derived category automorphisms from mirror symmetry inspired by the homological mirror symmetry conjecture of kontsevich 30 we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi projective variety msc 2000 18e30 14j32

the rubin stark conjecture for imaginary abelian fields of odd prime power conductor we build upon ideas developed in 9 as well as results of greither on a strong form of brumer s conjecture 2 4 and prove rubin s integral version of stark s conjecture for imaginary abelian extensions of q of odd prime power conductor

miniversal deformations of pairs of skew symmetric matrices under congruence for each pair of complex symmetric matrices a b we provide a normal form with a minimal number of independent parameters to which all pairs of complex symmetric matrices a over tilde b ove

stark s conjecture over complex cubic number fields jjsystematic computation of stark units over nontotally real base fields is carried out for the first time since the information provided by stark s conjecture is significantly less in this situation than the information provided over totally real base fields new techniques are required precomputing stark units in relative quadratic extensions where the conjecture is already known to hold and coupling this information with the fincke pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding stark units in larger extensions where the conjecture is still unproven stark s conjecture is verified in each case for these stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields including hilbert class fields are obtained from the minimal polynomials of these new stark units

on the stability of the extreme point set in linear optimization this paper analyzes the stability properties of the set of extreme points of a closed convex set described by means of a given linear inequality system sigma we assume that all the coefficients of sigma can be arbitrarily perturbed maintaining the possibly infinite index set as well as the finite dimension of the space of variables and we measure the size of these perturbations by means of the pseudometric of the uniform convergence the paper characterizes the nonemptiness of the extreme point set under sufficiently small perturbations and the berge lower semicontinuity of the extreme point set mapping at sigma it also gives necessary as well as sufficient conditions for the closedness and for the berge upper semicontinuity at sigma which provide characterizations of these properties for finite systems

fiedler companion linearizations for rectangular matrix polynomials the development of new classes of linearizations of square matrix polynomials that generalize the classical first and second frobenius companion forms has attracted much attention in the last decade research in this area has two main goals finding linearizations that retain whatever structure the original polynomial might possess and improving properties that are essential for accurate numerical computation such as eigenvalue condition numbers and backward errors however all recent progress on linearizations has been restricted to square matrix polynomials since rectangular polynomials arise in many applications it is natural to investigate if the new classes of linearizations can be extended to rectangular polynomials in this paper the family of fiedler linearizations is extended from square to rectangular matrix polynomials and it is shown that minimal indices and bases of polynomials can be recovered from those of any linearization in this class via the same simple procedures developed previously for square polynomials fiedler linearizations are one of the most important classes of linearizations introduced in recent years but their generalization to rectangular polynomials is nontrivial and requires a completely different approach to the one used in the square case to the best of our knowledge this is the first class of new linearizations that has been generalized to rectangular polynomials

generalization of roth s solvability criteria to systems of matrix equations w e roth 1952 proved that the matrix equation ax xb c has a solution if and only if the matrices graphics and graphics are similar a dmytryshyn and b kagstrom 2015 extended roth s c

generic skew symmetric matrix polynomials with fixed rank and fixed odd grade we show that the set of m m complex skew symmetric matrix polynomials of odd grade d i e of degree at most d and normal rank at most 2r is the closure of the single set of matrix polynomials

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

on class groups of random number fields the main aim of the present paper is to disprove the cohen lenstra martinet heuristics in two different ways and to offer possible corrections we also recast the heuristics in terms of arakelov class groups giving an explanation for the probability weights appearing in the general form of the heuristics we conclude by proposing a rigorously formulated cohen lenstra martinet conjecture

stickelberger ideals and fitting ideals of class groups for abelian number fields in this paper we determine completely the initial fitting ideal of the minus part of the ideal class group of an abelian number field over q up to the 2 component this answers an open question of mazur and wiles invent math 76 179 330 1984 up to the 2 component and proves conjecture 0 1 in kurihara j reine angew math 561 39 86 2003 we also study brumer s conjecture and prove a stronger version for a cm field assuming certain conditions in particular on the galois group

on a conjecture of brumer in a series of papers iwasawa proposed and studied analogues for number fields of constructions due to weil for curves over finite fields cf 11 in particular he conjectured an analogue for weil s theorem relating the zeta function of a non singular curve over a finite field with the characteristic polynomial of its frobenius automorphism one strong piece of evidence for this came from the classical theorem of stickelberger however iwasawa s original conjecture was proved for f q in mw without use of stickelberger s theorem for a general totally real number field brumer proposed a generalization of stickelberger s theorem in w2 we proved iwasawa s conjecture for an arbitrary totally real field and here we use this to recover a version of brumer s conjecture we note however that iwasawa s conjecture alone is not sufficient to prove our theorem owing to the phenomenon of trivial zeroes the reason is that in iwasawa theory one habitually interpolates the special values of lfunctions which are imprimitive the key part of this paper is the development of a technique introduced in w2 of adjoining an auxiliary variable to the p adic b functions in a way that permits the study of these primitive lfunctions i am grateful to the referee for correcting some obscure arguments let f be a totally real number field and let k be a cm field which is

computing stark units for totally real cubic fields a method for computing provably accurate values of partial zeta functions is used to numerically confirm the rank one abelian stark conjecture for some totally real cubic fields of discriminant less than 50000 the results of these computations are used to provide explicit hilbert class fields and some ray class fields for the cubic extensions

equivalences of derived catgories of sheaves on smooth stacks we extend orlov s representability theorem on the equivalence of derived categories of sheaves to the case of smooth stacks associated to normal projective varieties with only quotient singularities

twisted fourier mukai transforms for holomorphic symplectic four folds we apply the methods of c a ldararu to construct a twisted fourier mukai transform between a pair of holomorphic symplectic four folds which are fibred by lagrangian abelian surfaces more precisely we obtain an equivalence between the derived category of coherent sheaves on a certain lagrangian fibration and the derived category of twisted sheaves on its mirror partner as a corollary we extend the original fourier mukai transform to degenerations of abelian surfaces another consequence of the general theory is that the holomorphic symplectic four fold and its mirror are connected by a one parameter family of deformations through lagrangian fibrations

cohomology of units and l values at zero let klk be a finite galois extension of number fields with galois group g and let s be a finite g stable set of primes of k containing all archimedean primes we denote the g module of s units of k by e es and let as be the kernel of the augmentation map zs z which sends each basis element p c s to 1 we are concerned with invariants of k k which are associated to a g homomorphism p as e inducingq0 as cq0e these invariants were defined by tate ta2 and chinburg chl when s is large i e when s contains all ramified primes of k k and the s class group cl cls of k is trivial there are two of them and each is a function of the complex characters x of g the first aw has a x equal to the tate regulator at x divided by the leading coefficient c x of the taylor expansion at s 0 of the artin l function l s x with the euler factors at primes of s omitted according to stark s conjecture st in tate s form a takes algebraic values which satisfy a o x5 a x

fourier mukai transforms for gorenstein schemes we extend to singular schemes with gorenstein singularities or fibered in schemes of that kind bondal and orlov s criterion for an integral functor to be fully faithful we also prove that the original condition of characteristic zero cannot be removed by providing a counterexample in positive characteristic we contemplate a criterion for equivalence as well in addition we prove that for locally projective gorenstein morphisms a relative integral functor is fully faithful if and only if its restriction to each fibre is also fully faithful these results imply the invertibility of the usual relative fourier mukai transform for an elliptic fibration as a direct corollary

on the equivariant tamagawa number conjecture for tate motives let l be a finite abelian extension of and let k be any subfield of l for each integer r with r 0 we prove the equivariant tamagawa number conjecture for the pair h0 spec l r gal l k

complex surfaces with equivalent derived categories we examine the extent to which a smooth minimal complex projective surface x is determined by its derived category of coherent sheaves dx to do this we find for each such surface x the set of surfaces y for which there exists a fourier mukai transform d y to d x

jordan structures of alternating matrix polynomials alternating matrix polynomials that is polynomials whose coefficients alternate between symmetric and skew symmetric matrices generalize the notions of even and odd scalar polynomials we investigate the smith forms of alternating matrix polynomials showing that each invariant factor is an even or odd scalar polynomial necessary and sufficient conditions are derived for a given smith form to be that of an alternating matrix polynomial these conditions allow a characterization of the possible jordan structures of alternating matrix polynomials and also lead to necessary and sufficient conditions for the existence of structure preserving strong linearizations most of the results are applicable to singular as well as regular matrix polynomials

d equivalence and k equivalence let x and y be smooth projective varieties over they are called d equivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories and k equivalent if they are birationally equivalent and the pull backs of their canonical divisors to a common resolution coincide we expect that the two equivalences coincide for birationally equivalent varieties we shall provide a partial answer to the above problem in this paper

on galois structure invariants associated to tate motives we establish the equivalence of two definitions of invariants measuring the galois module structure of k groups of rings of integers in number fields due to chinburg et al on the one hand and the authors on the other we also make some remarks concerning the possibility of yet another such definition via lichtenbaum complexes

fitting ideals of l adic realizations of picard 1 motives and class groups of global function fields in previous work we proved a result on the equivariant fitting ideal of the adic realization of the picard 1 motive attached to an abelian covering of curves defined over a finite field in this paper we build upon this work to deduce results on the equivariant fitting ideal of the tate modules of the jacobian of the top curve and on the equivariant fitting ideal of the dual of the degree zero class group of the top curve at the end we discuss refinements of brumer s conjecture and consider examples

fourier mukai transforms for elliptic surfaces we compute a large number of moduli spaces of stable bun dles on a general algebraic elliptic surface using a new class of relative fourier mukai transforms

the change of the jordan structure of a mtrix under small perturbations abstract in this paper a complete description including multiplicity is given for the jordan structure of a matrix which is a small perturbation of a matrix of known jordan structure the problem solved here was solved independently as noted and the other solution has been published in english however the solutions are different the problem is important and it seems desirable to have this solution widely available

generic complete eigenstructures for sets of matrix polynomials with bounded rank and degree abstract the set pol d r m n of m n complex matrix polynomials of grade d and normal rank at most r in a complex d 1 m n dimensional space is studied for r 1 min m n 1 we show that pol d r m n is the union of the closures of the r d 1 sets of matrix polynomials with rank r degree exactly d and explicitly described complete eigenstructures in addition for the full rank rectangular polynomials i e r min m n and m n we show that pol d r m n coincides with the closure of a single set of the polynomials with rank r degree exactly d and the described complete eigenstructure these complete eigenstructures correspond to generic m n matrix polynomials of grade d and rank at most r

the iwasawa conjecture for totally real fields let f be a totally real number field let p be a prime number and for any integer n let fun denote the group of nth roots of unity let 41 be a p adic valued artin character for f and let f be the extension of f attached to 4 i e so that 4 is the character of a faithful representation of gal f f we will assume that f is also totally real for a number field k let k denote the cyclotomic zp extension of k following greenberg we say that 4 is of type s if f n fc f and of type w if 4 is one dimensional with f p c fcc deligne and ribet in dr following kubota and leopoldt for the case f q have proved the existence of a p adic l function associated to a one dimensional artin character 4 with f totally real this function lp s 4 is continuous for s e zp 1 and even at s 1 if 4 is not trivial and satisfies the following interpolation property

simplest miniversal deformations of matrices matrix pencils and contragredient matrix pencils abstract for a family of linear operators a u u over c that smoothly depend on parameters 1 k v i arnold obtained the simplest normal form of their matrices relative to a smoothly depending on change of a basis in u we solve the same problem for a family of linear operators a u u over r for a family of pairs of linear mappings a u v b u v over c and r and for a family of pairs of counter linear mappings a u v b v u over c and r

stickelberger elements fitting ideals of class groups of cm fields and dualisation in this paper we systematically construct abelian extensions of cm fields over a totally real field whose stickelberger elements are not in the fitting ideals of the class groups our evidence indicates that pontryagin duals of class groups behave better than the class groups themselves we also explore the behaviour of fitting ideals under projective limits and dualisation in a somewhat broader context

stratification of full rank polynomial matrices we show that perturbations of polynomial matrices of full normal rank can be analyzed viathe study of perturbations of companion form linearizations of such polynomial matrices it is proved that a

computation of p units in ray class fields of real quadratic number fields let k be a real quadratic field let p be a prime number which is inert in k and let k p be the completion of k at p as part of a ph d thesis we constructed a certain p adic invariant u e kp and conjectured that u is in fact a p unit in a suitable narrow ray class field of k in this paper we give numerical evidence in support of that conjecture our method of computation is similar to the one developed by dasgupta and relies on partial modular symbols attached to eisenstein series

change of the congruence canonical form of 2 by 2 and 3 by 3 matrices under perturbations and bundles of matrices under congruence we construct the hasse diagrams g2 and g3 for the closure ordering on the sets of congruence classes of 2 2 and 3 3 complex matrices in other words we construct two directed graphs whose vert

miniversal deformations of matrices under congruence and reducing transformations arnold 1971 1 constructed a miniversal deformation of a square complex matrix under similarity that is a simple normal form to which not only a given square matrix a but all matrices b close to it can be reduced by similarity transformations that smoothly depend on the entries of b we give miniversal deformations of matrices of sesquilinear forms that is of square complex matrices under congruence and construct an analytic reducing transformation to a miniversal deformation analogous results for matrices under congruence were obtained by dmytryshyn futorny and sergeichuk 2012 11

a note on the continuity of solutions of parametric linear programs this note shows that if a parametric linear program min cx ax b1 b2 x 0 has optimal solutions in an interval left underset raise0 3em hbox smash scriptscriptstyle theta bar theta right for then depending on degeneracy the solutions is a continuous vector function or a continuous point to set mapping in the latter case an algorithm is introduced to solve the problem and generate a continuous vector solution in left underset raise0 3em hbox smash scriptscriptstyle theta bar theta right

cyclotomic units and stickelberger ideals of global function fields in this paper we define the group of cyclotomic units and stickelberger ideals in any subfield of the cyclotomic function field we also calculate the index of the group of cyclotomic units in the total unit group in some special cases and the index of stickelberger ideals in the integral group ring

the lower semicontinuity of optimal solution sets abstract we provide a sufficient condition which guarantees the lower semicontinuity of the optimal solution set for a general nonlinear programming problem it is shown that the same condition is both sufficient and necessary for convex programming problems that satisfy the slater condition

structure preserving stratification of skew symmetric matrix polynomials we study how elementary divisors and minimal indices of a skew symmetric matrix polynomial of odd degree may change under small perturbations of the matrix coefficients we investigate these change

three dimensional flops and noncommutative rings limburgs univ ctr dept math b 3590 diepenbeek belgium van den bergh m limburgs univ ctr dept math univ campus b 3590 diepenbeek belgium vdbergh luc ac be

on the equivariant tamagawa number conjecture for tate motives and unconditional annihilation results dfg collaborative research center 701 spectral structures and topological methods in mathematics

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

on the rubin stark conjecture for a special class of cm extensions of totally real number fields we use greither s recent results on brumer s conjecture to prove rubin s integral version of stark s conjecture up to a power of 2 for an infinite class of cm extensions of totally real number fields called nice extensions under the assumption that a certain iwasawa invariant vanishes as a consequence and under the same assumption we show that the brumer stark conjecture is true for nice extensions up to a power of 2

skew symmetric matrix polynomials and their smith forms we characterize the smith form of skew symmetric matrix polynomials over an arbitrary field f showing that all elementary divisors occur with even multiplicity restricting the class of equivalence transformations to unimodular congruences a smith like skew symmetric canonical form for skew symmetric matrix polynomials is also obtained these results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skew symmetric matrices as well as the existence of structured linearizations for skew symmetric matrix polynomials by contrast with other classes of structured matrix polynomials e g alternating or palindromic polynomials every regular skew symmetric matrix polynomial is shown to have a structured strong linearization while there are singular skew symmetric polynomials of even degree for which a structured linearization is impossible for each odd degree we develop a skew symmetric companion form that uniformly provides a structured linearization for every regular and singular skew symmetric polynomial of that degree finally the results are applied to the construction of minimal symmetric factorizations of skew symmetric rational matrices

generalized stark formulae over function fields we establish formulae of stark type for the stickelberger elements in the function field setting our result generalizes a work of hayes and a conjecture of gross it is used to deduce a p adic version of rubin stark conjecture and burns conjecture

flops and equivalences of derived categories for threefolds with only terminal gorenstein singularities the main purpose of this paper is to show that bridgeland s moduli space of perverse point sheaves for certain flopping contractions gives the flops and the fourier mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories

integrality of stickelberger elements and the equivariant tamagawa number conjecture let l k be a nite galois cm extension of number elds with galois group g in an earlier paper the author has de ned a module sku l k over the center of the group ring zg which coincides with the sinnott kurihara ideal if g is abelian and in particular contains many stickelberger elements it was shown that a certain conjecture on the integrality of sku l k implies the minus part of the equivariant tamagawa number conjecture at an odd prime p for an in nite class of non abelian galois cm extensions of number elds which are at most tamely rami ed above p provided that iwasawa s invariant vanishes here we prove a relevant part of this integrality conjecture which enables us to deduce the equivariant tamagawa number conjecture from the vanishing of for the same class of extensions

the brumer stark conjecture in some families of extensions of specified degree as a starting point an important link is established between brumer s conjecture and the brumer stark conjecture which allows one to translate recent progress on the former into new results on the latter for example if k f is an abelian extension of relative degree 2p p an odd prime we prove the l part of the brumer stark conjecture for all odd primes l p with f belonging to a wide class of base fields in the same setting we study the 2 part and p part of brumer stark with no special restriction on f and are left with only two well defined specific classes of extensions that elude proof extensive computations were carried out within these two classes and a complete numerical proof of the brumer stark conjecture was obtained in all cases

haar measure and the artin conductor let g be a connected reductive group defined over a local non archimedean field k the group g k is locally compact and unimodular in gr a haar measure g was defined on g k using the theory of bruhat and tits in this note we give another construction of the measure g using the artin conductor of the motive m of g over k the equivalence of the two constructions is deduced from a result of prasad p 1 the root datum and motive of g in this section k is an arbitrary field and g is a connected reductive group over k we let k be an algebraic closure of k ks the separable closure of k in k and gal ks k let t b g be a maximal torus contained in a borel subgroup defined over ks let g b t be the based root datum defined by this choice we recall cf sp pg3 12 that x t t b x t t b 1 1 with x t and x t the character and cocharacter groups of t respectively and and the simple roots and coroots determined by b respectively let w ng t t be the weyl group of the finite group w acts as automorphisms of x t and is generated by the reflections s x x x 1 2 for the galois group acts as automorphisms of ie as automorphisms of the group x t preserving the finite set as follows if then we can find g g ks such that int g t g t g t and int g b g b g b with g well defined up to left multiplication by t ks hence it induces a well defined automorphism x t x t date 14th january 1997 1 2 benedict h gross and wee teck gan preserving hence we get a group homomorphism aut via acts on aut by inner automorphisms similarly if f g g is any automorphism of g over ks it induces an automorphism f of which depends only on the image of f in the quotient group outks g of outer automorphisms the resulting map outks g aut is an isomorphism which respects the respective galois actions on the two groups cf sp pg10 the galois group also acts on w via the formula s s 1 3 and the semi direct product w acts on the rational vector space e x t q 1 4 let r sym e w which is a graded q module let r be the ideal of elements of positive degree in r and define v r r 2 d 1vd 1 5 this is a graded q module and chevalley proved that dim v dim e cf ch steinberg extended the proof to show that e and v are isomorphic modules cf st pg22 we sketch a proof of this result that does not involve the classification of irreducible root systems proposition 1 6 the q modules e and v are isomorphic proof by the criterion in se pg 104 it suffices to show that for all the fixed spaces e and v have the same dimension for any graded module a am we define the poincare series of by p a t tr am t then p a b p a p b steinberg showed that there is an isomorphism of graded modules s e s vd a here a is finite dimensional with basis bw w w and action given by bw b w the degree of bw is the length l w of w with respect to the generators s furnished by this isomorphism yields the following identity of poincare series det 1 t e d 1 det 1 t vd

computing fitting ideals of iwasawa modules this paper determines in an equivariant sense the fitting ideals of several iwasawa modules including the most canonical one the connection between the modules themselves which are usually not of finite projective dimension and the required auxiliary modules of finite projective dimension is made rather explicit the resulting fitting ideals look like stickelberger ideals and there is a close relation with recent work of kurihara as an application we obtain new evidence in favor of the brumer stark conjecture

computation of stark tamagawa units let k be a totally real number field and let l denote an odd prime number we design an algorithm which computes strong numerical evidence for the validity of the equivariant tamagawa number conjecture for the q g equivariant motive h0 spec l where l k is a cyclic extension of degree l and group g this conjecture is a very deep refinement of the classical analytic class number formula in the course of the algorithm we compute a set of special units which must be considered as a generalization of the conjecturally existing stark units associated to first order vanishing dirichlet l functions

twisting derived equivalences we introduce a new method for twisting relative equivalences of derived categories of sheaves on two spaces over the same base the first aspect of this is that the derived categories of sheaves on the spaces are twisted they become derived categories of sheaves on gerbes living over spaces that are locally on the base isomorphic to the original spaces secondly this is done in a compatible way so that the equivalence is maintained we apply this method by proving the conjectures of donagi and pantev on dualities between gerbes on genus one fibrations and comment on other applications to families of higher genus curves we also include a related conjecture in mirror symmetry

an explicit description of the irreducible components of the set of matrix pencils with bounded normal rank abstract the set of m n singular matrix pencils with normal rank at most r is an algebraic set with r 1 irreducible components these components are the closure of the orbits under strict equivalence of r 1 matrix pencils which are in kronecker canonical form in this paper we provide a new explicit description of each of these irreducible components which is a parametrization of each component therefore one can explicitly construct any pencil in each of these components the new description of each of these irreducible components consists of the sum of r rank 1 matrix pencils namely a column polynomial vector of degree at most 1 times a row polynomial vector of degree at most 1 where we impose one of these two vectors to have degree zero the number of row vectors with zero degree determines each irreducible component

miniversal deformations of matrices of bilinear forms arnold v i arnold on matrices depending on parameters russian math surveys 26 2 1971 29 43 constructed miniversal deformations of square complex matrices under similarity that is a simp

minus class groups of the fields of the l th roots of unity we show that for any prime number l 2 the minus class group of the field of the l th roots of unity q p l admits a finite free resolution of length 1 as a module over the ring z g 1 i here i denotes complex conjugation in g gal q p l q p z lz moreover for the primes l 509 we show that the minus class group is cyclic as a module over this ring for these primes we also determine the structure of the minus class group

non commutative fitting invariants and annihilation of class groups abstract one can associate to each finitely presented module m over a commutative ring r an r ideal fitt r m which is called the zeroth fitting ideal of m over r and which is an important natural invariant of m we generalize this notion to o orders in separable algebras where o is a complete commutative noetherian local ring as an application we construct annihilators of class groups assuming the validity of the equivariant tamagawa number conjecture for a certain motive attached to a galois cm extension of number fields

refined iwasawa theory and kolyvagin systems of gauss sum type in this paper we establish a refinement of the usual iwasawa main conjecture for the ideal class groups of cm fields over a totally real field using higher fitting ideals

adjoints to a fourier mukai functor abstract given a fourier mukai functor in the general setting of singular schemes under various hypotheses we provide both left and a right adjoints to and also give explicit formulas for them these formulas are simple and natural and recover the usual formulas when the fourier mukai kernel is a perfect complex this extends previous work of 1 12 13 and has applications to the twist autoequivalences of 9

spectral equivalence of matrix polynomials and the index sum theorem abstract the concept of linearization is fundamental for theory applications and spectral computations related to matrix polynomials however recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive because some structured polynomials do not have any linearization with the same structure this phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations motivated by this fact we introduce equivalence relations that allow the possibility of matrix polynomials with coefficients in an arbitrary field to be equivalent with the same spectral structure but have different sizes and degrees these equivalence relations are directly modeled on the notion of linearization and consequently inherit the simplicity applicability and most relevant properties of linearizations simultaneously though they are much more flexible in the possible degrees of equivalent polynomials this flexibility allows us to define in a unified way the notions of quadratification and l ification to introduce the concept of companion form of arbitrary degree and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second frobenius companion forms the properties of l ifications are studied in depth in this process a fundamental result on matrix polynomials the index sum theorem is recovered and extended to arbitrary fields although this result is known in the systems theory literature for real matrix polynomials it has remained unnoticed by many researchers it establishes that the sum of the finite and infinite partial multiplicities together with the left and right minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial the index sum theorem turns out to be a key tool for obtaining a number of significant results on the possible sizes and degrees of l ifications and companion forms on the minimal index preservation properties of companion forms of arbitrary degree as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree this paper presents many new results blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields therefore we have written the paper in an expository and self contained style that makes it accessible to a wide variety of readers

skew symmetric matrix pencils codimension counts and the solution of a pair of matrix equations the homogeneous system of matrix equations x t a ax xb b t bx 0 0 where a b is a pair of skew symmetric matrices of the same size is considered we establish the general solution

elliptic units for real quadratic fields 1 a review of the classical setting 2 elliptic units for real quadratic fields 2 1 p adic measures 2 2 double integrals 2 3 splitting a two cocycle 2 4 the main conjecture 2 5 modular symbols and dedekind sums 2 6 measures and the bruhat tits tree 2 7 indefinite integrals 2 8 the action of complex conjugation and of up 3 special values of zeta functions 3 1 the zeta function 3 2 values at negative integers 3 3 the p adic valuation 3 4 the brumer stark conjecture 3 5 connection with the gross stark conjecture 4 a kronecker limit formula 4 1 measures associated to eisenstein series 4 2 construction of the p adic l function 4 3 an explicit splitting of a two cocycle 4 4 generalized dedekind sums 4 5 measures on zp zp 4 6 a partial modular symbol of measures on zp zp 4 7 from zp zp to x 4 8 the measures and invariance

divisibility properties of values of partial zeta functions at non positive integers we conjecture a new bound on the exact denominators of the values at non positive integers of imprimitive partial zeta functions associated with an abelian extension of number fields at s 0 this conjecture is closely connected to a conjecture of david hayes we prove the new conjecture assuming that the coates sinnott conjecture holds for the extension

the eigenstructure of an arbitrary polynomial matrix computational aspects we give a new numerical method to compute the eigenstructure i e the zero structure the polar structure and the left and right null space structure of a polynomial matrix p these structural elements are of fundamental importance in several systems and control problems involving polynomial matrices the approach is more general than previous numerical methods because it can be applied to an arbitrary m n polynomial matrix p with normal rank r smaller than m and or n the algorithm is then shown to compute the structure of the left and right null spaces of p as well the speed and accuracy of this new approach are also discussed

on the equivariant tamagawa number conjecture in tame cm extensions let l k be a finite galois cm extension with galois group g the equivariant tamagawa number conjecture etnc for the pair h 0 rm spec l 0 mathbb z g naturally decomposes into p parts where p runs over all rational primes if p is odd these p parts in turn decompose into a plus and a minus part let l k be tame above p we show that a certain ray class group of l defines an element in k 0 mathbb z p g mathbb q p which is determined by a corresponding stickelberger element if and only if the minus part of the etnc at p holds for this we use the lifted root number conjecture for small sets of places which is equivalent to the etnc in the number field case for abelian g we show that the minus part of the etnc at p implies the strong brumer stark conjecture at p we prove the minus part of the etnc at p for almost all primes p

noncommutative deformations and flops we prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 33 fold is representable and hence we associate to every such curve a noncommutative deformation algebra aconacon this new invariant extends and unifies known invariants for flopping curves in 33 folds such as the width of reid and the bidegree of the normal bundle it also applies in the settings of flips and singular schemes we show that the noncommutative deformation algebra aconacon is finite dimensional and give a new way of obtaining the commutative deformations of the curve allowing us to make explicit calculations of these deformations for certain 3 1 3 1 curves we then show how our new invariant aconacon also controls the homological algebra of flops for any flopping curve in a projective 33 fold with only gorenstein terminal singularities we construct an autoequivalence of the derived category of the 33 fold by twisting around a universal family over the noncommutative deformation algebra aconacon and prove that this autoequivalence is an inverse of bridgeland s flop flop functor this demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 33 fold and thus the bridgeland stability manifold

computing the hilbert class field of real quadratic fields using the units appearing in stark s conjectures on the values of l functions at s 0 we give a complete algorithm for computing an explicit generator of the hilbert class field of a real quadratic field