p converse to a theorem of gross zagier kolyvagin and rubin let e be a cm elliptic curve over the rationals and p 3 a good ordinary prime for e we show that begin aligned mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 implies mathrm ord s 1 l s e mathbb q 1 end aligned for the p infty selmer group mathrm sel p infty e mathbb q and the complex l function l s e mathbb q in particular the tate shafarevich group hbox x e mathbb q is finite whenever mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 we also prove an analogous p converse for cm abelian varieties arising from weight two elliptic cm modular forms with trivial central character for non cm elliptic curves over the rationals first general results towards such a p converse theorem are independently due to skinner a converse to a theorem of gross zagier and kolyvagin arxiv 1405 7294 2014 and zhang camb j math 2 2 191 253 2014

priors and component structures in autoregressive time series models new approaches to prior specification and structuring in autoregressive time series models are introduced and developed we focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series these new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends quasi periodic subprocesses and high frequency residual noise components of observed series the class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series as it does for predictions of future values posterior analysis involves easily implemented iterative simulation methods developed and described here one motivating field of application is climatology where the evaluation of latent structure especially quasi periodic structure is of critical importance in connection with issues of global climatic variability we explore the analysis of data from the southern oscillation index one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators

time series decomposition a constructive result on time series decomposition is presented and illustrated developed through dynamic linear models the decomposition is useful in analysis of an observed time series through inference about underlying latent component series that may have physical interpretations particular special cases include state space autoregressive component models in which the decomposition is useful for isolating latent quasi cyclical components in particular brief summaries of analyses of some geological records related to climatic change illustrate the result

relating invariant linear form and local epsilon factors via global methods we use the recent proof of jacquet s conjecture due to harris and kudla hk and the burger sarnak principle see bs to give a proof of the relationship between the existence of trilinear forms on representations of gl 2 k u for a non archimedean local field k u and local epsilon factors which was earlier proved only in the odd residue characteristic by this author in p1 theorem 1 4 the method used is very flexible and gives a global proof of a theorem of saito and tunnell about characters of gl 2 using a theorem of waldspurger w theorem 2 about period integrals for gl 2 and also an extension of the theorem of saito and tunnell by this author in p3 theorem 1 2 which was earlier proved only in odd residue characteristic in the appendix to this article h saito gives a local proof of lemma 4 which plays an important role in the article

bessel models for gsp 4 methods of theta correspondence are used to analyze local and global bessel models for gsp 4 proving a conjecture of gross and prasad which describes these models in terms of local epsilon factors in the local case and the non vanishing of central critical l value in the global case

invariant tori of duffing type equations abstract in this paper we prove the existence of invariant tori and thus the boundedness of all solutions and the existence of quasiperiodic solutions for a class of duffing type equations d 2 x dt 2 x 2n 1 j 0 2n p j t x j 0 n 1 wherepj t j 0 1 n are 1 periodicc 1 functions and wherepj t j n 1 2n are 1 periodicc 2 functions

time series decomposition into oscillation components and phase estimation many time series are naturally considered as a superposition of several oscillation components for example electroencephalogram eeg time series include oscillation components such as alpha beta and gamma we propose a method for decomposing time series into such oscillation components using state space models based on the concept of random frequency modulation gaussian linear state space models for oscillation components are developed in this model the frequency of an oscillator fluctuates by noise time series decomposition is accomplished by this model like the bayesian seasonal adjustment method since the model parameters are estimated from data by the empirical bayes method the amplitudes and the frequencies of oscillation components are determined in a data driven manner also the appropriate number of oscillation components is determined with the akaike information criterion aic in this way the proposed method provides a natural decomposition of the given time series into oscillation components in neuroscience the phase of neural time series plays an important role in neural information processing the proposed method can be used to estimate the phase of each oscillation component and has several advantages over a conventional method based on the hilbert transform thus the proposed method enables an investigation of the phase dynamics of time series numerical results show that the proposed method succeeds in extracting intermittent oscillations like ripples and detecting the phase reset phenomena we apply the proposed method to real data from various fields such as astronomy ecology tidology and neuroscience

local e factors and characters of gl 2 introduction the representation theory of gl 2 k for k a nonarchimedean local field is somewhat more complicated than the corresponding theory for gl 2 r in particular the formulas for the characters of discrete series representation of gl 2 r have a concise closed form while the published character formulas for supercuspidal representations of gl 2 k occupy several pages 10 14 the purpose of this paper is to give a concise formula for the character of certain infinite dimensional representations of gl 2 k the representations considered are admissible in the sense that the stabilizer of a vector in the representation space is open in gl 2 k and the subspace stabilized by an open compact subgroup of gl 2 k is finite dimensional irreducible admissible representations ir of gl 2 k satisfy schur s lemma so that the center k of gl 2 k acts by multiplication by a quasicharacter there exists a locally constant character function ch defined on a dense open subgroup of gl 2 k which determines the isomorphism class of ir 7 section 7 jacquet and langlands 7 section 2 introduced factors e 10 x attached to the twist of ir by quasicharacters x of k which describe the action of r in a specific model for each separable quadratic extension l of k there exists an irreducible admissible representation bcl k 1r of gl 2 l the base change of ir to l 9 section 2 this representation of gl 2 l enters into the expression for the character the main result of this paper is the following formula recall that conjugacy classes of nonsplit cartan subgroups are parameterized by quadratic separable field extensions l of k via embeddings of l in gl 2 k

multiplicity theorems modulo p for mathbf gl _ 2 mathbf q _p let f be a non archimedean local field pi an admissible irreducible mathbf gl 2 f representation with complex coefficients for a quadratic extension l f and an l times character chi a classical result of tunnell and saito establish a precise connection between the dimension of the hom space hbox hom l times big pi vert l times chi big and the normalized local factor of the pair pi chi the study of analogous hom spaces for complex valued representations has recently been generalized to mathbf gl n in aizenbud et al ann math 172 1407 1434 2010 and their connections with local factors have been established by work of moeglin and waldspurger la conjecture locale de gross prasad pour les groupes speciaux orthogonaux le cas general preprint available at http www math jussieu fr moeglin gp pdf 2009 in this paper we approach the analogous problem in the context of the p modular langlands correspondence for mathbf gl 2 mathbf q p we describe the restriction to cartan subgroups of an irreducible p modular representation pi of mathbf gl 2 mathbf q p and deduce generalized multiplicity results on the dimension of the ext spaces mathrm ext fancyscript o l times i big pi vert fancyscript o l times chi big where fancyscript o l times is the ring of integers of a quadratic extension of mathbf q p and chi a smooth character of fancyscript o l times

cube sum problem and an explicit gross zagier formula a nonzero rational number is called a it cube sum if it is of the form a 3 b 3 with a b in bbb q times in this paper we prove that for any odd integer k geq 1 there exist infinitely many cube free odd integers n with exactly k distinct prime factors such that 2n is a cube sum resp not a cube sum we present also a general construction of heegner points and obtain an explicit gross zagier formula which is used to prove the birch and swinnerton dyer conjecture for certain elliptic curves related to the cube sum problem

unboundedness in a duffing equation with polynomial potentials abstract in this paper a continuous periodic function p t is constructed such that the duffing equation d2x dt2 x2n 1 p t xl 0 with p t c0 s1 n 2 2n 1 l n 2 possesses an unbounded solution which shows that the smoothness of the coefficients of the perturbation does influence the boundedness of solutions

theta dichotomy for unitary groups some recent work of gross and prasad 14 suggests that the root numbers attached to certain symplectic representations of the weil deligne group of a local field f control certain branching rules for representations of orthogonal groups over f on a global level this local phenomenon should have implications for the structure of the value at the center of symmetry for certain l functions of arithmetic interest 12 this conjectural picture is based to some extent on the now classic work of tunnell 47 and waldspurger 49 as well as on the case of the triple product l function 13 34 17 in all of these examples the local root number detects the existence of a certain type of invariant linear functional it is possible to set up analogous conjectures for the isometry groups of hermitian or quaternion hermitian forms it turns out that root numbers also play a role in the local and global theta correspondence roughly speaking certain local root numbers should control the occurrence of representations in the local theta correspondence between groups of the same size e g for dual pairs of the form sp n o 2n 1 and u n u n as will be seen the theta correspondence for such pairs is connected with a certain induced representation in s at the point s 0 on the unitary axis by contrast the correspondence for the pairs sp n o 2n and sp n o 2n 2 discussed by prasad 35 is connected to the behavior of a similar induced representation at the points 12 in which case no epsilon factor arises in this paper we consider the local theta correspondence for unitary groups in the non archimedean case let f be a non archimedean local field of characteristic not equal to 2 and let e be a quadratic extension of f for further notation see the notation section at the end of this introduction let v and w be e vector spaces of dimensions m and n equipped respectively with a hermitian form v v e and a skew hermitian form w w e then the isometry groups g v and g w of the spaces v and w form a dual reductive pair in the symplectic group sp w where w v e w viewed as an f vector space of dimension 2mn and equipped with the symplectic form

generalized heegner cycles and p adic rankin l series this article studies a distinguished collection of so called generalized heegner cycles in the product of a kuga sato variety with a power of a cm elliptic curve its main result is a p adic analogue of the gross zagier formula which relates the images of generalized heegner cycles under the p adic abel jacobi map to the special values of certain p adic rankin l series at critical points that lie outside their range of classical interpolation

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

local orders root numbers and modular curves let f be a locally compact field and let wr be an irreducible admissible representation of gl2 f on an infinite dimensional complex vector space let c be the central quasicharacter of wx let k be a separable quadratic field extension of f and let x be a quasicharacter of k such that xw 1 on f the character x gives a 1 dimensional representation of the weil group wk of k via the isomorphism wab k of local classfield theory let xw be the admissible irreducible representation of gl2 f which corresponds in the sense of langlands to the 2 dimensional induced representation indwfx of the weil group of f the representation

from laplace to langlands via representations of orthogonal groups in the late 1960s robert langlands proposed a new and far reaching connection between the representation theory of lie groups over real and p adic fields and the structure of the galois groups of these fields 24 even though this local langlands correspondence remains largely conjectural the relation that it predicts between representation theory and number theory has profoundly changed our views of both fields moreover we now know enough about the correspondence to address and sometimes solve traditional problems in representation theory that were previously inaccessible roughly speaking the local langlands correspondence predicts that complex irreducible representations of a reductive group g over a local field k should be

lagrange stability for impulsive pendulum type equations in the present paper we prove the boundedness of all solutions for some impulsive pendulum type equations by moser s twist theorem we also prove the existence of quasi periodic solutions for the impulsive pendulum type equations

lagrange stability for duffing type equations abstract in this paper the duffing type equation d 2 x dt 2 x 2 n 1 p 2 n t x 2 n p 1 t x p 0 t 0 is studied where the p j s are 1 periodic it is shown that all solutions of this equation are bounded provided that the p j t 0 j n are of bounded variation in 0 1 and that the derivatives of p j t n j 2 n are lipschitzian it is also shown that there exist p j s being discontinuous everywhere such that all solutions of the equation are bounded this implies that the continuity of p j s is not necessary for the boundedness of solutions of the equation

lagrange stability for impulsive duffing equations abstract this work discusses the boundedness of solutions for impulsive duffing equation with time dependent polynomial potentials by kam theorem we prove that all solutions of the duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many positive lebesgue measure quasi periodic solutions clustering at infinity this result extends some well known results on duffing equations to impulsive duffing equations

bayesian non parametric signal extraction for gaussian time series abstract we consider the problem of unobserved components in time series from a bayesian non parametric perspective the identification conditions are treated as unknown and analyzed in a probabilistic framework in particular informative prior distributions force the spectral decomposition to be in an identifiable region then the likelihood function adapts the prior decompositions to the data a full bayesian analysis of unobserved components will be presented for financial high frequency data particularly a three component model long term intra daily and short term will be analyzed to emphasize the importance and the potential of this work when dealing with the value at risk analysis a second astronomical application will show how to deal with multiple periodicities

boundedness for solutions of nonlinear hill s equations with periodic forcing terms via moser s twist theorem we consider the nonlinear hill s equation with periodic forcing term x x2n 1 a1 ea t x p t n 1 where a t and p t are continuous and 1 periodic a1 and are positive constants and e is a small parameter the purpose of this paper is to prove that all solutions of the above mentioned equation are bounded for t r and that there are infinitely many quasi periodic solutions and an infinity of periodic solutions of minimal period m for each positive integer m

quasiperiodic motions in superquadratic time periodic potentials it is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most in a precise sense motions are in fact quasiperiodic the class of potentials includes as very particular examples the exponential polynomial and much more this extends earlier results and gives an answer to a problem posed by littlewood in the mid 1960 s along the way machinery is developed for estimating the action angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived

an exact multivariate model based structural decomposition we propose a simple and structured procedure for decomposing a vector of time series into trend cycle seasonal and irregular components contrary to common practice we do not assume these components to be orthogonal conditional on their past however the state space representation employed ensures that their estimates converge to values with null variances and covariances null variances are very important as they ensure that the components do not change when the sample increases this lack of revisions is the most important feature of our method in comparison with most alternative procedures on the other hand null covariances provide a solid statistical foundation for the decomposition as it ensures that a given component can be analyzed and interpreted independently of any other component s other convenient properties of our method derive from the use of a state space approach first defining the problem in state space avoids dependence on particular model specifications so the same proc

local factors of nongeneric supercuspidal representations of gsp _4 we compute the local factors of the nongeneric supercuspidal representations of gsp 4 constructed by piatetski shapiro over a nonarchemedian local field of residue characteristic not equal to two the results are in terms of the local factors of the irreducible representations of quaternion division algebra

bessel periods and the non vanishing of yoshida lifts modulo a prime we give an explicit construction of vector valued yoshida lifts and derive a formula of the bessel periods of yoshida lifts by which we prove the non vanishing modulo a prime of yoshida lifts attached to a pair of elliptic modular newforms as a consequence we obtain a new proof of the non vanishing of yoshida lifts

horizontal non vanishing of heegner points and toric periods abstract let f be a totally real number field and a a modular gl 2 type abelian variety over f let k f be a cm quadratic extension let be a class group character over k such that the rankin selberg convolution l s a is self dual with root number 1 we show that the number of class group characters with bounded ramification such that l 1 a 0 increases with the absolute value of the discriminant of k we also consider a rather general rank zero situation let be a cuspidal cohomological automorphic representation over gl 2 a f let be a hecke character over k such that the rankin selberg convolution l s is self dual with root number 1 we show that the number of hecke characters with fixed type and bounded ramification such that l 1 2 0 increases with the absolute value of the discriminant of k the gross zagier formula and the waldspurger formula relate the question to horizontal non vanishing of heegner points and toric periods respectively for both situations the strategy is geometric relying on the zariski density of cm points on self products of a quaternionic shimura variety the recent result 26 31 1 on the andre oort conjecture is accordingly fundamental to the approach