rational points on k3 surfaces a new canonical height a fundamental tenet of diophantine geometry is that the geometric properties of an algebraic variety should determine its basic arithmetic properties this is certainly true for curves where the sign of the euler characteristic of c determines whether the set of rational points on c is finite x c 0 for higher dimensional varieties there are some precise conjectures due to bombieri lang and vojta 15 which predict when the rational points on a variety should be finite or degenerate i e not zariski dense and some conjectures of manin et al 2 5 on the distribution of rational points in those cases when they are zariski dense but except for abelian varieties their subvarieties and some fano varieties varieties for which the anticanonical bundle is ample there are very few general theorems in this paper we will study the rational points on a certain class of k 3 surfaces defined over a number field k the moduli space of marked algebraic k3 surfaces is a countable union of 19 dimensional quasi projective varieties we are going to look at the 18 dimensional family studied by wehler 1 17 wehler s family consists of k3 surfaces s whose automorphism group aut s contains a subgroup d isomorphic to the free product 7 2 2 of two cyclic groups of order 2 we will use the geometric information provided by this infinite automorphism group to study the k rat ional points on s for any point pes we can look at the orbit of p under the action of d

eisenstein series on loop groups maass selberg relations 2 we prove that truncated pseudo eisenstein series have compact support on the arithmetic quotient

an asymptotic estimate for heights of algebraic subspaces we count the number of subspaces of affine space with a given dimension defined over an algebraic number field with height less than or equal to b we give an explicit asymptotic estimate for the number of such subspaces as b goes to infinity where the constants involved depend on the classical invariants of the number field degree discriminant class number etc the problem is reformulated as an estimate for the number of lattice points in a certain bounded domains

exploiting negative curvature in deterministic and stochastic optimization this paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature although prior work has established convergence results for algorithms that integrate both descent and negative curvature steps there has not yet been extensive numerical evidence showing that such methods offer consistent performance improvements in this paper we present new frameworks for combining descent and negative curvature directions alternating two step approaches and dynamic step approaches the aspect that distinguishes our approaches from ones previously proposed is that they make algorithmic decisions based on estimated upper bounding models of the objective function a consequence of this aspect is that our frameworks can in theory employ fixed stepsizes which makes the methods readily translatable from deterministic to stochastic settings for deterministic problems we show that instances of our dynamic framework yield gains in performance compared to related methods that only follow descent steps we also show that gains can be made in a stochastic setting in cases when a standard stochastic gradient type method might make slow progress

homogeneous asymptotic limits of haar measures of semisimple linear groups and their lattices we show that haar measures of connected semisimple groups embedded via a representation into a matrix space have a homogeneous asymptotic limit when viewed from far away and appropriately rescaled this is still true if the haar measure of the semisimple group is replaced by the haar measure of a irreducible lattice of the group and the asymptotic measure is the same in the case of an almost simple group of rank greater than 2 a remainder term is also obtained this extends and makes precise anterior results of duke rudnick and sarnak and eskin mcmullen in the case of a group variety

varieties with too many rational points we investigate fano varieties defined over a number field that contain subvarieties whose number of rational points of bounded height is comparable to the total number on the variety

balanced line bundles on fano varieties a conjecture of batyrev and manin relates arithmetic properties of varieties with ample anticanonical class to geometric invariants in particular counting functions defined by metrized ample line bundles and the corresponding asymptotics of rational points of bounded height are interpreted in terms of cones of effective divisors and certain thresholds with respect to these cones this framework leads to the notion of balanced line bundles whose counting functions conjecturally capture generic distributions of rational points we investigate balanced line bundles in the context of the minimal model program with special regard to the classification of fano threefolds

height uniformity for integral points on elliptic curves we recall the result of d abramovich and its generalization by p pacelli on the uniformity for stably integral points on elliptic curves it says that the lang vojta conjecture on the distribution of integral points on a variety of logarithmic general type implies the uniformity for the numbers of stably integral points on elliptic curves in this paper we will investigate its analogue for their heights under the assumption of the vojta conjecture basically we will show that the vojta conjecture gives a naturally expected simple uniformity for their heights

cox rings of cubic surfaces and fano threefolds abstract we determine the cox rings of the minimal resolutions of cubic surfaces with at most rational double points of blow ups of the projective plane at non general configurations of six points and of three dimensional smooth fano varieties of picard numbers one and two

inhomogeneous cubic congruences and rational points on del pezzo surfaces for given non zero integers a b q we investigate the density of solutions x y 2 z 2 to the binary cubic congruence ax 2 by 3 0 mod q and use it to establish the manin conjecture for a singular del pezzo surface of degree 2 dened over q

leclate du plan projectif en quatre points dont deux conjugues we state manin s conjecture in the particular case of the blow up of the projective plan in four points where two are conjugated we show that for an open set u v nu q b card p 2 u q h p 6 b cb log b b 1 the constant c is the one conjectured by peyre

nombre de points de hauteur bornee sur les surfaces de del pezzo de degre 5 we state manin s conjecture in the particular case of the split del pezzo s surfaces of degree 5 over q we show that for an open set u v nu q b card p u q h p b c b log b 4 b the constant c is the one conjectured by e peyre sommaire

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

on the distribution of points of bounded height on equivariant compactifications of vector groups we prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space

a stabilized sqp method superlinear convergence stabilized sequential quadratic programming ssqp methods for nonlinear optimization generate a sequence of iterates with fast local convergence regardless of whether or not the active constraint gradients are linearly dependent this paper concerns the local convergence analysis of an ssqp method that uses a line search with a primal dual augmented lagrangian merit function to enforce global convergence the method is provably well defined and is based on solving a strictly convex quadratic programming subproblem at each iteration it is shown that the method has superlinear local convergence under assumptions that are no stronger than those required by conventional stabilized sqp methods the fast local convergence is obtained by allowing a small relaxation of the optimality conditions for the quadratic programming subproblem in the neighborhood of a solution in the limit the line search selects the unit step length which implies that the method does not suffer from the maratos effect the analysis indicates that the method has the same strong first and second order global convergence properties that have been established for augmented lagrangian methods yet is able to transition seamlessly to ssqp with fast local convergence in the neighborhood of a solution numerical results on some degenerate problems are reported

universal torsors of del pezzo surfaces and homogeneous spaces let cox sr be the homogeneous coordinate ring of the blow up sr of in r general points i e a smooth del pezzo surface of degree 9 r we prove that for r 6 7 proj cox sr can be embedded into gr pr where gr is an algebraic group with root system given by the primitive picard lattice of sr and pr gr is a certain maximal parabolic subgroup

rational points on diagonal quartic surfaces we searched up to height 107 for rational points on diagonal quartic surfaces the computations fill several gaps in earlier lists computed by pinch swinnerton dyer and bright

covers and envelopes in grothendieck categories flat covers of complexes with applications abstract in the general setting of grothendieck categories with enough projectives we prove theorems that make possible to restrict the study of the problem of the existence of f covers and envelopes to the study of some properties of the class f we then prove the existence of flat covers and cotorsion envelopes of complexes giving some examples this generalizes the earlier work j algebra201 1998 86 102 and finishes the problem of the existence of flat covers of complexes

counting algebraic numbers with large height ii let denote the field of rational numbers open image in new window an algebraic closure of and h open image in new window the absolute multiplicative weil height for each positive integer d and real number mathcal h geqslant 1 it is well known that the number open image in new window of points in open image in new window having degree d over and satisfying h left alpha right leqslant mathcal h is finite this is the one dimensional case of northcott s theorem 8 see also 5 page 59 the systematic study of the counting function open image in new window and that of related functions in higher dimensions was begun by schmidt 10 it is relatively easy to prove the existence of a positive constant c c d such that open image in new window 1 and also the existence of positive constants c c d and mathcal h 0 mathcal h 0 left d right such that open image in new window 2

computing modified newton directions using a partial cholesky factorization the effectiveness of newton s method for finding an unconstrained minimizer of a strictly convex twice continuously differentiable function has prompted the proposal of various modified newton methods for the nonconvex case linesearch modified newton methods utilize a linear combination of a descent direction and a direction of negative curvature if these directions are sufficient in a certain sense and a suitable linesearch is used the resulting method will generate limit points that satisfy the second order necessary conditions for optimality the authors propose an efficient method for computing a descent direction and a direction of negative curvature that is based on a partial cholesky factorization of the hessian this factorization not only gives theoretically satisfactory directions but also requires only a partial pivoting strategy i e the equivalent of only two rows of the schur complement needs to be examined at each step

on a certain senary cubic form a strong form of the manin peyre conjecture with a power saving error term is proved for a certain cubic fourfold

integral points of bounded height on partial equivariant compactifications of vector groups we establish asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups

density of rational points on a quadric bundle in mathbb p 3 times mathbb p 3 we establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain zariski dense subset of the biprojective hypersurface x1y12 x4y42 0 in p3 p3 this confirms the modified manin conjecture for this variety in which the removal of a thin set of rational points is allowed

homological dimensions and regular rings abstract a question of avramov and foxby concerning injective dimension of complexes is settled in the affirmative for the class of noetherian rings a key step in the proof is to recast the problem on hand into one about the homotopy category of complexes of injective modules analogous results for flat dimension and projective dimension are also established

flat covers in abelian and in non abelian categories abstract a general existence theorem for flat covers in e g quasi abelian locally finitely presented categories is obtained from an additive ramsey type theorem in the abelian case it is shown that flat covers always exist applications to categories of separated presheaves or sheaves localizations of bousfield type torsion free classes of finite type and categories of filtered objects or complexes are given

cohomology theories based on flats abstract let a be an associative ring with identity k flat a the homotopy category of flat modules and k p flat a the full subcategory of pure complexes the quotient category k flat a k p flat a called here the pure derived category of flats was introduced by neeman in this category flat resolutions are unique up to homotopy and so can be used to compute cohomology we develop theories of tate and complete cohomology in the pure derived category of flats these theories extend naturally to sheaves over semi separated noetherian schemes where there are not always enough projectives but we do have enough flats as applications we characterize rings with finite sfli and schemes which are locally gorenstein

rational points of bounded height on general conic bundle surfaces a conjecture of manin predicts the asymptotic distribution of rational points of bounded height on fano varieties in this paper we use conic bundles to obtain correct lower bounds for a wide class of surfaces over number fields for which the conjecture is still far from being proved for example we obtain the conjectured lower bound of manin s conjecture for any del pezzo surface whose picard rank is sufficiently large or for arbitrary del pezzo surfaces after possibly an extension of the ground field of small degree

a solver for nonconvex bound constrained quadratic optimization we present a new algorithm for nonconvex bound constrained quadratic optimization in the strictly convex case our method is equivalent to the state of the art algorithm by dostal and schoberl comput optim appl 30 2005 pp 23 43 unlike their method however we establish a convergence theory for our algorithm that holds even when the problems are nonconvex this is achieved by carefully addressing the challenges associated with directions of negative curvature in particular those that may naturally arise when applying the conjugate gradient algorithm to an indefinite system of equations our presentation and analysis deal explicitly with both lower and upper bounds on the optimization variables whereas the work by dostal and schoberl considers only strictly convex problems with lower bounds to handle this generality we introduce the reduced chopped gradient that is analogous to the reduced free gradient previously used the reduced chopped gradient leads to a new condition that is used to

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

gorenstein injective complexes of modules over noetherian rings abstract a complex c is called gorenstein injective if there exists an exact sequence of complexes i 1 i 0 i 1 such that each i i is injective c ker i 0 i 1 and the sequence remains exact when hom e is applied to it for any injective complex e we show that over a left noetherian ring r a complex c of left r modules is gorenstein injective if and only if c m is gorenstein injective in r mod for all m z also gorenstein injective dimensions of complexes are considered

tamagawa numbers of diagonal cubic surfaces numerical evidence a refined version of manin sconjecture aboutthe asymptotics of points of bounded height on fano varieties has been developped by batyrev and the authors we test numerically this refined conjecture for some diagonal cubic surfaces

counting rational points on biquadratic hypersurfaces abstract an asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables the proof uses the hardy littlewood circle method

translate of horospheres and counting problems let g be a semisimple lie group without compact factors gamma be an irreducible lattice in g in the first part of the article we give the necessary and sufficient condition under which a sequence of translates of probability horospherical measures is convergent and the limiting measure is also determined when it is convergent see theorems 1 and 2 for the precise statements in the second part two applications are presented the first one is of geometric nature and the second one gives an alternative way to count the number of rational points on a flag variety

the relationship between homological properties and representation theoretic realization of artin algebras we will study the relationship of quite different object in the theory of artin algebras namely auslander regular rings of global dimension two torsion theories categories and almost abelian categories we will apply our results to characterization problems of auslander reiten quivers 0 1 there exists a bijection between equivalence classes of krull schmidt categories c with additive generators m and morita equivalence classes of semiperfect rings which is given by c 7 c m m and the converse is given by 7 pr for the category pr of finitely generated projective modules although th is bijection itself is rather formal it will be very fruitful to study the relationship of a d below the object of this paper is to study it under the assumption that is an arti n algebra

on pure acyclic complexes abstract in this paper we study the pure acyclic complexes of modules we obtain several characterizations of these complexes extending results that are known for the pure acyclic complexes of flat modules in particular we extend neeman s characterization 18 of the pure acyclic complexes of flat modules in terms of complexes of projective modules by considering more generally complexes of pure projective modules as a consequence we obtain simson s result 22 on the pure projectivity of pure periodic modules

sieving rational points on varieties an upper bound sieve for rational points on suitable varieties isdeveloped together with applications tocounting rational points in thin sets to local solubility in families and to the notion of friable rational pointswith respect to divisors in the special case of quadrics sharper estimates areobtained by developing a version of the selberg sieve for rational points

hardy littlewood varieties and semisimple groups we are interested in counting integer and rational points in affine algebraic varieties also under congruence conditions we introduce the notions of a strongly hardy littlewood variety and a relatively hardy littlewood variety in terms of counting rational points satisfying congruence conditions the definition of a strongly hardy littlewood variety is given in such a way that varieties for which the hardy littlewood circle method is applicable are strongly hardy littlewood

hauteurs de sous espaces sur les corps non commutatifs we study heights of subspaces of d n where d is a finite dimensional rational division algebra and n a positive integer we define them in terms of volumes of euclidean lattices by extending a formula of w schmidt so that we recover the classical height if d is commutative we review basic properties prove a siegel lemma over d a duality theorem and a new formula for the degree of certain abelian varieties we further give matrix versions and compare our notion with the height defined through algebraic groups by j franke y manin and y tschinkel

on the relation between k flatness and k projectivity abstract krause 19 has proved that the homotopy category k r pureproj of pure projective modules over an associative ring r is compactly generated and equivalent to the verdier quotient k r t r where t r is the triangulated subcategory of the homotopy category k r consisting of the pure acyclic complexes we provide an alternative proof of this result when restricted to the homotopy category of projective modules this equivalence reduces to that obtained by neeman 26 as an application of this description of k r pureproj we show that the k flat complexes defined by spaltenstein 29 are up to homotopy equivalence the filtered colimits of k projective complexes of pure projective modules

quadratic families of elliptic curves and unirationality of degree 1 conic bundles we study families of elliptic curves whose coefficients are degree 2 polynomials in a variable t all such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers we prove that such conic bundles are unirational as a consequence one obtains that for infinitely many values of t the resulting elliptic curve has rank at least 1

some adjoints in homotopy categories let r be a ring in a previous paper 11 we found a new description for the category k r proj it is equivalent to the verdier quotient k r flat s for some suitable s c k r flat in this article we show that the quotient map from k r flat to k r flat s always has a right adjoint this gives a new fully faithful embedding of k r proj into k r flat its virtue is that it generalizes to nonaffine schemes

flat covers of complexes abstract in this article we define and study flat complexes over any ring also we prove that any complex over a commutative noetherian ring with finite krull dimension has a flat cover and a dg flat cover

integral points on symmetric varieties and satake compatifications let v be an affine symmetric variety defined over q we compute the asymptotic distribution of the angular components of the integral points in v this distribution is described by a family of invariant measures concentrated on the satake boundary of v in the course of the proof we describe the structure of the satake compactifications for general affine symmetric varieties and compute the asymptotic of the volumes of norm balls

minimal semi flat cotorsion replacements and cosupport abstract over a commutative noetherian ring r of finite krull dimension we show that every complex of flat cotorsion r modules decomposes as a direct sum of a minimal complex and a contractible complex moreover we define the notion of a semi flat cotorsion complex as a special type of semi flat complex and provide functorial ways to construct a quasi isomorphism from a semi flat complex to a semi flat cotorsion complex consequently every r complex can be replaced by a minimal semi flat cotorsion complex in the derived category over r furthermore we describe structure of semi flat cotorsion replacements by which we recover classic theorems for finitistic dimensions in addition we improve some results on cosupport and give a cautionary example we also explain that semi flat cotorsion replacements always exist and can be used to describe the derived category over any associative ring

auslander type conditions and cotorsion pairs we study the properties of rings satisfying auslander type conditions if an artin algebra satisfies the auslander condition that is is an gorenstein artin algebra then we construct two kinds of subcategories which form functorially finite cotorsion pairs

gorenstein flat covers of modules over gorenstein rings abstract a left and right noetherian ring r is called gorenstein if both r r and r r have finite injective dimensions these rings were studied by bass for the commutative case and iwanaga for the noncommutative case in this paper we define gorenstein flat modules over a gorenstein ring these modules are a generalization of flat modules after discussing the properties of modules over a gorenstein ring we prove that every module over a gorenstein ring has a gorenstein flat cover or a minimal right approximation by a gorenstein flat module in auslander and reiten s terminology

rational points on quartic hypersurfaces let x be a projective non singular quartic hypersurface of dimension 39 or more which is defined over we show that x is non empty provided that x is non empty and x has p adic points for every prime p 2009 walter de gruyter

triangulated equivalence between a homotopy category and a triangulated quotient category abstract given two complete hereditary cotorsion pairs q r and q r in a bicomplete abelian category g such that q q and q r q r becker showed that there exists a hereditary abelian model structure m q w r on g where w is a thick subcategory of g we prove that the homotopy category ho m of m is triangulated equivalent to the triangulated quotient category d b g q r k b q r where d b g q r is the subcategory of d b g consisting of all homology bounded complexes with both finite q dimension and r dimension and k b q r is the bounded homotopy category of q r core objects applications are given in the category of modules it is shown that the homotopy category of the gorenstein flat resp ding projective and gorenstein ac projective model structure on the category of modules established by gillespie and his coauthors can be realized as a certain triangulated quotient category

manin s conjecture for certain spherical threefolds abstract we prove manin s conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for a spherical threefold with canonical singularities and two infinite families of spherical threefolds with log terminal singularities moreover we show that one of these families does not satisfy a conjecture of batyrev and tschinkel on the leading constant in the asymptotic formula our proofs are based on the universal torsor method using brion s description of cox rings of spherical varieties

rational points on compactifications of semi simple groups we prove manin s conjecture concerning the distribution of rational points of bounded height and its refinement by peyre for wonderful compactifications of semi simple algebraic groups over number fields the proof proceeds via the study of the associated height zeta function and its spectral expansion

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

the homotopy category of flat modules and grothendieck duality let r be a ring we prove that the homotopy category k r proj is always aleph 1 compactly generated and depending on the ring r it may or may not be compactly generated we use this to give a description of k r proj as a quotient of k r flat the remarkable fact is that this new description of k r proj generalizes to non affine schemes this will appear in murfet s thesis

the density of rational points on non singular hypersurfaces ii for any integers d n geq 2 let x subset mathbb p n be a non singular hypersurface of degree d that is defined over the rational numbers the main result in this paper is a proof that the number of rational points on x which have height at most b is o b n 1 varepsilon for any varepsilon 0 the implied constant in this estimate depends at most upon d varepsilon and n

height zeta functions of equivariant compactifications of unipotent groups we prove manin s conjecture for bi equivariant compactifications of unipotent groups 2015 wiley periodicals inc

counting rational points on quartic del pezzo surfaces with a rational conic upper and lower bounds of the expected order of magnitude are obtained for the number of rational points of bounded height on any quartic del pezzo surface over mathbb q that contains a conic defined over mathbb q

covers and envelopes over gr gorenstein rings abstract in this paper we study the existence of gorenstein injective envelopes and gorenstein projective and flat covers in the category of graded modules and we relate them with the corresponding envelopes and covers in the category of modules

eisenstein series on loop groups maass selberg relations 3 in this paper we prove the maass selberg relations for the inner product of truncated pseudo eisenstein series one can then extract the maass selberg relations for truncated eisenstein series themselves from those for truncated pseudo eisenstein series a task which we shall complete in a subsequent paper