic class of an l adic etale sheaf using a cohomological pairing due to verdier sga5 as a consequence of the lefschetz verdier trace formula its trace computes the euler poincare characteristic of the sheaf we compare the characteristic class to two other invariants arising from ramification theory one is the swan class of kato saito 17 and the other is the 0 cycle class defined by kato for rank 1 sheaves in 16

geometric criteria for tame ramification we prove an a campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field k as a first application we relate the orders of the tame monodromy eigenvalues on the l adic cohomology of a k curve to the geometry of a relatively minimal sncd model and we show that the semi stable reduction theorem and saito s criterion for cohomological tameness are immediate consequences of this result as a second application we compute the error term in the trace formula for smooth and proper k varieties we see that the validity of the trace formula would imply a partial generalization of saito s criterion to arbitrary dimension

the characteristic cycle and the singular support of a constructible sheaf we define the characteristic cycle of an etale sheaf as a cycle on the cotangent bundle of a smooth variety in positive characteristic using the singular support recently defined by beilinson we prove a formula a la milnor for the total dimension of the space of vanishing cycles and an index formula computing the euler poincare characteristic generalizing the grothendieck ogg shafarevich formula to higher dimension an essential ingredient of the construction and the proof is a partial generalization to higher dimension of the semi continuity of the swan conductor due to deligne laumon we prove the index formula by establishing certain functorial properties of characteristic cycles

superlinear convergence of krylov subspace methods for self adjoint problems in hilbert space the conjugate gradient and minimum residual methods for self adjoint problems in hilbert space are considered linear and superlinear convergence results with respect to both q and r rates are reviewed new results on ell step q superlinear and r superlinear convergence for the minimum residual method are provided and examples are considered to underscore the relevance of a hilbert space theory

termination and equivalence results for conjugate gradient algorithms this paper considers the problem of minimizing a functionf x when linear equality constraints onx could be present and when an algorithm is to be used which may employ both conjugate gradient and quasi newton steps the conjugate gradient algorithm is examined when a preconditioner is used which may be only positive semi definite quite a general theorem is given about the equivalence between conjugate gradient steps and quasi newton steps which includes several previously published results as special cases some results are also given for the case where some line searches are inexact

ramification theory for varieties over a perfect field for an l adic sheaf on a variety of arbitrary dimension over a perfect field we define the swan class measuring the wild ramification as a 0 cycle class supported on the ramification locus we prove a lefschetz trace formula for open varieties and a generalization of the grothendieck ogg shararevich formula using the swan class

chow group of 0 cycles with modulus and higher dimensional class field theory one of the main results of this article is a proof of the rank one case of an existence conjecture on lisse q over bar l sheaves on a smooth variety u over a finite field due to deligne and drinfeld the problem is translated into the language of higher dimensional class field theory over finite fields which describes the abelian fundamental group of u by chow groups of 0 cycles with moduli a key ingredient is the construction of a cycle theoretic avatar of a refined artin conductor in ramification theory originally studied by kazuya kato

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

logarithmic good reduction of abelian varieties let k be a field which is complete for a discrete valuation we prove a logarithmic version of the neron ogg shafarevich criterion if a is an abelian variety over k which is cohomologically tame then a has good reduction in the logarithmic setting i e there exists a projective log smooth model of a over mathcal o k this implies in particular the existence of a projective regular model of a generalizing a result of kunnemann the proof combines a deep theorem of gabber with the theory of degenerations of abelian varieties developed by mumford faltings chai et al

general fixed point formula for an algebraic surface and the theory of swan representations for two dimensional local rings 07755 1986 03 17 7755