bicyclic algebras of prime exponent over function fields we examine some properties of bicyclic algebras i e the tensor product of two cyclic algebras defined over a purely transcendental function field in one variable we focus on the following problem when does the set of local invariants of such an algebra coincide with the set of local invariants of some cyclic algebra although we show this is not always the case we determine when it happens for the case where all degeneration points are defined over the ground field our main tool is faddeev s theory we also study a geometric counterpart of this problem pencils of severi brauer varieties with prescribed degeneration data

on hormander s solution of the bar partial equation i we explain how hormander s classical solution of the bar partial equation in the plane with a weight which permits growth near infinity carries over to the rather opposite situation when we ask for decay near infinity here however a natural condition on the datum needs to be imposed the condition is not only natural but also necessary to have the result at least in the fock weight case the norm identity which leads to the estimate is related to general area type results in the theory of conformal mappings

homological stability for topological chiral homology of completions abstract by proving that several new complexes of embedded disks are highly connected we obtain several new homological stability results our main result is homological stability for topological chiral homology on an open manifold with coefficients in certain partial framed e n algebras using this we prove a special case of a conjecture of vakil and wood on homological stability for complements of closures of particular strata in the symmetric powers of an open manifold and we prove that the bounded symmetric powers of closed manifolds satisfy homological stability rationally

groupe de chow de codimension deux des varietes definies sur un corps de nombres un theoreme de finitude pour la torsion letx be a smooth projective variety defined over a number field and let ch2 x denote the chow group of codimension two cycles modulo rational equivalence we show that if the cohomology grouph2 x ox vanishes then the torsion subgroup of ch2 x is a finite group this result covers all previous results in this direction the hypothesish2 x ox 0 is used to lift line bundles

insufficiency of the brauer manin obstruction applied to etale covers let k be any global field of characteristic not 2 we construct a k variety x such that x k is empty but for which the emptiness cannot be explained by the brauer manin obstruction or even by the brauer manin obstruction applied to finite etale covers

weighted bergman spaces and the integral means spectrum of conformal mappings the classical theory of conformal mappings involves best possible pointwise estimates of the derivative thus supplying a measure of the extremal expansion contraction possible for a conformal mapping it is natural to consider also the integral means of phi t along circles z r where phi is the conformal mapping in question and t is a real parameter 0 1 of the integral means which follows from the classical pointwise estimates is by far too fast better estimates were found by clunie makarov pommerenke bertilsson shimorin and others here we introduce a new method based on area type estimates which discards as little as possible of the information supplied by the area methods the result is a considerable improvement in the estimates of the integral means spectrum known tip to this point

batalin vilkovisky algebras and two dimensional topological field theories by a batalin vilkovisky algebra we mean a graded commutative algebraa together with an operator a a 1 such that 2 0 and a a is a graded derivation ofa for alla a in this article we show that there is a natural structure of a batalin vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions we make use of a technique from algebraic topology the theory of operads

the arithmetic of zero cycles on surfaces with geometric genus and irregularity zero let x be a smooth projective geometrically irreducible surface over a perfect field f throughout this paper it will be assumed that the geometric genus pg and the irregularity q of x both vanish denote the separable closure of f by f let x x be the surface obtained from x by base extension it will also be assumed that the group ao x of rational equivalence classes of zero cycles of degree zero on x vanishes this is a technical hypothesis which could presumably be eliminated at the expense of working with ker ao x ao x for want of a better name and for ease in stating various results any surface which satisfies these three hypotheses will be called a pseudo rational surface bloch 2 has conjectured that the vanishing of ao x should follow from the assumption that po q 0 this was proven by bloch kas and lieberman 5 for all such surfaces which are not of general type i e which have kodaira dimension less than 2 it has also been proven for particular surfaces of general type by inose and mizukami 23 barlow 1 and keum 25 consequently the class of pseudo rational surfaces includes rational surfaces enriques surfaces elliptic surfaces with q 0 the classical godeaux surface burniat inoue surfaces campedelti surfaces and the surfaces of barlow and of keum this paper will study ao x for pseudo rational surfaces defined over fields of number theoretic interest bloch 4 introduced k theoretic techniques into the study of zero cycles on rational surfaces his work was extended 7 12 15 27 28 32 36 to achieve a thorough understanding of such cycles colliot th616ne and raskind 10 developed this machinery further to study codimension two cycles on any variety the author 14 applied these techniques to enriques surfaces raskind 30 used them to study zero cycles on pseudo rational surfaces one of the main results of this paper is a new proof of the following theorem of raskind 31

linear growth for châtelet surfaces an upper bound of the expected order of magnitude is established for the number of mathbb q rational points of bounded height on ch telet surfaces defined over mathbb q

effectivity of brauer manin obstructions on surfaces abstract we study brauer manin obstructions to the hasse principle and to weak approximation on algebraic surfaces over number fields a technique for constructing azumaya algebra representatives of brauer group elements is given and this is applied to the computation of obstructions

central stability for the homology of congruence subgroups and the second homology of torelli groups abstract we prove a representation stability result for the second homology groups of torelli subgroups of mapping class groups and automorphism groups of free groups this strengthens the results of boldsen hauge dollerup and day putman we also prove a new representation stability result for the homology of certain congruence subgroups partially improving upon the work of putman sam these results follow from a general theorem on syzygies of certain modules with finite polynomial degree

integral estimates of conformal derivatives and spectral properties of the neumann laplacian abstract in this paper we study integral estimates of derivatives of conformal mappings d of the unit disc d c onto bounded domains that satisfy the ahlfors condition these integral estimates lead to estimates of constants in sobolev poincare inequalities and by the rayleigh quotient we obtain spectral estimates of the neumann laplace operator in non lipschitz domains quasidiscs in terms of the quasi conformal geometry of the domains specifically the lower estimates of the first non trivial eigenvalues of the neumann laplace operator in some fractal type domains snowflakes were obtained

random conformal snowflakes in many problems of classical analysis extremal configurations appear to exhibit complicated fractal structures making it hard to describe them and to attack such problems this is particularly true for questions related to the multifractal analysis of harmonic measure we argue that searching for extremals in such problems one should work with random fractals rather than deterministic ones we introduce a new class of fractals random conformal snowflakes and investigate their properties developing tools to estimate spectra and showing that extremals can be found in this class as an application we significantly improve known estimates from below on the extremal behavior of harmonic measure showing how to construct a rather simple snowflake which has a spectrum quite close to the conjectured extremal value

rational points on pencils of conics and quadrics with many degenerate fibres for any pencil of conics or higher dimensional quadrics over q with all degenerate bres dened over q we show that the brauer manin obstruction controls weak approximation the proof is based on the hasse principle and weak approximation for some special intersections of quadrics over q which is a consequence of recent advances in additive combinatorics

the rost nilpotence theorem for geometrically rational surfaces we show that the rost nilpotence theorem is true for geometrically rational surfaces over fields of characteristic zero

boundary properties of green functions in the plane we study the boundary properties of the green function of bounded simply connected domains in the plane essentially this amounts to studying the conformal mapping taking the unit disk onto the domain in question our technique is inspired by a 1995 article of jones and makarov 11 the main tools are an integral identity as well as a uniform sobolev embedding theorem the latter is in a sense dual to the exponential integrability of marcinkiewicz zygmund integrals we also develop a grunsky identity which contains the information of the classical grunsky inequality this grunsky identity is the case where p 2 of a more general grunsky identily for l p spaces

on the rational homotopy type of function spaces the main result of this paper is the construction of a minimal model for the function space f x y of continuous functions from a finite type finite dimensional space x to a finite type nilpotent space y in terms of minimal models for x and y for the component containing the constant map 7r f x y 0 q 7r y 0 h x q in positive dimensions when x is formal there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for y and the coproduct of h x q we also give a version of the main result for the space of cross sections of a fibration

higher hochschild homology topological chiral homology and factorization algebras we study the higher hochschild functor factorization algebras and their relationship with topological chiral homology to this end we emphasize that the higher hochschild complex is a functor sset cdga where sset and cdga are the 1 categories of simplicial sets and commutative differential graded algebras and give an axiomatic characterization of this functor from the axioms we deduce several properties and computational tools for this functor we study the relationship between the higher hochschild functor and factorization algebras by showing that in good cases the hochschild functor determines a constant commutative factorization algebra conversely every constant commutative factorization algebra is naturally equivalent to a hochschild chain factorization algebra similarly we study the relationship between the above concepts and topological chiral homology in particular we show that on their common domains of definition the higher hochschild functor is naturally equivalent to topological chiral homology finally we prove that topological chiral homology determines a locally constant factorization algebra and further that this functor induces an equivalence between locally constant factorization algebras on a manifold and local system of en algebras

e_n cell attachments and a local to global principle for homological stability we define bounded generation for e n algebras in chain complexes and prove that this property is equivalent to homological stability for n ge 2 using this we prove a local to global principle for homological stability which says that if an e n algebra a has homological stability or equivalently the topological chiral homology int mathbb r n a has homology stability then so has the topological chiral homology int m a of any connected non compact manifold m using scanning we reformulate the local to global homological stability principle so that it applies to compact manifolds we also give several applications of our results

manin s conjecture for quartic del pezzo surfaces with a conic fibration an asymptotic formula is established for the number of q rational points of bounded height on a non singular quartic del pezzo surface with a conic bundle structure

hasse principle for pencils of curves of genus one whose jacobians have rational 2 division points une serie d articles exploite une nouvelle technique qui mene a des conditions suffisantes d existence et de densite des points rationnels sur certaines surfaces fibrees en courbes de genre un au dessus de la droite projective dans les premiers articles de cette serie plusieurs articles de swinnerton dyer un article en collaboration de skorobogatov swinnerton dyer et l auteur la jacobienne de la fibre generique des surfaces considerees a tous ses points d ordre 2 rationnels un article recent de bender et swinnerton dyer traite de cas ou cette jacobienne possede seulement un point d ordre 2 non trivial pour que la methode fonctionne il semble necessaire que la jacobienne possede un point de torsion rationnel non trivial le present article est une reecriture de celui de bender et swinnerton dyer la principale contribution est une reformulation plus abstraite des hypotheses principales des theoremes la premiere hypothese est formulee de facon entierement algebrique certains groupes de selmer algebrico geometriques sont supposes petits et la seconde hypothese est simplement il n y a pas d obstruction de brauer manin verticale comme dans la plupart des articles de cette serie les resultats dependent de deux conjectures difficiles l hypothese de schinzel et la finitude des groupes de tate shafarevich

local global principle for 0 cycles on fibrations over rationally connected bases abstract we study the brauer manin obstruction to the hasse principle and to weak approximation for 0 cycles on algebraic varieties that possess a fibration structure the issue is to establish the exactness of a local to global sequence e of chow groups of 0 cycles recently harpaz and wittenberg proved the exactness of e for rationally connected fibrations whose smooth fibres verify the exactness of e and whose base is either the projective space or a curve verifying the exactness of e in the present paper we prove the exactness of e for fibrations whose bases are ch telet surfaces or smooth proper models of homogeneous spaces of connected linear algebraic groups with connected stabilizers we require that either the fibration is valuatively split in codimension 1 and most closed fibres satisfy weak approximation for 0 cycles or the generic fibre has a 0 cycle of degree 1 and e is exact for most fibres

zero cycles sur les fibrations au dessus d une courbe de genre quelconque resume soit x une variete propre et lisse sur un corps de nombres k des conjectures sur l image du groupe de chow des zero cycles de x dans le produit des memes groupes sur tous les completes de k ont ete proposees par colliot thelene kato et saito nous demontrons ces conjectures pour l espace total de fibrations en varietes rationnellement connexes verifiant l approximation faible au dessus de courbes dont le groupe de tate shafarevich est fini sous une hypothese d abelianite sur les fibres singulieres

on the fibration method for zero cycles and rational points conjectures on the existence of zero cycles on arbitrary smooth projective varieties over number elds were proposed by colliot th el ene sansuc kato and saito in the 1980 s we prove that these conjectures are compatible with brations for brations into rationally connected varieties over a curve in particular they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers we prove the analogous result for rational points conditionally on a conjecture on locally split values of polynomials which a recent work of matthiesen establishes in the case of linear polynomials over the rationals

stabilization for mapping class groups of 3 manifolds we prove that the homology of the mapping class group of any 3manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3 manifold when both manifolds are compact and orientable the stabilization also holds for the quotient group by twists along spheres and disks and includes as particular cases homological stability for symmetric automorphisms of free groups automorphisms of certain free products and handlebody mapping class groups our methods also apply to manifolds of other dimensions in the case of stabilization by punctures the main result of this paper is a homological stability theorem for mapping class groups of 3 manifolds where the stabilization is by connected sum with an arbitrary 3 manifold more precisely we show that given any two compact connected oriented 3 manifolds n and p with n 6 the homology group hi 0diff n p p rel n z is independent of the number n of copies of p in the connected sum as long as n 2i 2 i e each homology group stabilizes with p we also prove an analogous result for boundary connected sum and a version for the quotient group of the mapping class group by twists along spheres and disks a group closely related to the automorphism group of the fundamental group of the manifold homological stability theorems were first found in the sixties for symmetric groups by nakaoka 36 and linear groups by quillen and now form the foundation of modern algebraic k theory see for example 28 part iv and 42 stability theorems for mapping class groups of surfaces were obtained in the eighties by harer and ivanov 14 25 and recently turned out to be a key ingredient to a solution of the mumford conjecture about the homology of the riemann moduli space 30 the other main examples of families of groups for which stability has been known are braid groups 1 and automorphism groups of free groups 17 18 the present paper extends significantly the class of groups for which homological stability is known to hold it suggests that it is a widespread phenomenon among families of groups containing enough symmetries in addition to the already mentioned stability theorems for mapping class groups of 3 manifolds corollaries of our main result include stability for handlebody subgroups of surface mapping class groups symmetric automorphism groups of free groups and automorphism groups of free products ng for many groups g using similar techniques we obtain stability results also for mapping class groups 0diff m n points rel m for m any m dimensional manifold with boundary m 2 even the case m 2 is new here as well as for the fundamental group 1conf m n of the configuration space of n unordered points in m our paper thus also unifies previous known results as we recover stability for braid groups as 1conf d n symmetric groups as 1conf d n and automorphism groups of free groups as aut nz

simultaneous integer values of pairs of quadratic forms we prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent almost all pairs of integers that satisfy the necessary local conditions provided that the forms satisfy a suitable nonsingularity condition in particular such forms simultaneously attain prime values if the obvious local conditions hold the proof uses the circle method and in particular pioneers a two dimensional version of a kloosterman refinement

on the hilbert property and the fundamental group of algebraic varieties in this paper we link the so called hilbert property hp for an algebraic variety over a number field with its fundamental group in a perspective which appears new the notion of hp comes from hilbert s irreducibility theorem and has important implications for instance towards the inverse galois problem we shall observe that the hp is in a sense opposite to the chevalley weil theorem this shall immediately entail the result that the hp can possibly hold only for simply connected varieties in the appropriate sense in turn this leads to new counterexamples to the hp involving enriques surfaces in this view we shall also formulate an alternative related property possibly holding in full generality and some conjectures which unify part of what is known in the topic these predict that for a variety with a zariski dense set of rational points the validity of the hp which is defined arithmetically is indeed of purely topological nature also a consequence of these conjectures would be a positive solution of the inverse galois problem in the paper we shall also prove the hp for a k3 surface related to the above enriques surface providing what appears to be the first example of a variety non rational over mathbb c for which the hp can be proved in an appendix we shall further discuss among other things the hp also for kummer surfaces all of this basically exhausts the study of the hp for surfaces with a zariski dense set of rational points

arithmetic of del pezzo surfaces of degree 4 and vertical brauer groups we show that brauer classes of a locally solvable degree 4 del pezzo surface x are vertical for some projection away from a plane g x p 1 i e that every brauer class is obtained by pullback from an element of br k p 1 as a consequence we prove that a brauer class obstructs the existence of a k rational point if and only if all k fibers of g fail to be locally solvable or in other words if and only if x is covered by curves that each have no adelic points using work of wittenberg we deduce that for certain quartic del pezzo surfaces with nontrivial brauer group the algebraic brauer manin obstruction is sufficient to explain all failures of the hasse principle conditional on schinzel s hypothesis and the finiteness of tate shafarevich groups the proof of the main theorem is constructive and gives a simple and practical algorithm distinct from that in 5 for computing all classes in the brauer group of x modulo constant algebras c 2014 elsevier inc all rights reserved

universal torsors and values of quadratic polynomials represented by norms let k k be an extension of number fields and let p t be a quadratic polynomial over k let x be the affine variety defined by p t n k k mathbf z we study the hasse principle and weak approximation for x in three cases for k k 4 and p t irreducible over k and split in k we prove the hasse principle and weak approximation for k mathbb q with arbitrary k we show that the brauer manin obstruction to the hasse principle and weak approximation is the only one for k k 4 and p t irreducible over k we determine the brauer group of smooth proper models of x in a case where it is non trivial we exhibit a counterexample to weak approximation

cantor boundary behavior of analytic functions let mathbb d be the open unit disc and let partial mathbb d be the boundary of mathbb d for f z analytic in mathbb d and continuous on overline mathbb d it follows from the open mapping theorem that partial f mathbb d subset f partial mathbb d these two sets have very rich and intrigue geometric properties when f z is univalent then they are equal and there is a large literature to study their boundary behaviors our interest is on the class of analytic functions f z for which the image curves f partial mathbb d form infinitely many loops everywhere they are not univalent of course we formulate this as the cantor boundary behavior we give sufficient conditions for such property making use of the distribution of the zeros of f and the mean growth rate of f examples includes the complex weierstrass functions and the cauchy transform of the canonical hausdorff measure on the sierpiski gasket

rational homotopy of the space of sections of a nilpotent bundle we show that an algebraic construction proposed by sullivan is indeed a model for the rational homotopy type of the space of sections of a nilpotent bundle in his paper uhomologie des espaces fonctionnels r thom studied the homotopy type of the space f of continuous maps of x into f homotopic to a given map starting from a postnikov decomposition of f he built the functional space f step by step he also indicated how one could construct a differential graded algebra describing the rational homotopy type of f later on sullivan gave an algebraic model which mirrors this construction in terms of a dg algebra representing x and the minimal model of f the aim of this paper is to show following the method of thom that the model of sullivan is indeed a model for the functional space under suitable restrictions as in 3 we consider the slightly more general problem of the determination of the rational homotopy type of the space of sections ts of a nilpotent fiber space p y x homotopic to a given section s in 1 we explain thorn s geometric construction in 2 we describe an algebraic model for an abelian galois covering of a nilpotent space in 3 we show how the model of sullivan fits with the geometry i thank the referee for many improvements of this paper 1 a postnikov factorization of the space of sections 1 1 let g be a finitely generated abehan group and let x be a path connected space whose cohomology groups hk x g are finitely generated for each k to avoid difficulties with the topologies cf 4 we can work in the category of simplicial sets proposition thom 4 the space k g m x of continuous maps of x in the eilenberg mac lane complex k g m is homotopically equivalent to the product nr ra fthe eilenberg mac lane spaces ki k hm x g i more precisely let x g hm k g m g be the fundamental class of k g m if e k g m x xx k g m

weighted integrability of polyharmonic functions abstract to address the uniqueness issues associated with the dirichlet problem for the n harmonic equation on the unit disk d in the plane we investigate the l p integrability of n harmonic functions with respect to the standard weights 1 z 2 the question at hand is the following if u solves n u 0 in d where stands for the laplacian and d u z p 1 z 2 d a z must then u z 0 here n is a positive integer is real and 0 p d a is the usual area element the answer will generally speaking depend on the triple n p the most interesting case is 0 p 1 for a given n we find an explicit critical curve p n p a piecewise affine function such that for n p there exist nontrivial functions u with n u 0 of the given integrability while for n p only u z 0 is possible we also investigate the obstruction to uniqueness for the dirichlet problem that is we study the structure of the functions in ph n p d when this space is nontrivial we find a new structural decomposition of the polyharmonic functions the cellular decomposition which decomposes the polyharmonic weighted l p space in a canonical fashion corresponding to the cellular expansion is a tiling of part of the p plane into cells the above uniqueness for the dirichlet problem may be considered for any elliptic operator of order 2n however the above mentioned critical integrability curve will depend rather strongly on the given elliptic operator even in the constant coefficient case for n 1

rationally connected varieties over local fields let x be a proper variety defined over a field k following ma we say that two points x x x k are r equivalent if they can be connected by a chain of rational curves defined over k cf 4 1 in essence two points are r equivalent if they are obviously rationally equivalent several authors have proved finiteness results over local and global fields cubic hypersurfaces ma sd linear algebraic groups ct sa vo1 gi vo2 rational surfaces ct co ct1 ct sk1 quadric bundles and intersections of two quadrics ct sa sd pa su r equivalence is only interesting if there are plenty of rational curves on x at least over k such varieties have been studied in the series of papers komi mo1 ko mi mo3 see also ko1 there are many a priori different ways of defining what plenty of rational curves should mean fortunately many of these turn out to be equivalent and this leads to the notion of rationally connected varieties see ko mi mo2 ko1 iv 3 ko2 4 1 2

bloch functions and asymptotic tail variance let p denote the bergman projection on the unit disk dd p z d w 1 zw 2da w z d where dada is normalized area measure we prove that if z 1 z 1 on dd then the integral i a r 02 exp ar4 p rei 2log 11 r2 d 2 0 r 1 has the bound i a r c a 10 1 a 3 2i a r c a 10 1 a 3 2 for 0 1a 1 no such uniform bound is possible we interpret the theorem in terms the asymptotic tail variance of such a bergman projection p p by the way the asymptotic tail variance induces a seminorm on the bloch space this improves upon earlier work of makarov which covers the range 0 a 264 0 1542 we then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k quasiconformal extension for 0 k 10 k 1 the estimate reads for t ct c and 0 k 10 k 1 b k t 14k2 t 2 1 7k 2 for t 2k 1 7k 2 k t 1 1 7k 2 for t 2k 1 7k 2 which should be compared with the conjecture by prause and smirnov to the effect that for real t with t 2 k t 2 k we should have b k t 14k2t2

bloch functions asymptotic variance and geometric zero packing motivated by a problem in quasiconformal mapping we introduce a problem in complex analysis with its roots in the mathematical physics of the bose einstein condensates in superconductivity the p

on the arithmetic of del pezzo surfaces of degree 2 we study the arithmetic of certain del pezzo surfaces of degree 2 we produce examples of brauer manin obstruction to the hasse principle coming from 2 and 4 torsion elements in the brauer group

the brauer group and the second abel jacobi map for 0 cycles on algebraic varieties this paper focuses on the connection between the brauer group and the 0 cycles of an algebraic variety we give an alternative construction of the second l adic abel jacobi map for such cycles linked to the algebraic geometry of severi brauer varieties on x this allows us then to relate this abel jacobi map to the standard pairing between 0 cycles and brauer groups see m l completing results from m in this direction second for surfaces it allows us to present this map according to the more geometrical approach devised by m green in the framework of arithmetic mixed hodge structures see g needless to say this paper owes much to the work of u jannsen and especially to his recently published older letter j4 to b gross

zero cycles on rational surfaces over number fields let x be a smooth projective geometrically integral surface defined over a perfect field k let kbe an algebraic closure of k and f be the function field of x k x then x is said to be rational if f is a purely transcendental extension of k the tame symbols define a map from ke f to the sum of function fields h k 7 of all irreducible divisors 7 on x let m be the cokernel of this map if k is a number field and kv the completion at a place v of k let kv be an algebraic closure of kv and m the corresponding cokernel for k x x further let i i i l k m be the kernel of the natural map from hl gal k k m to i h1 gal k v kv my in 5 w colliot th616ne and sansuc conjectured that

rational homotopy of spaces of maps into spheres and complex projective spaces we investigate the rational homotopy classification problem for the components of some function spaces with sn or cpn as target space

rationally connected varieties over finite fields let x be a geometrically rational or more generally separably rationally connected variety over a finite field k we prove that if k is large enough then x contains many rational curves defined over k as a consequence we prove that r equivalence is trivial on x if k is large enough these results imply the following conjecture of j l colliot thelene let y be a rationally connected variety over a number field f for a prime p let y p denote the corresponding variety over the local field f p then for almost all primes p the chow group of 0 cycles on y p is trivial and r equivalence is trivial on y p

towards the brauer manin obstruction on varieties fibred over the projective line abstract recently dasheng wei proved that the brauer manin obstruction is the only obstruction to the hasse principle for 0 cycles of degree 1 on some fibrations over the projective line defined by bi cyclic normic equations in the present paper we prove the exactness of the global to local sequence for chow groups of 0 cycles of such varieties this signifies that the brauer manin obstruction is also the only obstruction to weak approximation for 0 cycles of arbitrary degree our main theorem also generalizes several existing results

on manin s conjecture for a family of châtelet surfaces the manin conjecture is established for ch telet surfaces over q arising as minimal proper smooth models of the surface y 2 z 2 f x where f is a totally reducible polynomial of degree 3 without repeated roots these surfaces do not satisfy weak approximation

nevanlinna domains with large boundaries abstract nevanlinna domains are an important class of bounded simply connected domains in the complex plane they are images of the unit disc under mappings by univalent functions belonging to model spaces i e the subspaces of the hardy space h 2 invariant with respect to the backward shift operator nevanlinna domains play a crucial role in recent progress in problems of uniform approximation of functions on compact sets in c by polyanalytic polynomials and polyanalytic rational functions we give a complete solution to the following problem posed in the early 2000s how large in the sense of dimension can be the boundaries of nevanlinna domains we establish the existence of nevanlinna domains with large boundaries in particular these domains can have boundaries of positive planar measure the sets of accessible points can be of any hausdorff dimension between 1 and 2 as a quantitative counterpart of these results we construct rational functions univalent in the unit disc with extremely long boundaries for a given amount of poles

descent on fibrations over the projective line we formulate a general set up for the descent method of j l colliot thelene and j j sansuc applied to varieties fibred over the projective line over a number field this makes it possible to prove that the manin obstruction to the hasse principle and weak approximation is the only one provided there are only few degenerate fibres usually two three in some cases and that sufficiently many smooth fc fibres satisfy the hasse principle and weak approximation we introduce a new concept of what should be called a degenerate fibre the so called split fibres the property which depends only on the generic fibre and not on the choice of a particular model this yields a simpler and more general approach to the previous results of that kind 0 introduction motivated by hasse s theorem on quadratic forms one says that the hasse principle holds for a family of geometrically integral smooth and proper varieties over a number field k9 if for every member of this family the existence of rational points over all the completions of k implies the existence of a rational point the fact that the hasse principle holds for some family signif icantly simplifies the study of its arithmetic because the problem of determining whether or not there is a rational point is then reduced to a finite number of

contractibility of the space of rational maps we define an algebro geometric model for the space of rational maps from a smooth curve x to an algebraic group g and show that this space is homologically contractible as a consequence we deduce that the moduli space operatorname bun g of g bundles on x is uniformized by the appropriate rational version of the affine grassmannian where the uniformizing map has contractible fibers