cyclic length in the tame brauer group of the function field of a p adic curve let f be the function field of a smooth curve over the p adic number field qp we show that for each prime to p number n the n torsion subgroup h 2 f n nbr f is generated by z n cyclic classes in fact the z n length is equal to two it follows that the brauer dimension of f is two first proved in 20 and any f division algebra of period n and index n 2 is decomposable

the period index problem for fields of transcendence degree 2 using geometric methods we prove the standard period index conjecture for the brauer group of a field of transcendence degree 2 over fp

hasse principle for hermitian spaces over semi global fields abstract in a recent paper colliot thelene parimala and suresh conjectured that a local global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p adic curves in this paper we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew hermitian spaces over central simple algebras with involutions the proof implements patching techniques of harbater hartmann and krashen as an application we obtain a springer type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p adic curves

the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

the u invariant of the function fields of p adic curves the u invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field it is an open question whether the u invariant of function fields of p aidc curves is 8 in this paper we answer this question in the affirmative for function fields of nondyadic p adic curves

r equivalence in adjoint classical groups over fields of virtual cohomological dimension 2 let f be a field of characteristic not 2 whose virtual cohomological dimension is at most 2 let g be a semisimple group of adjoint type defined over f let rg f denote the normal subgroup of g f consisting of elements r equivalent to identity we show that if g is of classical type not containing a factor of type d n g f rg f 0 if g is a simple classical adjoint group of type d n we show that if f and its multi quadratic extensions satisfy strong approximation property then g f rg f 0 this leads to a new proof of the r triviality of f rational points of adjoint classical groups defined over number fields

rationality problem for generic tori in simple groups abstract we prove that except for several already known cases the generic torus of a simple adjoint or simply connected group is not stably rational this confirms a conjecture by le bruyn on generic norm tori

polarized variation of hodge structures of calabi yau type and characteristic subvarieties over bounded symmetric domains in this paper we extend the construction of the canonical polarized variation of hodge structures over tube domain considered by gross math res lett 1 1 9 1994 to bounded symmetric domain and introduce a series of invariants of infinitesimal variation of hodge structures which we call characteristic subvarieties we prove that the characteristic subvariety of the canonical polarized variations of hodge structures over irreducible bounded symmetric domains are identified with the characteristic bundles defined by mok ann math 125 1 105 152 1987 we verified the generating property of gross for all irreducible bounded symmetric domains which was predicted in gross math res lett 1 1 9 1994

local global principles for norm one tori over semi global fields let k be a complete discretely valued field with the residue field kappa let f be the function field of a smooth projective geometrically integral curve over k and mathscr x be a regular proper model of f such that the reduced special fibre x is a union of regular curves with normal crossings suppose that the graph associated to mathscr x is a tree e g f k t let l f be a galois extension of degree n such that n is coprime to char kappa suppose that kappa is an algebraically closed field or a finite field containing a primitive n mathrm th root of unity then we show that the local global principle holds for the norm one torus associated to the extension l f with respect to discrete valuations on f i e an element in f times is a norm from the extension l f if and only if it is a norm from the extensions l otimes f f nu f nu for all discrete valuations nu of f

on galois cohomology of semisimple groups over local and global fields of positive characteristic ii we show that the recent results of prasad and rapinchuk adv math 207 2 646 660 2006 on the existence and uniqueness of certain global forms of semisimple algebraic groups with given local behaviour in the case of number fields still hold in the case of global function fields

applications of patching to quadratic forms and central simple algebras this paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields in particular we use a patching approach to reprove and generalize a recent result of parimala and suresh in preprint arxiv 0708 3128 2007 on the u invariant of p adic function fields p 2 the strategy relies on a local global principle for homogeneous spaces for rational algebraic groups combined with local computations

a pointwise quasi newton method for unconstrained optimal control problems for a class of unconstrained optimal control problems we propose a quasi newton method that exploits the structure of the problem we define a new type of superlinear convergence for sequences in function spaces and prove superlinear convergence of the iterates generated by the quasi newton method in this sense

the kernel of the rost invariant serre s conjecture ii and the hasse principle for quasi split groups 3 6d4 e6 e7 we prove that for a simple simply connected quasi split group of type 3 6d4 e6 e7 defined over a perfect field f of characteristic 2 3 the rost invariant has trivial kernel in certain cases we give a formula for the rost invariant it follows immediately from the result above that if cd f 2 resp vcd f 2 then serre s conjecture ii resp the hasse principle holds for such a group for a c2 field in particular x y we prove the stronger result that serre s conjecture ii holds for all not necessary quasi split exceptional groups of type 3 6d4 e6 e7

admissibility of groups over function fields of p adic curves let k be a field and g a finite group the question of admissibility of g over k was originally posed by schacher who gave partial results in the case k q in this paper we give necessary conditions for admissibility of a finite group g over function fields of curves over complete discretely valued fields using this criterion we give an example of a finite group which is not admissible over qp t we also prove a certain hasse principle for division algebras over such fields

strongly inner anisotropic forms of simple algebraic groups the definition of the index of a semisimple algebraic group over a field will be recalled in 1 55 more precisely we show in proposition 2 b cf 3 1 that if over a given field k there exists a central division algebra of degree 4 i e of dimension 16 and of order 4 in the brauer group then there exists a group of type ei t over a purely transcendental extension of degree 1 k t of k this will be deduced in two different ways from theorem 1 of section 2 whose precise statement is too technical to be given in this introduction but whose spirit is that it enables one from the existence of groups of a given type with preassigned properties of a geometric and cohomological nature over a field k to infer the existence of groups of a more complicated type meeting similar requirements over k t that theorem has other interesting applications to classification problems for instance we also deduce from it again in two different ways proposition 3 b in 3 2 that the existence of a division algebra of degree 3 over k implies the existence of groups with index

le groupe fondamental etale d un espace homogene d un groupe algebrique lineaire let x be a homogeneous space of a connected linear algebraic group g defined over an algebraic closed field k of characteristic exponent p let x in x k we denote by h the stabilizer of x in g and we assumed that h is connected or abelian in this text we compute explicitely the prime to p part of the etale fundamental group pi 1 acute mathrm e mathrm t x x in terms of the character groups of g and h as an application we prove a variant of the section conjecture for homogeneous spaces

polarized mixed hodge structures and the local monodromy of a variation of hodge structure a variation of polarized hodge structures on the complement of a divisor with normal crossings gives rise locally on the divisor to a commuting set ni of nilpotent endomorphisms of the vector space underlying the variation the logarithms of the unipotent parts of the various picard lefschetz transformations these reflect properties of the singularities of the period map associated to the variation cf 5 7 9 and sect 3 for example a variation depending on a single parameter defines asymptotically a mixed hodge structure whose weight filtration is the monodromy weight filtration of the corresponding endomorphism n 9 11 this paper is mainly concerned with the properties of the set n arising in the several parameters case in particular theorem 3 3 asserts that all the elements in the open polyhedral cone c spanned over r by the n s determine the same monodromy weight filtration it also describes the relationship between this common filtration and those associated to the faces of the cone this statement was conjectured by deligne based on his analogous result 4 1 9 2 for the q cone associated to those variations which arise from families of polarized non singular algebraic varieties in w 1 we discuss some general properties of commuting sets of nilpotent endomorphisms and their associated filtrations section2 focuses on the case when each element in such set defines in the sense of 2 4 a polarized mixed hodge structure with a fixed hodge filtration f as is the case for the monodromy cone c of a variation these conditions imply cf 2 16 a generic uniqueness for the monodromy weight filtration we also consider those pola

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

the elementary obstruction and homogeneous spaces let k be a field of characteristic zero and let k be an algebraic closure of k for a geometrically integral variety x over k we write k x for the function field of x x k k if x has a smooth k point the natural embedding of multiplicative groups k k x admits a galois equivariant retraction in the first part of this article equivalent conditions to the existence of such a retraction are given over local and then over global fields those conditions are expressed in terms of the brauer group of x in the second part of the article we restrict attention to varieties that are homogeneous spaces of connected but otherwise arbitrary algebraic groups with connected geometric stabilizers for k local or global and for such a variety x in many situations but not all the existence of a galois equivariant retraction to k k x ensures the existence of a k rational point on x for homogeneous spaces of linear algebraic groups the technique also handles the case where k is the function field of a complex surface resume soient k un corps de caracteristique nulle et k une cloture algebrique de k pour une k variete x geometriquement integre on note k x le corps des fonctions de x x k k si x possede un k point lisse le plongement naturel de groupes multiplicatifs k k x admet une retraction equivariante pour l action du groupe de galois de k sur k dans la premiere partie de l article sur les corps locaux puis sur les corps globaux on donne des conditions equivalentes a l existence d une telle retraction equivariante ces conditions s expriment en terme du groupe de brauer de la variete x dans la seconde partie de l article on considere le cas des espaces homogenes de groupes algebriques connexes non necessairement lineaires avec groupes d isotropie duke mathematical journal vol 141 no 2 c 2008 received 6 november 2006 revision received 23 april 2007 2000 mathematics subject classification primary 14g05 11g99 12g05 secondary 11e72 14f22 14k15

division algebras over surfaces abstract we consider here a number of topics concerning the theory of division algebras over the function field of a surface one result relates the obstruction for ramification data to be from a division algebra and third etale cohomology another result shows this obstruction is always zero when the surface is spec of a regular local ring with some mild assumptions at the same time we study the brauer group of this function field as it relates to the brauer group of the function field of the henselization finally we prove a result which says that brauer group elements which look like they are of prime index q unequal to any characteristic must have all their ramification split by a cyclic galois extension of the same degree this last result requires a primitive q root of one

galois cohomology and forms of algebras over laurent polynomial rings 3 loop torsors 8 3 1 the witt tits index of a loop torsor 9 3 2 reducibility 12 3 3 almost commutative subgroups 14 3 4 almost commuting families of elements of finite order 16 3 5 almost commuting pairs and their invariants 17 3 6 failure in the anisotropic case 24

on the u invariant of function fields of curves over complete discretely valued fields abstract let k be a complete discretely valued field with residue field if char k 0 char 2 and 2 d we prove that there exists an integer n depending on d such that the u invariant of any function field in one variable over k is bounded by n the method of proof is via introducing the notion of uniform boundedness for the p torsion of the brauer group of a field and relating the uniform boundedness of the 2 torsion of the brauer group to the finiteness of the u invariant we prove that the 2 torsion of the brauer group of function fields in one variable over k is uniformly bounded

indecomposable and noncrossed product division algebras over function fields of smooth p adic curves we construct indecomposable and noncrossed product division algebras over function fields of connected smooth curves x over zp this is done by defining an index preserving morphism s br k x br k x which splits res br k x br k x where k x is the completion of k x at the special fiber and using it to lift indecomposable and noncrossed product division algebras over k x

cyclic algebras over p adic curves abstract in this paper we study division algebras over the function fields of curves over q p the first and main tool is to view these fields as function fields over nonsingular s which are projective of relative dimension 1 over the p adic ring z p a previous paper showed such division algebras had index bounded by n 2 assuming the exponent was n and n was prime to p in this paper we consider algebras of prime degree and hence exponent q p and show these algebras are cyclic we also find a geometric criterion for a brauer class to have index q

motives with exceptional galois groups and the inverse galois problem we construct motivic l adic representations of textup gal overline mathbb q mathbb q into exceptional groups of type e 7 e 8 and g 2 whose image is zariski dense this answers a question of serre the construction is uniform for these groups and is inspired by the langlands correspondence for function fields as an application we solve new cases of the inverse galois problem the finite simple groups e 8 mathbb f ell are galois groups over mathbb q for large enough primes l

on the behaviour of brauer p dimensions under finitely generated field extensions abstract the present paper shows that if q p or q 0 where p is the set of prime numbers then there exist characteristic q fields e q k k n of brauer dimension brd e q k k and infinite absolute brauer p dimensions abrd p e q k for all p p not dividing q 2 q this ensures that brd p f q k p q 2 q for every finitely generated transcendental extension f q k e q k we also prove that each sequence a p b p p p satisfying the conditions a 2 b 2 and 0 b p a p equals the sequence abrd p e brd p e p p for a field e of characteristic zero

hermitian analogue of a theorem of springer with involutions and in fact in a much more general setupse for algebraic groups the weak form of springer s theorem forhermitian forms was proved by bayer fluckiger and lenstra in bl andthe analogue of springer s theorem for hermitian forms was raised as anopen question in bst the aim of this paper is to settle this question inthe affirmative for hermitian forms over quaternion algebras with respectto involutions of the first kind we prove that if

classification theorems for hermitian forms the rost kernel and hasse principle over fields with cd2 k 3 abstract in this paper we prove classification theorems for hermitian forms over some central simple algebras with involution over a field k with cd 2 k 3 we apply these results to show the triviality of the kernel of the rost invariant for the classical algebraic groups associated to such hermitian forms over k we also deduce a hasse principle for algebraic groups defined over function fields of curves over p adic fields thus proving a conjecture due to colliot thelene parimala suresh for a large class of groups

lois de reciprocite superieures et points rationnels let k c x y or k c x y let g be a connected linear algebraic group over k under the assumption that the k variety g is k rational i e that the function field is purely transcendental it was proved that a principal homogeneous space of g has a rational point over k as soon as it has one over each completion of k with respect to a valuation in this paper we show that one cannot in general do without the k rationality assumption to produce our examples we introduce a new type of obstruction it is based on higher reciprocity laws on a 2 dimensional scheme we also produce a family of principal homogeneous spaces for which the refined obstruction controls exactly the existence of rational points

quasi abelian crossed modules and nonabelian cohomology abstract we extend the work of m borovoi on the nonabelian galois cohomology of linear reductive algebraic groups over number fields to a general base scheme as an application we obtain new results on the arithmetic of such groups over global function fields

noncyclic division algebras over fields of brauer dimension one abstract let k be a complete discretely valued field of rank one with residue field q p it is well known that period equals index in br k we prove that when p 2 there exist noncyclic k division algebras of every 2 power degree divisible by four otherwise every k division algebra is cyclic this gives the first published example of a field whose brauer dimension and cyclic length are not equal

colliot thelene s conjecture and finiteness of u invariants let k be a number field and f a function field in one variable over k we prove that the ramification of a p torsion element in br f on a regular proper model over the ring of integers in k can be split in an extension of degree p 3 using this result we show that colliot thelene s conjecture on 0 cycles of degree 1 implies finiteness for the u invariant of the function field of a curve over a totally imaginary number field and period index bounds for the brauer groups of arbitrary fields of transcendence degree 1 over the rational numbers

an implicit filtering algorithm for optimization of functions with many local minima in this paper we describe and analyze an algorithm for certain box constrained optimization problems that may have several local minima a paradigm for these problems is one in which the function to be minimized is the sum of a simple function such as a convex quadratic and high frequency low amplitude terms that cause local minima away from the global minimum of the simple function our method is gradient based and therefore the performance can be improved by use of quasi newton methods

grothendieck s theorem on non abelian h 2 and local global principles a theorem of grothendieck asserts that over a perfect field k of cohomological dimension one all non abelian h 2 cohomology sets of algebraic groups are trivial the purpose of this paper is to establish a formally real generalization of this theorem the generalization to the context of perfect fields of virtual cohomological dimension one takes the form of a local global principle for the h 2 sets with respect to the orderings of the field this principle asserts in particular that an element in h 2 is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of k our techniques provide a new proof of grothendieck s original theorem an application to homogeneous spaces over k is also given

quasi newton methods and unconstrained optimal control problems we prove a mesh independence result for the bfgs method in hilbert space and apply it to a class of unconstrained optimal control problems a new fourth order discretization scheme is used in the i

isotropic trialitarian algebraic groups abstract by modifying a construction from knus et al we construct all isotropic algebraic groups of type 3 d 4 and 6 d 4 over an arbitrary field of characteristic 2 we also provide a nice isomorphism criterion for such groups the results of this paper extend the main results of allison using entirely different methods to fields of nonzero characteristic and algebraic groups

a springer theorem for hermitian forms throughout this paper we let d be a central f division algebra with involution such that f d f d d is a henselian valued field by 11 the valuation on f extends uniquely to a valuation on d we denote this valuation by v moreover we assume that the characteristic of the residue field open image in new window is not 2 if the valuation on f is discrete then any quadratic form q can be written as q q1 q2 where is a uniformizer and qi are unit forms springer s theorem states that q is isotropic if and only if at least one of the residue forms open image in new window and open image in new window is isotropic in this paper we generalize this result to hermitian forms in section 4 we use the connection between involutions on algebras and hermitian forms to prove an analog of the springer theorem for involutions

convergence theorems for least change secant update methods the purpose of this paper is to present a convergence analysis of least change secant methods in which part of the derivative matrix being approximated is computed by other means the theorems and proofs given here can be viewed as generalizations of those given by broyden dennis more j inst math appl 12 1973 pp 223 246 and by dennis more math comp 28 1974 pp 549 560 the analysis is done in the orthogonal projection setting of dennis schnabel siam rev 21 1980 pp 443 459 and many readers might feel that it is easier to understand the theorems here readily imply local and q superlinear convergence of all the standard methods in addition to proving these results for the first time for the sparse symmetric method of marwil and toint and the nonlinear least squares method of dennis gay welsch

period index bounds for arithmetic threefolds the standard period index conjecture for brauer groups of p adic surfaces s predicts that mathrm ind alpha mathrm per alpha 3 for every alpha in mathrm br mathbf q p s using gabber s theory of prime to ell alterations and the deformation theory of twisted sheaves we prove that mathrm ind alpha mathrm per alpha 4 for alpha of period prime to 6p giving the first uniform period index bounds over such fields

local global principles for torsors over arithmetic curves we consider local global principles for torsors under linear algebraic groups over function fields of curves over complete discretely valued fields the obstruction to such a principle is a version of the tate shafarevich group and for groups with rational components we compute it explicitly and show that it is finite this yields necessary and sufficient conditions for local global principles to hold our results rely on first obtaining a mayer vietoris sequence for galois cohomology and then showing that torsors can be patched we also give new applications to quadratic forms and central simple algebras

variations of hodge structure and orbits in flag varieties abstract period domains the classifying spaces for pure polarized hodge structures and more generally mumford tate domains arise as open g r orbits in flag varieties g p we investigate hodge theoretic aspects of the geometry and representation theory associated with these flag varieties in particular we relate the griffiths yukawa coupling to the variety of lines on g p under a minimal homogeneous embedding construct a large class of polarized g r orbits in g p and compute the associated hodge theoretic boundary components an emphasis is placed throughout on adjoint flag varieties and the corresponding families of hodge structures of levels two and four

partitioned dynamic programming for optimal control parallel algorithms for the solution of linear quadratic optimal control problems are described the algorithms are based on a straightforward decomposition of the domain of the problem and are related to multiple shooting methods for two point boundary value problems their arithmetic cost is approximately twice that of the serial dynamic programming approach however they have the advantage that they can be efficiently implemented on a wide variety of parallel architectures extension to the case in which there are box constraints on the controls is simple the algorithms can be used to solve linear quadratic subproblems arising from the application of newton s method or two metric gradient projection methods to nonlinear problems

period index and u invariant questions for function fields over complete discretely valued fields let k be a complete discretely valued field with residue field and f the function field of a curve over k let p be the characteristic of and l a prime not equal to p if the brauer l dimensions of all finite extensions of are bounded by d and the brauer l dimensions of all extensions of of transcendence degree at most 1 are bounded by d 1 then it is known that the brauer l dimension of f is at most d 2 lieblich in j reine angew math 659 1 41 2011 saltman in j ramanujan math soc 12 25 47 1997 harbater et al in invent math 178 231 263 2009 in this paper we give a bound for the brauer p dimension of f in terms of the p rank of as an application we show that if is a perfect field of characteristic 2 then any quadratic form over f in at least 9 variables is isotropic this leads to the fact that every element in h 3 f mathbb z 2 mathbb z is a symbol if is a finite field of characteristic 2 u f 8 is a result of heath brown leep heath brown in compos math 146 271 287 2010 leep in j reine angew math 2013 to appear

r equivalence and special unitary groups abstract a norm homomorphism for the group of r equivalence classes of all simply connected semisimple classical algebraic groups is constructed the group of r equivalence classes for special unitary groups su b is computed it is proved that the variety of su b is rational if ind b 3 and the stable birational type of su b depends only on the brauer class of b and does not depend on the involution

a quasi newton method for elliptic boundary value problems for some boundary value problems for equations of the form nabla 2 u f x u nabla u 0 with linear boundary conditions we give a convergence analysis for a quasi newton method that converges superlinearly in the c 1 norm this scheme for infinite dimensional problems becomes a method for the finite dimensional problems that are produced by discretizing the differential equation this method is a generalization of a method proposed by hart and soul for discretizations of two point boundary value problems