distribution of periodic torus orbits and duke s theorem for cubic fields we study periodic torus orbits on spaces of lattices using the action of the group of adelic points of the underlying tori we define a natural equivalence relation on these orbits and show that the equivalence classes become uniformly distributed this is a cubic analogue of duke s theorem about the distribution of closed geodesics on the modular surface suitably interpreted the ideal classes of a cubic totally real field are equidistributed in the modular 5 fold sl3 z sl3 r so3 in particular this proves a stronger form of the folklore conjecture that the collection of maximal compact flats in sl3 z sl3 r so3 of volume infinity

the theory of n v k v l is undecidable we first give a simple encoding for k automata in n vk where vk is the function from n 0 to n mapping x to the largest power of k which divides x this with a result of hodgson gives a proof of buchi s theorem which states that a subset of nn is k recognizable by automata if and only if it is definable in n vk our proof also shows that every formula of n vk is equivalent to a formula furthermore solving a problem of a joyal we show that the first order theory of n vk vl is undecidable for k l multiplicatively independent

representations of integers by an invariant polynomial and unipotent flows we study a refined version of linnik s problem on the asymptotic behavior of the number of representations of integers m by an integral polynomial as m tends to infinity assuming that the polynomials arise from invariant theory we reduce the question to the study of limiting behavior of measures invariant under unipotent flows our main tool is then ratner s theorem on the uniform distribution of unipotent flows in a form refined by dani and margulis dm2

decidability of the theory of the natural integers with the cantor pairing function and the successor abstract the binary cantor pairing function c from n n into n is defined by c x y 1 2 x y x y 1 y we consider the theory of natural integers equipped with the cantor pairing function and an extra relation or function x on n when x is equal either to multiplication or coprimeness or divisibility or addition or natural ordering it can be proved that the theory th n c x is undecidable let s be the successor function we provide an algorithm solving the decision problem for th n c s

high action orbits for tonelli lagrangians and superlinear hamiltonians on compact configuration spaces abstract multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds given by lagrangians or hamiltonians which have quadratic growth in the velocities or in the momenta such results are based on the richness of the topology of the space of curves satisfying the given boundary conditions in this note we show how these results can be extended to the classical setting of tonelli lagrangians lagrangians which are c 2 convex and superlinear in the velocities or to hamiltonians which are superlinear in the momenta and have a coercive action integrand

corrigendum to the conley conjecture for hamiltonian systems on the cotangent bundle and its analogue for lagrangian systems j funct anal 256 9 2009 2967 3034 abstract in lines 8 11 of lu 2009 18 p 2977 we wrote for integer m 3 if m is c m smooth and c m 1 smooth l r t m r satisfies the assumptions l1 l3 then the functional l is c 2 smooth bounded below satisfies the palais smale condition and all critical points of it have finite morse indexes and nullities see 1 prop 4 1 4 2 and 4 however as proved in abbondandolo and schwarz 2009 2 the claim that l is c 2 smooth is true if and only if for every t q the function v l t q v is a polynomial of degree at most 2 so the arguments in lu 2009 18 are only valid for the physical hamiltonian in 1 2 and corresponding lagrangian therein in this note we shall correct our arguments in lu 2009 18 with a new splitting lemma obtained in lu 2011 20

periodic solutions of lagrangian systems on a compact manifold abstract let m be a smooth n dimensional manifold and let tm be its tangent bundle we consider a time periodic lagrangian of period t l t tm r and we seek t periodic solutions of the lagrange equations which in local coordinates are d dt l q t q q l q t q q 0 i 1 n our main result states that if the fundamental group of m is finite then has infinitely many t periodic solutions provided that l t satisfies certain physically reasonable assumptions

estimates for rankin selberg l functions and quantum unique ergodicity abstract subconvex bounds in the weight aspect for rankin selberg l functions associated to two cusp forms are established these bounds are applied to prove the equidistribution of mass conjecture for cm forms

a bifurcation result for semi riemannian trajectories of the lorentz force equation we obtain a bifurcation result for solutions of the lorentz equation in a semi riemannian manifold such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi riemannian metric and the electromagnetic field the flow of the jacobi equation along each solution preserves the so called electromagnetic symplectic form and the corresponding curve in the symplectic group determines an integer valued homology class called the maslov index of the solution we study electromagnetic conjugate instants with symplectic techniques and we prove at first an analogous of the semi riemannian morse index theorem see calculus of variations prentice hall englewood cliffs nj usa 1963 by using this result together with recent results on the bifurcation for critical points of strongly indefinite functionals see j funct anal 162 1 1999 52 we are able to prove that each non degenerate and non null electromagnetic conjugate instant along a given solution of the semi riemannian lorentz force equation is a bifurcation point

equidistribution de sous varietes speciales de points speciaux les points a multiplication complexe et un ensemble de sous varietes speciales que l on appelle sous varietes de type de hodge les definitions qui seront donnees plus tard dans le texte sont presentees de maniere tres agreable dans le papier de moonen 8 dans ce cadre andre et oort font la conjecture suivante soit y une sousvariete de s il existe un ensemble fini s1 sr de sous varietes speciales avec si c y pour tout i tel que toute variete speciale z de s contenue dans y est en fait contenue dans un des si le resultat le plus profond dans la direction de cette conjecture a ete obtenu par edixhoven et yafaev 5 on definit dans ce texte une classe assez large de sous varietes speciales que nous appellerons fortement speciales par manque d une terminologie plus adequate decrivons les sous varietes fortement speciales soit s une variete de shimura associee a une donnee de shimura g x pour un groupe algebrique adjoint sur q et une g r classe de conjugaison x de morphismes

optical hamiltonians and symplectic twist maps abstract this paper concentrates on optical hamiltonian systems of t t n i e those for which hpp is a positive definite matrix and their relationship with symplectic twist maps we present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems the novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system

floer homology for magnetic fields with at most linear growth on the universal cover abstract the floer homology of a cotangent bundle is isomorphic to loop space homology of the underlying manifold as proved by abbondandolo and schwarz salamon and weber and viterbo in this paper we show that in the presence of a dirac magnetic monopole which admits a primitive with at most linear growth on the universal cover the floer homology in atoroidal free homotopy classes is again isomorphic to loop space homology as a consequence we prove that for any atoroidal free homotopy class and any sufficiently small 0 any magnetic flow associated to the dirac magnetic monopole has a closed orbit of period belonging to the given free homotopy class in the case where the dirac magnetic monopole admits a bounded primitive on the universal cover we also prove the conley conjecture for hamiltonians that are quadratic at infinity i e we show that such hamiltonians have infinitely many periodic orbits

periodic solutions for dynamical systems on non complete riemannian manifolds abstract we look for periodic solutions in a dynamical system on a non complete riemannian manifold if the potential is bounded and suitable convexity assumptions hold the existence of infinitely many solutions can be proved by means of variational methods penalization arguments and classical ljusternik schnirelman theory

closed geodesies on compact lorentzian manifolds of splitting type in this paper we consider the problem of the existence of a spacelike closed geodesic on compact lorentzian manifolds tipler and galloway proved that under suitable topological properties of the manifold there exists a closed timelike geodesic in their proofs they use the hypothesis that the time coordinate of one timelike geodesic has derivative always different from zero this clearly fails for spacelike geodesies using variational methods and applying the relative category theory we prove the existence of a closed spacelike geodesic on a compact manifold of splitting type observe that thanks to the previous results the existence of at least two geometrically distinct closed geodesies on follows

the conley conjecture for hamiltonian systems on the cotangent bundle and its analogue for lagrangian systems abstract in this paper the conley conjecture which was recently proved by franks and handel j franks m handel periodic points of hamiltonian surface diffeomorphism geom topol 7 2003 713 756 for surfaces of positive genus hingston n hingston subharmonic solutions of hamiltonian equations on tori ann math in press for tori and ginzburg v l ginzburg the conley conjecture arxiv math sg 0610956v1 for closed symplectically aspherical manifolds is proved for c 1 hamiltonian systems on the cotangent bundle of a c 3 smooth compact manifold m without boundary of a time 1 periodic c 2 smooth hamiltonian h r t m r which is strongly convex and has quadratic growth on the fibers namely we show that such a hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations if h also satisfies h t q p h t q p for any t q p r t m it is shown that the time 1 map of the hamiltonian system if exists has infinitely many periodic points siting in the zero section of t m if m is c 5 smooth and dim m 1 h is of c 4 class and independent of time t then for any 0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of such that any one of them cannot be obtained from others by iterations or rotations these results are obtained by proving similar results for the lagrangian system of the fenchel transform of h l r t m r which is proved to be strongly convex and to have quadratic growth in the velocities yet

a combinatorial approach to the theory of ω automata a combinatorial lemma is proved and used here to derive new results on automata and to give simpler proofs of known ones in particular we reprove mcnaughton s fundamental theorem characterizing the regular sequence sets without having to construct a sophisticated automaton the theorem is obtained by coding the behaviour of automata in a second order language and a simple application of the lemma in close analogy now referring to a first order language a theory of counter free automata is developed it is shown that these automata are appropriate for characterizing the star free sequence sets finally the lemma is applied in mathematical logic here new normal form theorems and also decidability results are proved for the first order and the monadic second order theory of certain structures over the ordering of natural numbers

periodic orbits and subharmonics of dynamical systems on non compact riemannian manifolds where dt x t is the covariant derivative of x along the direction of x and r the riemannian gradient has been studied when m is a noncontractible manifold see 2 8 9 14 assuming if m is non compact the existence of a function on m convex at infinity when v is bounded the difficulties arise from the lack of compactness of m indeed in this case the action functional does not satisfy the palais smale compactness condition on the other side if v is unbounded the action functional is unbounded both from below and from above therefore neither min max methods and linking arguments can be used since the loop space is not linear nor the ljusternik schnirelmann category theory can be applied although m has a non trivial topology article no 0136

fourier expansion of eisenstein series on the hilbert modular group and hilbert class fields in this paper we consider the eisenstein series for the hilbert modular group of a general number field we compute the fourier expansion at each cusp explicitly the fourier coefficients are given in terms of completed partial hecke l series and from their functional equations we get the functional equation for the eisenstein vector that is we identify the scattering matrix when we compute the determinant of the scattering matrix in the principal case the dedekind function of the hilbert class field shows up a proof in the imaginary quadratic case was given in efrat and sarnak and for totally real fields with class number one a proof was given in efrat

determinant expression of selberg zeta functions ii we will prove that for psl 2 r and its cofinite subgroup the selberg zeta function is expressed by the determinant of the laplacian we will also give an explicit calculation in case of congruence subgroups and deduce that the part of the determinant of the laplacian composed of the continuous spectrum is expressed by dirichlet functions the first discovery of the relation between the selberg zeta function and the determinant of the laplacian was by physicists 3 4 7 sarnak 14 and voros 15 obtained the determinant expression of selberg zeta functions for compact riemann surfaces with torsionfree fundamental groups in those cases all the spectrum of the laplacians are discrete the determinant was defined via the holomorphy at the origin of the spectral zeta function of minakshisundaram and pleijel 13 for noncompact but finite riemann surfaces with torsion free fundamental groups these results are generalized by efrat 5 in this case there exist both discrete and continuous spectrum he constructs the spectral zeta function composed not only of eigenvalues but some values concerning continuous spectrum which are decided by all the poles of the scattering de terminant in the selberg trace formula the determinant of the laplacian is defined by the standard method with the holomorphy of the spectral zeta func tion at the origin the aim of the present paper is to generalize his results to the case with any fundamental group y 3 and to give some arithmetic examples of the determinant of the laplacians 4 in 4 we restrict ourselves to the case when y is a congruence subgroup of psl 2 z in this case the partial spectral zeta function composed of only eigenvalues is also holomorphic at the origin 11 theorem 3 3 hence we have a decomposition of the determinant into parts corresponding to the discrete and continuous spectrum the scatter ing determinant is expressed very explicitly by huxley 8 in terms of dirichlet 7 functions i x almost all the poles of the scattering determinant are de scribed by nontrivial zeros of l s x the continuous part of the determinant

on the theory of hilbert modular functions i arithmetic groups and eisenstein series one of the objectives of this paper is to prepare for a relatively elementary treatment of the reciprocity laws for special values of hilbert modular functions our attention was drawn to this subject by a paper of eichler 151 in which a significant portion of the theory of complex multiplication and reciprocity laws for special values of the j function was developed by elementary methods and by a modified version of eichler s treatment due to chevalley lo eichler s paper caused us to look more carefully at hecke s thesis and habilitationsschrift nos 1 and 4 of his mathematische werke where it was shown that the special values of arithmetic modular functions for the hilbert modular group of a real quadratic number field generate abelian extensions of certain cm fields and a reciprocity law for some of these special values was established subsequently sugawara 39 studied the fields generated by division values of suitably normalized abelian functions with complex multiplication however both of hecke s papers though basically correct in their thrust contained mistakes in function theory and sugawara s work though useful in some ways was marred by deficiencies therefore in the case of modular functions of several variables it remained for shimura and taniyama 37 40 to make pioneering breakthroughs in developing a general theory of complex multiplication based on the theory of abelian varieties and their fields of moduli this was developed much further by shimura 36 in providing a general theory of canonical models and deriving from it a very general reciprocity law for special values of modular functions of several variables on classical domains of symplectic type after

geodesics in stationary spacetimes and classical lagrangian systems abstract we state a fundamental correspondence between geodesics on stationary spacetimes and the equations of classical particles on riemannian manifolds accelerated by a potential and a magnetic field by variational methods we prove some existence and multiplicity theorems for fixed energy solutions joining two points or periodic of the above described riemannian equation as a consequence we obtain existence and multiplicity results for geodesics with fixed energy connecting a point to a line or periodic trajectories in standard stationary spacetimes

multiplicity of forced oscillations for the spherical pendulum acted on by a retarded periodic force abstract we prove a multiplicity result for forced oscillations of a spherical pendulum that is a massive point moving on a sphere subject to a periodic action with or without friction allowed to depend on the whole past of the motion the approach is based on topological methods in particular when the unperturbed forcing term is the gravity we obtain two harmonic forced oscillations regardless of the presence of friction and of the form of the perturbing force field

a list of arithmetical structures complete with respect to the first order definability a structure with base set n is complete with respect to the first order definability in the class of arithmetical structures if and only if the operations are definable in it a list of such structures is presented although structures with pascal s triangles modulo n are preferred a little an effort was made to collect as many simply formulated results as possible

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points ii we prove a general subconvex bound in the level aspect for rankin selberg l functions associated with two primitive holomorphic or maass cusp forms over q we use this bound to establish the equidistribution of incomplete galois orbits of heegner points on shimura curves associated with indefinite quaternion algebras over q

logically defined subsets of n k we give a characterization in terms of a restriction of semi simple sets of the class of subsets of in k definable in an extension of first order logic obtained by adjoining quantifiers which count modulo an integer it is shown that this class strictly contains the class of recognizable subsets of in k and is strictly contained in the class of rational subsets of in k links with the parallel complexity class acc0 are discussed

sparse equidistribution problems period bounds and subconvexity we introduce a geometric method to bound periods of automorphic forms the key features of this method are the use of equidistribution results in place of mean value theorems and the systematic use of mixing and the spectral gap applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of cm points to subconvexity of the triple product period in the level aspect over number fields which implies subconvexity for certain standard and rankin selberg l functions and to bounding fourier coefficients of automorphic forms

periodic trajectories on stationary lorentzian manifolds in this paper we present an existence and multiplicity result for periodic trajectories on stationary lorentzian manifolds possibly with boundary whose proof is based on a morse theory approach see 5 we recall that a lorentzian manifold is a smooth connected nite dimensional manifold m equipped with a 0 2 tensor eld g such that for any z m g z is a nondegenerate symmetric bilinear form on the tangent space tzm having exactly one negative eigenvalue moreover relativistic spacetimes are a particular class of lorentzian manifolds of dimension four

on the floer homology of cotangent bundles this paper concerns floer homology for periodic orbits and for a lagrangian intersection problem on the cotangent bundle t m of a compact orientable manifold m the first result is a new l estimate for the solutions of the floer equation which allows us to deal with a larger and more natural class of hamiltonians the second and main result is a new construction of the isomorphism between the floer homology and the singular homology of the free loop space of m in the periodic case or of the based loop space of m in the lagrangian intersection problem the idea for the construction of such an isomorphism is to consider a hamiltonian that is the legendre transform of a lagrangian on t m and to construct an isomorphism between the floer complex and the morse complex of the classical lagrangian action functional on the space of w1 2 free or based loops on m 2005 wiley periodicals inc

complexity of logical theories involving coprimality upper bounds on the computational complexity of some logical theories are proved the theory th pf n of finite subsets of n is proved to be in c 0 atime alt 2cn n the same result is proved for the theories thn th n x and th n x 2x x2 2x where denotes coprimality of numbers the theory th n is proved to be in c 0 atime alt 2cn2 n

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points keywords rankin selberg l function subconvexity heegner points reference tan article 2004 002doi 10 4007 annals 2004 160 185 record created on 2008 11 14 modified on 2017 05 12

zeta functions of binary hermitian forms and special values of eisenstein series on three dimensional hyperbolic space the present work contains a systematic investigation of zeta functions of binary hermitian forms over the ring o of integers of an imaginary quadratic number field k tl with discriminant d 0 the resulting relations between these zetafunctions have numerous applications to the computation of special values of the eisenstein series for the group psl 2 o acting on the three dimensional hyperbolic space i i in particular we reprove many known formulae for representation numbers for quaternary quadratic forms over z and we obtain many explicit new results for such representation numbers in sect 2 we develop a theory of representations of numbers by binary hermitian forms over o which parallels the classical theory of representations of integers by binary quadratic forms over a key result is a certain bijection described in theorem 2 3 this bijection means that the sum of the numbers of representations modulo sl 2 o units of an integer k 0 by a set of representatives of the sl 2 o classes of binary hermitian forms over o with discriminant a is equal to the number of cosets 2 ko with 2 o 22 a 0 m o d k the number of these cosets is explicitly computed in sect 2 in all cases hence a certain mean value of the representation numbers is known whereas simple formulae for the individual representation numbers do not exist we apply the formulae of sect 2 to the study of zeta functions of definite or indefinite hermitian forms for simplicity we restrict in this introduction to the case of positive definite forms over o although the results of chap i are valid in the indefinite case as well let f be a positive definite binary hermitian form over o with sl 2 o unit group gl f then we define