uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

periods of residual automorphic forms abstract we develop a technique for computing period integrals of noncuspidal automorphic forms applications include a formula for the integral of a square integrable automorphic form on gl n a e over gl n f gl n a f 1 for a quadratic extension e of a number field f and a characterization of residual automorphic representations on which the period integral is nonvanishing

a comparison of numerical algorithms for fourier extension of the first second and third kinds the range of fourier methods can be significantly increased by extending a nonperiodic function f x to a periodic function f on a larger interval when f x is analytically known on the extended interval the extension is straightforward when f x is unknown outside the physical interval there is no standard recipe worse still like a radarless aircraft groping through fog the algorithm may wreck on the mountain in fog problem a function f x which is perfectly well behaved on the physical interval may very well have singularities in the extended domain in this article we compare several algorithms for successfully extending a function f x into the fog even when the analytic extension is singular the best third kind extension requires singular value decomposition with iterative refinement but achieves accuracy close to machine precision

the local langlands conjecture for gsp 4 we prove the local langlands conjecture for gsp4 f where f is a nonarchimedean local eld of characteristic zero

torsion homology growth and cycle complexity of arithmetic manifolds let m be an arithmetic hyperbolic 3 manifold such as a bianchi manifold we conjecture that there is a basis for the second homology of m where each basis element is represented by a surface of low genus and give evidence for this we explain the relationship between this conjecture and the study of torsion homology growth

an integral transform and its applications this was later extended by deshouillers and iwaniec 9 and their work was then used by bombieri et al 2 3 to prove some spectacular results in number theory in an unpublished note zagier gave a new proof of the kuznietsov trace formula cf iwaniec 13 his method can be generalized to the gl n case as pointed out by friedberg 10 and was later carried out by joyner 18 in order to extend the kuznietsov trace formula to real quadratic number fields the generalization to the group gl n was also done by goldfeld 11 from which he expressed the principal cuspidal contribution to the selberg trace formula in terms of special values of quadratic l functions an adelic version of the kuznietsov trace formula was formulated by the author in 24 and by jacquet and rallis 14 comparing the kuznietsov

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

on the tate and mumford tate conjectures in codimension 1 for varieties with h2 0 1 we prove the tate conjecture for divisor classes and the mumford tate conjecture for the cohomology in degree 2 for varieties with h 1 over a finitely generated field of characteristic 0 under a mild assumption on their moduli as an application of this general result we prove the tate and mumford tate conjectures for several classes of algebraic surfaces with pg 1 ams 2010 mathematics subject classification 14c 14d 14j20

on the ramanujan conjecture and finiteness of poles for certain l functions in this paper we prove a number of general results about automorphic l functions which appear in the constant terms of eisenstein series for an arbitrary quasi split connected reductive algebraic group over a number field these l functions were first introduced by langlands for chevalley groups 20 and through them he was led to make some of his deep conjectures 21 one significance of these l functions is that all the automorphic l functions studied so far are among them there are three general results first we obtain a uniform line of absolute convergence for all of these l functions theorem 5 1 a uniform estimate for the hecke eigenvalues of generic cusp forms on many absolutely simple quasi split connected reductive algebraic groups over number fields corollary 5 4 and an improvement on the best available estimate for the fourier coefficients of maass wave forms corollary 5 5 also follow next we establish a meromorphic continuation and functional equation for each of these l functions theorem 6 1 finally in theorem 6 2 we prove finiteness of poles on the whole complex plane for an important class of these l functions corollaries 6 6 through 6 10 more precisely let g be a quasi split connected reductive algebraic group over a number field f set g g a f where a f is the ring of adeles of f fix a borel subgroup b of g over f and let u be its unipotent radical let p be a maximal f parabolic subgroup of g with p d b in the context of the problems studied here nothing new will be obtained if one drops the maximality condition on p write p mn a levi decomposition n c u and let b u p m and n be the corresponding groups of adelic points for every place v of f let gv g fj similarly we have bv us pv me and nv let x xv be a generic character of u f u cf section 3 then each xv is generic let r ov 7tv be a cusp form on m we shall say 7t is

tate cycles on some quaternionic shimura varieties mod p let f be a totally real field in which a prime number p 2 is inert we continue the study of the generalized goren oort strata on quaternionic shimura varieties over finite extensions of fp we prove that when the dimension of the quaternionic shimura variety is even the tate conjecture for the special fiber of the quaternionic shimura variety holds for the cuspidal isotypical component as long as the two unramified satake parameters at p are not differed by a root of unity

zero cycles on hilbert blumenthal surfaces introduction the main object of study in this paper with respect to zero cycles is a special class of hilbert blumenthal surfaces x which are defined over q as smooth compactifications of quasi projective varieties s q more precisely of coarse moduli schemes s that represent the moduli stack of polarized abelian surfaces with real multiplication by the ring of integers in a real quadratic field f q d we assume that d q is a prime 1 4 and that the class number of f is 1 then s c the complex points of s can be described as h h sl2 of where h is the upper halfplane in the early seventies hirzebruch and zagier hz defined for each integer n a curve tn on s called hirzebruch zagier cycles and showed that their intersection numbers occur as fourier coefficients of modular forms of level q with nebentypes eq the quadratic character of f q in this connection with modular forms hirzebruchzagier cycles reveal very similar properties to hecke correspondences on the selfproduct of the modular curve x0 q this crucial observation of hirzebruch and zagier together with tunnell s proof of the tate conjecture for a product of modular curves inspired harder langlands and rapoport hlr to prove the tate conjecture for divisors on hilbert blumenthal surfaces over abelian number fields the proof of the tate conjecture was then accomplished by klingenberg kl and murty and ramakrishnan mr in the general case from the new strategy to study torsion zero cycles on algebraic surfaces as developed in ls and used in l1 l2 compare also lr it is clear that a crucial point is the tate conjecture in characteristic p at good reduction primes as one of our main results we prove the tate conjecture in characteristic p for primes p that split in f for a certain class of hilbert blumenthal surfaces for this we recall that to each modular cusp form f of weight 2 level q and nebentypes eq that is f s2 0 q eq w e associate a hilbert modular cusp form f s2 sl2 of under the doi naganuma

completion of a partial integral matrix to a unimodular matrix abstract we first characterize submatrices of a unimodular integral matrix we then prove that if n entries of an n n partial integral matrix are prescribed and these n entries do not constitute a row or a column then this matrix can be completed to a unimodular matrix consequently an n n partial integral matrix with n 1 prescribed entries can always be completed to a unimodular matrix

parameter selection and numerical approximation properties of fourier extensions from fixed data fourier extensions have been shown to be an effective means for the approximation of smooth nonperiodic functions on bounded intervals given their values on an equispaced or in general scattered grid related to this method are two parameters these are the extension parameter t the ratio of the size of the extended domain to the physical domain and the oversampling ratio h the number of sampling nodes per fourier mode the purpose of this paper is to investigate how the choice of these parameters affects the accuracy and stability of the approximation our main contribution is to document the following interesting phenomenon namely if the desired condition number of the algorithm is fixed in advance then the particular choice of such parameters makes little difference to the algorithm s accuracy as a result one is free to choose t without concern that it is suboptimal in particular one may use the value t 2 which corresponds to the case where the extended domain is precisely twice the size of the physical domain for which there is known to be a fast algorithm for computing the approximation in addition we determine the resolution power points per wavelength of the approximation to be equal to email protected and address the trade off between resolution power and stability

hirzebruch zagier cycles and twisted triple product selmer groups let e be an elliptic curve over mathbb q and a another elliptic curve over a real quadratic number field we construct a mathbb q motive of rank 8 together with a distinguished class in the associated bloch kato selmer group using hirzebruch zagier cycles that is graphs of hirzebruch zagier morphisms we show that under certain assumptions on e and a the non vanishing of the central critical value of the twisted triple product l function attached to e a implies that the dimension of the associated bloch kato selmer group of the motive is 0 and the non vanishing of the distinguished class implies that the dimension of the associated bloch kato selmer group of the motive is 1 this can be viewed as the triple product version of kolyvagin s work on bounding selmer groups of a single elliptic curve using heegner points

on the completion of a partial integral matrix to a unimodular matrix abstract we prove that if a partial integral matrix has a free diagonal then this matrix can be completed to a unimodular matrix such a condition is necessary in a general sense consequently if an n n n 2 partial integral matrix has 2 n 3 prescribed entries and any n entries of these do not constitute a row or a column then it can be completed to a unimodular matrix this improves a recent result of zhan

on regular square integrable representations of p adic groups we study the problem of construction of noncuspidal irreducible square integrable representations of classical p adic groups and questions related to it starting from cuspidal reducibilities in the generalized rank one case we give a construction of regular irreducible square integrable representations of p adic gsp n sp n and so 2 n 1 under an assumption expected to hold in general we show that all such representations should come from this construction further we obtain a number of general results about places where square integrable representations can appear in parabolically induced representation among others we show that in irreducible cuspidal representations of general linear groups only the self contragredient ones play a role in the construction of square integrable representations of the above groups at the end we use our regular square integrable representations to study some basic properties of whittaker models we also show that there exists nondegenerate standard modules of classical groups which contain degenerate irreducible subrepresentations

on a uniqueness property of supercuspidal unipotent representations abstract the formal degree of a unipotent discrete series character of a simple linear algebraic group over a non archimedean local field in the sense of lusztig 17 is a rational function of q evaluated at q q the cardinality of the residue field the irreducible factors of this rational function are q and cyclotomic polynomials we prove that the formal degree of a supercuspidal unipotent representation determines its lusztig langlands parameter up to twisting by weakly unramified characters for split exceptional groups this result follows from the work of m reeder 28 and for the remaining exceptional cases this is verified in 7 in the present paper we treat the classical families the main result of this article characterizes unramified lusztig langlands parameters which support a cuspidal local system in terms of formal degrees the result implies the uniqueness of so called cuspidal spectral transfer morphisms as introduced in 22 between unipotent affine hecke algebras up to twisting by unramified characters in 23 the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case

algebraic cycles on a product of two hilbert modular surfaces in this paper we prove the tate conjecture for a product of two hilbert modular surfaces for non cm submotives

the fourier extension method and discrete orthogonal polynomials on an arc of the circle abstract the fourier extension method also known as the fourier continuation method is a method for approximating non periodic functions on an interval using truncated fourier series with period larger than the interval on which the function is defined when the function being approximated is known at only finitely many points the approximation is constructed as a projection based on this discrete set of points in this paper we address the issue of estimating the absolute error in the approximation the error can be expressed in terms of a system of discrete orthogonal polynomials on an arc of the unit circle and these polynomials are then evaluated asymptotically using riemann hilbert methods

impossibility of fast stable approximation of analytic functions from equispaced samples it is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions to avoid instability one must settle for root exponential convergence the proof combines a bernstein inequality of 1912 with an estimate due to coppersmith and rivlin in 1992

self dual representations with vectors fixed under an iwahori subgroup abstract let g be the group of f points of a split connected reductive f group over a non archimedean local field f of characteristic 0 let be an irreducible smooth self dual representation of g the space w of carries a non degenerate g invariant bilinear form which is unique up to scaling the form is easily seen to be symmetric or skew symmetric and we set e 1 accordingly in this article we show that e 1 when is a generic representation of g with non zero vectors fixed under an iwahori subgroup i

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

natural density of rectangular unimodular integer matrices abstract in this paper we compute the natural density of the set of k n integer matrices that can be extended to an invertible n n matrix over the integers as a corollary we find the density of rectangular matrices with hermite normal form o k n k i k connections with cesaro s theorem on the density of coprime integers and quillen suslin s theorem are also presented

on the sl 2 period integral let e f be a quadratic extension of number fields for a cuspidal representation of sl2 ae we study in this paper the integral of functions in on sl2 f sl2 af we characterize the nonvanishing of these integrals called period integrals in terms of having a whittaker model with respect to characters of e ae which are trivial on af we show that the period integral in general is not a product of local invariant functionals and find a necessary and sufficient condition when it is we exhibit cuspidal representations of sl2 ae whose period integral vanishes identically while each local constituent admits an sl2 invariant linear functional finally we construct an automorphic representation on sl2 ae which is abstractly sl2 af distinguished but for which none of the elements in the global l packet determined by it is distinguished by sl2 af

convergence analysis of krylov subspace iterations with methods from potential theory krylov subspace iterations are among the best known and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices these methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix this leads to an extremal problem in polynomial approximation theory how small can a monic polynomial of a given degree be on the spectrum this survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices it is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure then depending on the number of iterations the lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region which is characterized by means of a constrained equilibrium problem from potential theory the same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems

on multiplicity in restriction of tempered representations of p adic groups we establish an equality between two multiplicities one in the restriction of tempered representations of a p adic group to its closed subgroup with the same derived group and one occurring in their corresponding component groups in langlands dual sides so called mathcal s groups under working hypotheses about the tempered local langlands conjecture and the internal structure of tempered l packets this provides a formula of the multiplicity for p adic groups by means of dimensions of irreducible representations of their mathcal s groups

accurate high order representation of complex three dimensional surfaces via fourier continuation analysis we present a new method for construction of high order parametrizations of surfaces starting from point clouds the method we propose can be used to produce full surface parametrizations by sets of local charts each one representing a large surface patch which typically contains thousands of the points in the original point cloud for complex surfaces of scientific and engineering relevance the proposed approach accurately renders both smooth and non smooth portions of a surface it yields super algebraically convergent fourier series approximations to a given surface up to and including all points of geometric singularity such as corners edges conical points etc in view of their c smoothness except at true geometric singularities and their properties of high order approximation the surfaces produced by this method are suitable for use in conjunction with high order numerical methods for boundary value problems in domains with complex boundaries including pde solvers integral equation solvers etc our approach is based on a very simple concept use of fourier analysis to continue smooth portions of a piecewise smooth function into new functions which defined on larger domains are both smooth and periodic the continuation functions arising from a function f converge super algebraically to f in its domain of definition as discretizations are refined we demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance

period relations and the tate conjecture for hilbert modular surfaces introduction 319 a sketch of the proof 320 0 notation and preliminaries 322 1 statement of the main results 328 2 twisting correspondences 331 3 period relations 334 4 hodge cycles 339 5 tare cycles 340 6 poles of l functions 341 7 abelian varieties 342 8 concluding remarks and open questions 343 acknowledgements 344 bibliography 344

representations des groupes reductifs p adiques et de leurs sous groupes distingues cocompacts resume soient g et h deux groupes reductifs p adiques et h g une isogenie centrale a silberger a prouve qu une representation complexe admissible irreductible de g a pour restriction via une representation admissible semi simple de longueur finie de h nous prouvons que reciproquement toute representation complexe admissible irreductible de h est facteur direct d une telle restriction et s etend en une representation admissible d un sous groupe ouvert d indice fini de g nous etendons la portee de ces resultats en les reliant aux proprietes algebriques des categories de representations lisses de g et h let g h be reductive p adic groups and h g a central isogeny a silberger has proved that any admissible irreducible complex representation of g restricts via to a direct sum of finitely many admissible irreducible representations of h we prove that conversely any admissible irreducible complex representation of h occurs as a direct summand of such a restriction and extends to an admissible representation of a finite index open subgroup of g we extend those results by linking them with algebraic properties of the categories of smooth representations of g and h

on the resolution power of fourier extensions for oscillatory functions functions that are smooth but non periodic on a certain interval possess fourier series that lack uniform convergence and suffer from the gibbs phenomenon however they can be represented accurately by a fourier series that is periodic on a larger interval this is commonly called a fourier extension when constructed in a particular manner fourier extensions share many of the same features of a standard fourier series in particular one can compute fourier extensions which converge spectrally fast whenever the function is smooth and geometrically fast if the function is analytic much the same as the fourier series of a smooth analytic and periodic function with this in mind the purpose of this paper is to describe analyse and explain the observation that fourier extensions much like classical fourier series also have excellent resolution properties for representing oscillatory functions the resolution power or required number of degrees of freedom per wavelength depends on a user controlled parameter and as we show it varies between 2 and p the former value is optimal and is achieved by classical fourier series for periodic functions for example the latter value is the resolution power of algebraic polynomial approximations thus fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation

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

unipotent representations and unipotent classes in sl n to achieve the stabilization of the trace formula for a reductive group 18 there are two related problems that must first be solved one is the matching of the sorbital integrals of smooth functions on a reductive group g over a p adic field of characteristic zero with stable orbital integrals on its endoscopic groups h igusa theory clarifies the nature of this first problem 11 12 13 19 22 23 the second problem is the fundamental lemma which asserts the compatibility of the matching of smooth functions with the hecke algebras on g and h buildings have been used to study this problem 25 the statements of the problems are closely related and the main results of this paper give a method of reducing one problem to the other if the first problem matching smooth functions is solved for sl n then the second problem the fundamental lemma for the hecke algebras follows as a consequence at least for the identity of the algebra the solutions of the two problems are linked by making a detailed study of the unipotent representations and the unipotent classes of sl n representations with iwahori fixed vectors on sl n f are called unipotent in this paper to emphasize the close relationship between unipotent conjugacy classes and these representations not long after this paper was written waldspurger refined reformulated and greatly extended the results of this paper this paper made a modest contribution to this subsequent work of his he eliminated the hypothesis that functions on sl n have matching smooth functions by making clever use of a strengthened version of kazhdan s lemma the opinion of the experts is still divided will the solution of the fundamental lemma for other reductive groups come from the matching of smooth functions or will it come from even stronger versions of kazhdan s lemma this much is clear the fundamental lemma no longer needs to be viewed as an isolated combinatorial problem in buildings howe s conjecture uniform germ expansions homogeneity of germs compact traces unipotent orbits and kazhdan s density theorem give us a framework through which the fundamental lemma may be understood in the final portion of the introduction i would like to state a theorem that has considerable interest beyond its usefulness for the fundamental lemma this theorem will be used in section 1 to produce a uniform germ expansion the

towards a geometric jacquet langlands correspondence for unitary shimura varieties let g be a unitary group over a totally real field and let x be a shimura variety associated to g for certain primes p of good reduction for x we construct cycles x 0 i on the characteristic p fiber of x these cycles are defined as the loci on which the verschiebung map has small rank on particular pieces of the lie algebra of the universal abelian variety on x the geometry of these cycles turns out to be closely related to shimura varieties for a different unitary group g which is isomorphic to g at all finite places but not isomorphic to g at archimedean places more precisely each cycle x 0 i has a natural desingularization x 0 i which is almost isomorphic to a scheme parameterizing certain subbundles of the lie algebra of the universal abelian variety over a shimura variety x associated to g we exploit this relationship to construct an injection of the etale cohomology of x into that of x this yields a geometric construction of jacquet langlands transfers of automorphic representations of g to automorphic representations of g

bounds for polynomials with a unit discrete norm let e be the set of n equidistant points in 1 1 and fn e be the set of all polynomials p of degree n with max p c c e 1 we prove that 7t kn n x max p x clog p e arctan ky s x r v 1 n2 jv2 where n n and c is an absolute constant the result is essentially sharp bounds for kn z z g c uniform for n n are also obtained the method of proof of those results is a general one it allows one to obtain sharp or sharp up to a log n factor bounds for kn under rather general assumptions on e j5 n a model result is announced for a class of sets e main components of the method are discussed in some detail in the process of investigating the case of equally spaced points

on the sign of regular algebraic polarizable automorphic representations we remove a parity condition from the construction of automorphic galois representations carried out in the paris book project we subsequently generalize this construction to the case of mixed parity but still regular essentially self dual automorphic representations over totally real fields finding associated geometric projective representations finally we optimize some of our previous results on finding geometric lifts through central torus quotients of geometric galois representations and apply them to the previous mixed parity setting

on depth zero l packets for classical groups by computing reducibility points of parabolically induced representations we construct to within at most two unramified quadratic characters the langlands parameter of an arbitrary depth zero irreducible cuspidal representation formula presented of a classical group which may be not quasi split over a non archimedean local field of odd residual characteristic from this we can explicitly describe all the irreducible cuspidal representations in the union of one two or four formula presented packets containing formula presented these results generalize the work of debacker reeder in the case of classical groups from regular to arbitrary tame langlands parameters

on the fourier extension of nonperiodic functions we obtain exponentially accurate fourier series for nonperiodic functions on the interval 1 1 by extending these functions to periodic functions on a larger domain the series may be evaluated but not constructed by means of the fft a complete convergence theory is given based on orthogonal polynomials that resemble chebyshev polynomials of the first and second kinds we analyze a previously proposed numerical method which is unstable in theory but stable in practice we propose a new numerical method that is stable both in theory and in practice