cubic correspondences arising from g 2 we use the automorphic realization of the minimal representation the theta representation of the three fold cover of g 2 to construct a correspondence of automorphic forms between members of dual pairs inside g 2 the dual pairs under consideration are sl2 sl2 and sl3 z 3 in the first case we obtain the shimura correspondence between the three fold cover of sl2 and sl2 through explicit integral formulae in the second case we decompose theta over the three fold cover of sl3 both locally and globally the constituents are parametrized by their central characters and are cuspidal except essentially when the central character is trivial we thus obtain cuspidal theta representations and the unramified theta representation of the three fold cover of sl3

spherical characters on p adic symmetric spaces symmetric spaces over local fields and the harmonic analysis of their class one representations arise as the local calculation of jacquet s theory of the relative trace formula there is an extensive literature at the real place but few general results for p adic fields are known the objective here is to carry over to symmetric spaces as much as possible of queens and the prerequisite results of howe and to provide counterexamples for those things which do not generalize we derive the weyl integration formula local constancy of the spherical character on the regular set howe s conjecture for groups the germ expansion for spherical characters at the origin and a spherical version of howe s kirillov theory for compact p adic groups we find that the density property of regular orbital integrals fails some of the basic ideas are nascent in hakim s thesis written under herve jacquet

secondary terms in counting functions for cubic fields we prove the existence of secondary terms of order x5 6 in the davenport heilbronn theorems on cubic fields and 3 torsion in class groups of quadratic fields for cubic fields this confirms a conjecture of datskovsky wright and roberts we also prove a variety of generalizations including to arithmetic progressions where we discover a curious bias in the secondary term roberts s conjecture has also been proved independently by bhargava shankar and tsimerman in contrast to their work our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms developed by shintani and datskovsky wright

error estimates for the davenport heilbronn theorems we obtain the first known power saving remainder terms for the theorems of davenport and heilbronn on the density of discriminants of cubic fields and the mean number of 3 torsion elements in the class groups of quadratic fields in addition we prove analogous error terms for the density of discriminants of quartic fields and the mean number of 2 torsion elements in the class groups of cubic fields these results prove analytic continuation of the related dirichlet series to the left of the line r s 1

singular invariant tempered distributions on regular prehomogeneous vector spaces revue des espaces vectoriels prehomogenes distributions temperees invariantes distributions temperees avec un parametre meromorphe puissances complexes des invariants relatifs soit p x un polynome homogene irreductible a coefficients reels sur un espace vectoriel v soit s x v p x 0 se decompose en un nombre fini de g 1 orbites avec g 1 g gl v p g x p x on pose v 1 uv 2 u uv e v s la decomposition en composantes connexes on donne une condition necessaire et suffisante pour qu une distribution temperee g 1 invariante a support dans s puisse s obtenir comme une combinaison lineaire finie de coefficients de laurent negatifs de la distribution p x s vd i

a conjecture on coincidence among the zeta functions associated with the space of binary cubic forms about twenty five years ago t shintani defined and studied four dirichlet series whose coefficients are class numbers of integral binary cubic forms by using the theory of prehomogeneous vector spaces in this paper we give a conjecture that two of these dirichlet series are essentially the same as the remaining two series the conjecture is based on our calculation of the first two hundred nonzero coefficients of the four dirichlet series and is also consistent with the known functional equation and residues at poles if the conjecture is true we get simpler symmetric functional equations than previously known namely if we take a certain linear combination of shintani s dirichlet series that a single function is invariant under the variable change of s into 1 s

on the convergence of the adelic zeta functions associated to irreducible regular prehomogeneous vector spaces introduction let g x be an irreducible regular prehomogeneous vector space defined over an algebraic number field k d denotes the derived group of g x is an affine n space affn f x e k xl xn is the relative invariant f gx v g f x for every g e g where vi e hom g gm y x f 1 0 the subscript a denotes the adelization relative to k the subscript k denotes the taking of k rational points let aug be a haar measure on ga consider the following integral

regularized orbital integrals for representations of 2 given a finite dimensional representation of sl 2 f on a vector space v defined over a local field f of characteristic zero we produce a regularization of orbital integrals and determine when the resulting distribution is non trivial

on the classification of k involutions let g be a connected reductive algebraic group defined over a field k of characteristic not 2 an involution of g defined over k h a k open subgroup of the fixed point group of and gk resp hk the set of k rational points of g resp h the variety gk hk is called a symmetric k variety these varieties occur in many problems in representation theory geometry and singularity theory over the last few decades the representation theory of these varieties has been extensively studied for k r and c as most of the work in these two cases was completed the study of the representation theory over other fields like local fields and finite fields began the representations of a homogeneous space usually depend heavily on the fine structure of the homogeneous space like the restricted root systems with weyl groups etc thus it is essential to study first this structure and the related geometry in this paper we give a characterization of the isomorphy classes of these symmetric k varieties together with their fine structure of restricted root systems and also a classification of this fine structure for the real numbers p adic numbers finite fields and number fields

a polynomial time complexity bound for the computation of the singular part of a puiseux expansion of an algebraic function in this paper we present a refined version of the newton polygon process to compute the puiseux expansions of an algebraic function defined over the rational function field we determine an upper bound for the bit complexity of computing the singular part of a puiseux expansion by this algorithm and use a recent quantitative version of eisenstein s theorem on power series expansions of algebraic functions to show that this computational complexity is polynomial in the degrees and the logarithm of the height of the polynomial defining the algebraic function

a revised modified cholesky factorization algorithm a modified cholesky factorization algorithm introduced originally by gill and murray and refined by gill murray and wright is used extensively in optimization algorithms since its introduction in 1990 a different modified cholesky factorization of schnabel and eskow has also gained widespread usage compared with the gill murray wright algorithm the schnabel eskow algorithm has a smaller a priori bound on the perturbation added to ensure positive definiteness and some computational advantages especially for large problems users of the schnabel eskow algorithm however have reported cases from two different contexts where it makes a far larger modification to the original matrix than is necessary and than is made by the gill murray wright method this paper reports on a simple modification to the schnabel eskow algorithm that appears to correct all the known computational difficulties with the method without harming its theoretical properties or its computational behavior in any other cases in new computational tests the modifications to the original matrix made by the new algorithm appear virtually always to be smaller than those made by the gill murray wright algorithm sometimes by significant amounts the perturbed matrix is allowed to be more ill conditioned with the new algorithm but this seems to be appropriate in the known contexts where the underlying problem is ill conditioned

orbital integrals on symmetric spaces and spherical characters abstract in the first section of this paper we obtain an asymptotic expansion near semi simple elements of orbital integrals x f of c c functions f on symmetric spaces g h here g is a reductive p adic group and h is the group of fixed points of an involution on g this extends the germ expansion of j shalika and m f vigneras in the group case the main part of the paper studies examples of groups g with involution which have the property that the spherical characters associated with their spherical admissible representations are not identically zero on the regular set of g h these include g gl n m h gl n gl m for n m 1 or 2 and n 1 m 3 more general results had been obtained by j sekiguchi in the case of real symmetric spaces generalizing harish chandra s work in the group case over archimedean and non archimedean fields our interest is in the p adic case there the techniques are entirely different from sekiguchi s in fact we use the recent work of j hakim and c rader and s rallis who showed that the spherical character is smooth on the regular set and has asymptotic expansion in terms of fourier transforms of invariant distributions on the nilpotent cone as found by harish chandra in the group case our study of the nonvanishing of some spherical characters uses a construction of an explicit basis of the space of invariant distributions on the nilpotent cone this is done on regularizing spherical orbital integrals and taking suitable linear combinations this local work is motivated by concrete applications to the theory of deligne kazhdan lifting of spherical automorphic representations in some other examples concerning g gl 3 n and h gl n gl 2 n and g o 3 2 h o 2 2 we explicitly construct invariant distributions on the nilpotent cone which are equal to their fourier transform such examples do not exist in harish chandra s group case in the last two sections following harish chandra s simple proof in the group case we show that x f is locally constant on the regular set of x uniformly in f in some cases following d kazhdan s proof of his density theorem we show that an f which annihilates all spherical characters has x f 0 on the regular elliptic set

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

racines orthogonales formes quadratiques et orbites d algebres de lie simples graduees abstract we give a unified orbital decomposition for irreducible regular prehomogeneous vector spaces of parabolic type graded by strongly orthogonal roots of same length with one dimensional corresponding root space on a local or global field of characteristic 0 in terms of a class of quadratic forms related to a root system weyl action and the invariants of the prehomogeneous vector spaces

geometric side of a local relative trace formula following a scheme suggested by b feigon we investigate a local relative trace formula in the situation of a reductive p adic group g relative to a symmetric subgroup h underline h f where underline h is split over the local field f of characteristic zero and g underline g f is the restriction of scalars of underline h i e relative to a quadratic unramified extension e of f we adapt techniques of the proof of the local trace formula by j arthur in order to get a geometric expansion of the integral over h times h of a truncated kernel associated to the regular representation of g

the adelic zeta function associated with the space of binary cubic forms with coefficients in a function field in this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms defined over a function field we establish its rationality find its poles and residues and a simple functional equation that this zeta function satisfies introduction the subject of integral equivalence of binary cubic forms although not as popular or well developed as the theory of integral equivalence of binary quadratic forms has received a fair amount of attention the first paper on this topic 9 written by eisenstein appeared in 1844 later on several other first rate mathematicians tried their hand at the subject among others the list includes arndt cayley hermite and davenport the classical method of attacking this problem is to draw upon a one to one discriminant preserving relation between binary cubic and binary quadratic forms which assigns to a cubic form its hessian and then use the reduction theory for binary quadratic forms to obtain results on class numbers of binary cubic forms the best of the classical results are due to davenport 5 who showed that e h ir n 2 36x o x15 16 nax e h ir n v2 12x o x 5116 nax where h ir n is the class number of irreducible binary cubic forms of discriminant n the credit for a new surge of interest in the study of integral equivalence of binary cubic forms must indisputably go to shintani for it was his 1972 paper 25 that put this study on modern footing in that paper shintani introduced a certain zeta function which as it turns out decomposes into a linear combination of products of dirichlet series whose coefficients are weighted class numbers of binary cubic forms and certain mellin transforms by studying analytic properties of the zeta function shintani was able to improve the error term in davenport s result to o x576 it is of interest to note that the zeta function and the dirichlet series it received by the editors september 3 1985 1980 mathematics subject classification 1985 revision primary 12a70 secondary iocio 12a90 1987 american mathematical society 0002 9947 87 1 00 25 per page

finite order automorphisms and a parametrization of nilpotent orbits in p adic lie algebras let k be a field with a nontrivial discrete valuation which is complete and has perfect residue field let g be the group of k rational points of a reductive linear algebraic group bf g equipped with a finite cyclic group l of order m acting on bf g by algebraic automorphisms defined over k we assume that the lie algebra of g decomposes into a direct sum of eigenspaces under the action of l with generator theta we let frak g i denote the eigenspace corresponding to eigenvalue xi i where xi is a primitive m th root of unity that is x in frak g i if and only if theta x xi ix if bf h is a k subgroup of bf g l the group of l fixed points which contains the neutral component of bf g l then h bf h k acts on each eigenspace of frak g let r in bbb r under mild restrictions on the residual characteristic of k the set of nilpotent h orbits in the xi eigenspace frak g 1 is parametrized by equivalence classes of noticed moy prasad cosets of depth r which lie in frak g 1

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

prehomogeneous vector spaces and field extensions letk be an infinite field of characteristic not equal to 2 3 5 in this paper we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of galois extensions ofk which are splitting fields of equations of certain degrees and prove that the inverse image of this map corresponds bijectively with conjugacy classes of galois homomorphisms

twists of the iwasawa tate zeta function the intent of this paper is to develop the theory of a simple variation of the adelic zeta function first described by tate 14 and iwasawa 9 in independent efforts to simplify hecke s analysis of his l functions attached to gr6bencharakters our twists of the iwasawa tate zeta function arise in the framework of a more general theory of zeta functions associated to representations of algebraic groups many of the basic problems of this general theory have already been solved by yukie and the author see 20 and 21 1 but still await appearance in final written form thus it seems worthwhile to give here a general statement of these problems before proceeding to the very special case considered in this paper let k be a global field i e algebraic number field or function field of a curve defined over a finite field let n and a denote the ring of adeles and the group of ideles of k respectively let co be a quasicharacter of the idele group trivial on the embedded group k of nonzero elements of k the group f2 f2 k of all such quasicharacters co has a natural riemann surface structure let l i denote the usual idele norm on a and for s tl2 let co be the principal quasicharacter co t ltl let re co be the real part of co f2 defined as the unique real number a such that l o i co for any reductive linear algebraic group g defined over k let g denote the adelizat ion of g over k as defined in 16 and let gk be the group of k rational points of g embedded as a discrete group in ga for simplicity we shall fix g as g1 now although the theory to be described below can also be effected for any reductive algebraic group with at least one nontrivial k rational character let p g g1 v be an irreducible k rational representation of g in the finite dimensional vector space v of dimension m let i be the identity matrix in g by schur s lemma there is an integer d such that p ti tdp i for all scalars t we shall always assume that d 0 let dg be any convenient left invariant measure on the quotient ga gk let v be the schwartz bruhat space of functions on v for the time being let vk be any gk invariant subset of vk the zeta function associated to the representation p may now be defined

twisted relative trace formulae with a view towards unitary groups we introduce a twisted relative trace formula which simultaneously generalizes the twisted trace formula of langlands et al in the quadratic case and the relative trace formula of jacquet and lai certain matching statements relating this twisted relative trace formula to a relative trace formula are also proven including the relevant fundamental lemma in the biquadratic case using recent work of jacquet lapid and their collaborators and the rankin selberg integral representation of the asai l function obtained by flicker using the theory of jacquet piatetskii shapiro and shalika we give the following application let e f be a totally real quadratic extension with gal e f let u be a quasi split unitary group with respect to a cm extension m f and let u res e f u under suitable local hypotheses we show that a cuspidal cohomological automorphic representation of u whose asai l function has a pole at the edge of the critical strip is nearly equivalent to a cuspidal cohomological automorphic representation of u that is u distinguished in the sense that there is a form in the space of admitting a nonzero period over u this provides cohomologically nontrivial cycles of middle dimension on unitary shimura varieties analogous to those on hilbert modular surfaces studied by harder langlands and rapoport

modified cholesky algorithms a catalog with new approaches given an n n symmetric possibly indefinite matrix a a modified cholesky algorithm computes a factorization of the positive definite matrix a e where e is a correction matrix since the factorization is often used to compute a newton like downhill search direction for an optimization problem the goals are to compute the modification without much additional cost and to keep a e well conditioned and close to a gill murray and wright introduced a stable algorithm with a bound of e 2 o n 2 an algorithm of schnabel and eskow further guarantees e 2 o n we present variants that also ensure e 2 o n more and sorensen and cheng and higham used the block lbl t factorization with blocks of order 1 or 2 algorithms in this class have a worst case cost o n 3 higher than the standard cholesky factorization we present a new approach using a sandwiched ltl t lbl t factorization with t tridiagonal that guarantees a modification cost of at most o n 2

bessel identities in the waldspurger correspondence over a p adic field in this paper we develop the local theory for a jacquet s relative trace formula the local theory is essential to the application of the trace formula it is an identity of bessel and relative bessel distributions we show that the relative bessel distribution attached to a distinguished that is self contragradient unitary representation of gl 2 k where k is a p adic field is given by a locally integrable function namely the relative bessel function we compute the relative bessel function for principal series complementary series and special representations we also show that the bessel distributions associated to unitary irreducible admissible representations of the double covers of gl 2 k and sl 2 k are given by a locally integrable bessel functions we compute these bessel functions for principal series complementary series and special representations finally we obtain the waldspurger correspondence via bessel identities between relative bessel functions on gl 2 k and bessel functions on the double cover of sl 2 k the bessel identity also implies the identity between bessel and relative bessel distributions

fourier transform and the global gan gross prasad conjecture for unitary groups by the relative trace formula approach of jacquet rallis we prove the global gan gross prasad conjecture for unitary groups under some local restrictions for the automorphic representations

on linear periods let pi be a cuspidal automorphic representation of mathrm gl 2n mathbb a which is assumed to be the jacquet langlands transfer from a cuspidal automorphic representation pi of mathrm gl 2m d mathbb a where d is a division algebra so that mathrm gl 2m d is an inner form of mathrm gl 2n in this paper we consider the relation between linear periods on pi and pi we conjecture that the non vanishing of the linear period on pi would imply the non vanishing of that on pi we illustrate an approach using a relative trace formula towards this conjecture and prove the existence of smooth transfer over non archimedean local fields

z finite distributions on p adic groups abstract for a real reductive group g the center z u g of the universal enveloping algebra of the lie algebra g of g acts on the space of distributions on g this action proved to be very useful see e g 17 18 32 8 over non archimedean local fields one can replace the action of z u g by the action of the bernstein center z of g i e the center of the category of smooth representations however this action is not well studied in this paper we provide some tools to work with this action and prove the following results the wavefront set of any z finite distribution on g over any point g g lies inside the nilpotent cone of t g g g let h 1 h 2 g be symmetric subgroups consider the space j of h 1 h 2 invariant distributions on g we prove that the z finite distributions in j form a dense subspace in fact we prove this result in wider generality where the groups h i are spherical subgroups of certain type and the invariance condition is replaced by equivariance further we apply those results to density and regularity of relative characters the first result can be viewed as a version of howe s expansion of characters the second result can be viewed as a spherical space analog of a classical theorem on density of characters of finite length representations it can also be viewed as a spectral version of bernstein s localization principle in the archimedean case the first result is well known and the second remains open

factoring multivariate integral polynomials an algorithm is presented to factorize polynomials in several variables with integral coefficients that is polynomial time in the degrees of the polynomial to be factored for any fixed number of variables the algorithm generalizes the algorithm presented by a k lenstra et al to factorize integral polynomials in one variable

relations among dirichlet series whose coefficients are class numbers of binary cubic forms we study the zeta functions for the space of binary cubic forms introduced by shintani the zeta function is defined for each invariant lattice we classify the invariant lattices and investigate explicit relationships between the zeta functions associated with those lattices we also study the analytic properties of the zeta functions and rewrite shintani s functional equation in a self dual form using an explicit relation

preconditioning saddle point systems with applications in optimization saddle point systems arise in many applications areas in fact in any situation where an extremum principle arises with constraints the stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from pdes and in the area of optimization such problems are ubiquitous in this paper we present a framework into which many well known methods for solving saddle point systems fit based on this description we show how new approaches for the solution of saddle point systems arising in optimization can be derived from the bramble pasciak conjugate gradient approach widely used in pdes and more recent generalizations thereof in particular we derive a class of new solution methods based on the use of preconditioned conjugate gradients in nonstandard inner products and demonstrate how these can be understood through more standard machinery we show connections to constraint preconditioning and give the results of numerical computations on a number of standard optimization test examples

generalized harish chandra descent gelfand pairs and an archimedean analog of jacquet rallis s theorem in the first part of the paper we generalize a descent technique due to harish chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field f of characteristic zero our main tool is the luna slice theorem in the second part of the paper we apply this technique to symmetric pairs in particular we prove that the pairs gln k f gln f glk f and gln e gln f are gelfand pairs for any local field f and its quadratic extension e in the non archimedean case the first result was proven earlier by jacquet and rallis and the second by flicker we also prove that any conjugation invariant distribution on gln f is invariant with respect to transposition for non archimedean f the latter is a classical theorem of gelfand and kazhdan

on bessel distributions for quasi split groups we show that the bessel distribution attached by gelfand and kazhdan and by shalika to a generic representation of a quasi split reductive group over a local field is given by a function when it is restricted to the open bruhat cell as in the case of the character distribution this function is real analytic for archimedean fields and locally constant for non archimidean fields

a polynomial time complexity bound for computations on curves in this paper we use a recent quantitative version of eisenstein s theorem on power series expansions of algebraic functions to compute a polynomial time bit complexity bound for the computation of the genus of an algebraic curve over the rationals the result is extendable to any field of characteristic zero which is finitely generated over the rationals

orbits invariants and representations associated to involutions of reductive groups by an involution of an algebraic group or lie algebra we mean an automorphism 0 such that 0 2 is the identity automorphism let 0 be an involution of a complex reductive lie algebra q and let p denote the 1 eigenspace of 0 on q get g be the adjoint group of q and let k g glog go kostant and rallis 10 have made a detailed study of the action of k on p in particular they obtain very precise information concerning the orbits of k on p and the representation of k on c p the algebra of polynomial functions on p in this paper we consider a global analogue of the problem studied by kostant and rallis let g be a reductive affine algebraic group over an algebraically closed field k of characteristic 2 let 0 be an involution of g and let k gegi0 g g if g is semisimple and the base field is c then g k is the complexification of a riemannian symmetric space we study the action of k and of groups closely related to k by left translation on the coset space g k for technical reasons it is convenient to study a slightly different but equivalent problem let r g g be defined by r g g0 g i then z g p is a closed subvariety of g furthermore z is constant on left cosets of g modulo k and induces an isomorphism o g k p we have o k g k k o g k for k e k and geg thus p is stable under conjugation by k and a is a k isomorphism of affine k varieties we consider the action of k on p by conjugation let x e g have jordan decomposition x x x then we show that x e p if and only if both x and x are in p thus we have a jordan decomposition for elements of p this will play a key role in our results a torus s of g is o anisotropic if o s s 1 for every ses it is known that any two maximal 0 anisotropic tori of g are k let a be a maximal 0 anisotropic torus of g the following result characterizes semisimple elements of p