a non hypersingular time domain biem for 3 d transient elastodynamic crack analysis a three dimensional 3 d time domain boundary integral equation method biem is presented for transient elastodynamic crack analysis a non hypersingular traction bie formulation is used with the crack opening displacements and their derivatives as unknown quantities a collocation method in conjunction with a time stepping scheme is developed to solve the non hypersingular time domain bies to simplify the analysis and to describe the proper behaviour of the unknown quantities at the crack front a constant spatial shape function is applied for elements away from the crack front while a spatial square root crack tip shape function is adopted for elements near the crack front a linear temporal shape function is used in the time stepping scheme numerical calculations have been carried out for penny shaped and square cracks results for the elastodynamic stress intensity factors are presented as functions of the temporal and the spatial variables for the test examples considered good agreement between the present results and those of other authors is obtained

on the singular values of weber modular functions the minimal polynomials of the singular values of the classical weber modular functions give far simpler defining polynomials for the class fields of imaginary quadratic fields than the minimal polynomials of singular moduli of level 1 we describe computations of these polynomials and give conjectural formulas describing the prime decomposition of their resultants and discriminants extending the formulas of gross zagier for the level 1 case

diffraction of elastic waves by a penny shaped crack consider an infinite elastic solid containing a penny shaped crack we suppose that time harmonic elastic waves are incident on the crack and are required to determine the scattered displacement field u i in this paper we describe a new method for solving the corresponding linear boundary value problem for u i which we denote by s we begin by defining an elastic double layer we prove that any solution of s can be represented by an elastic double layer whose density satisfies certain conditions we then introduce various green functions and define a new crack green function g ij that is discontinuous across the crack next we use g ij to derive a fredholm integral equation of the second kind for the discontinuity in u i across the crack we prove that this equation always has a unique solution hence we are able to prove that the original boundary value problem s always possesses a unique solution and that this solution has an integral representation as an elastic double layer whose density solves an integral equation of the second kind

an arithmetic intersection formula for denominators of igusa class polynomials in this paper we prove an explicit formula for the arithmetic intersection number rm cm k rm g 1 ell on the siegel moduli space of abelian surfaces generalizing the work of bruinier yang and yang these intersection numbers allow one to compute the denominators of igusa class polynomials which has important applications to the construction of genus 2 curves for use in cryptography bruinier and yang conjectured a formula for intersection numbers on an arithmetic hilbert modular surface and as a consequence obtained a conjectural formula for the intersection number rm cm k rm g 1 ell under strong assumptions on the ramification of the primitive quartic cm field k yang later proved this conjecture assuming that cal o k is freely generated by one element over the ring of integers of the real quadratic subfield in this paper we prove a formula for rm cm k rm g 1 ell for more general primitive quartic cm fields and we use a different method of proof than yang we prove a tight bound on this intersection number which holds for it all primitive quartic cm fields as a consequence we obtain a formula for a multiple of the denominators of the igusa class polynomials for an arbitrary primitive quartic cm field our proof entails studying the embedding problem posed by goren and lauter and counting solutions using our previous article that generalized work of gross zagier and dorman to arbitrary discriminants

rank one drinfeld modules on elliptic curves the sgn normalized rank one drinfeld modules 0 associated with all elliptic curves e over fq for 4 q 13 are computed in explicit form such 0 for q 4 were computed previously these computations verify a conjecture of dorman on the norm of j q aq l and also suggest some interesting new properties of 0 we prove dorman s conjecture in the ramified case we also prove the formula deg n a q hk 1 q where n a is the norm of a and hk is the class number of k fq e we describe a remarkable conjectural property of the trace of a in even characteristic that holds in all the examples in his recent paper 1 on the factorization of norms of j invariants of ranktwo drinfeld modules with complex multiplications d dorman conjectured that such norms are monic elements of fq x dorman computed the jinvariants in several examples and found his conjecture valid in all of them for this purpose he used the rank one examples from 6 the first extensive computational test of the conjecture was carried out by one of us dummit who computed the rank one drinfeld modules associated with all elliptic curves over fq with q 13 see 3 below the results of these computations may be found on the microfiche card included at the end of this issue inspired by these computations one of us hayes proved dorman s conjecture when the infinite place is ramified see 2 below in 3 we describe an algorithm for computing the rank one drinfeld modules associated with any hyperelliptic curve over fqq and we prove some basic attributes of the algorithm in 4 we prove formulas for the degrees of the norm and trace of the j invariants of drinfeld modules associated with elliptic curves these formulas were first observed computationally they suggest a number of interesting questions for elliptic curves with complex multiplications in characteristic zero in 5 we state some conjectures about the remarkable form of the trace term in characteristic two these conjectures are supported by all our computations

special values of hypergeometric functions and periods of cm elliptic curves let x x 0 6 1 w 6 be the quotient of the shimura curve x 0 6 1 by all the atkin lehner involutions by realizing modular forms on x in two ways one in terms of hypergeometric functions and the other in terms of borcherds forms and using schofer s formula for values of borcherds forms at cm points we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over overline q with complex multiplication

a variational boundary integral equation method for an elastodynamic antiplane crack this paper investigates the transient wave scattering by a crack the author has developed a new formulation to solve the bie for the crack opening displacement the solution is represented by means of a retarded double layer potential and the resulting bie with the cod as unknown has a hypersingular kernel the corresponding difficulty is overcome by using a variational method

cm values of hilbert modular functions this is a joint work with jan bruinier and is a generalization of the well known work of gross and zagier on singular moduli gz here is the main result for detail please see by let p 1 mod 4 be a prime number and f q p we write of for the ring of integers of f and x 7 x for the conjugation in f let sl2 of be the hilbert modular group associated to f the corresponding hilbert modular surface x h2 is a normal quasi projective algebraic variety defined over q let k f be a non biquadratic quartic cm number field containing f with discriminant dk p q for some prime q 1 mod 4 technical condition let and be the complex embeddings of k given by and then 1 and 1 are two cm types let cm k be the formal sum of cm points in x of cm type k by ok then cm k cm k cm k is an 0 cycle on x defined over q if is a rational modular function on x then cm k is a rational number an interesting and in general very hard question is to find a factorization formula for this number we did it successfully when is a borcherds product or equivalently has its divisor supported on the hirzebruch zagier divisors which were constructed in their seminar work in 1970 s hz let k be the reflex field of k with real quadratic subfield f for a nonzero element t d 1 k f relative discriminant and a prime ideal l of f we define

using reciprocity to derive the far field displacements due to buried sources and scatterers it is shown that elastodynamic reciprocity provides a simpler approach for deriving the far field displacements due to buried sub surface sources in a half space compared with integral transform techniques the auxiliary fields employed in this approach are the fields associated with the reflection of plane waves of the three possible polarisations and the required far field can be expressed in terms of these well known auxiliary fields the crucial step in this approach is to evaluate a surface integral involving cross work terms between an outgoing spherical wavefront and the auxiliary fields consisting of incident and reflected plane waves this integral can be evaluated by the stationary phase approximation for the two dimensional case or by a generalisation of this approximation for the three dimensional case although this evaluation involves several distinct contributions the final result is shown to be very simple and it can be interpreted as a generalisation of a known result for the one d

nonvanishing of hecke l functions and the bloch kato conjecture in this paper we study the central values of l functions associated to a large class of algebraic hecke characters of imaginary quadratic fields when these central values are nonzero the bloch kato conjecture predicts an exact formula for the algebraic parts of the central values in terms of periods and arithmetic data most notably the selmer groups corresponding to the hecke characters we investigate the nonvanishing of these central values and prove the p part of the bloch kato conjecture in these cases for primes p which split in k

nonlocal lagrange formalism in the thermodynamics of irreversible processes variational procedures for kinetic equations abstract this paper is concerned with generalizations of the known local lagrange formalism of first order it will be applied to kinetic equations like the fokker planck equation and the boltzmann equation in the latter case nonlocal methods are necessary from the very beginning nevertheless in the framework of frechet s formalism the calculations are as easy as in the classical local case furthermore a rather general entropy concept can be established within nonlocal lagrange formalism for irreversible systems as a main result of this paper we derive within our general concept the known entropy balances of the boltzmann theory and the fokker planck theory respectively it will be emphasized that our general concept may be applied to a very wide class of irreversible systems in principle

a p adic algorithm to compute the hilbert class polynomial classically the hilbert class polynomial p z x of an imaginary quadratic discriminant a is computed using complex analytic techniques in 2002 couveignes and henocq suggested a p adic algorithm to compute p unlike the complex analytic method it does not suffer from problems caused by rounding errors in this paper we give a detailed description of the algorithm in the paper by couveignes and henocq and our careful study of the complexity shows that if the generalized riemann hypothesis holds true the expected runtime of the p adic algorithm is o log 8 e instead of o 1 e we illustrate the algorithm by computing the polynomial p 639 using a 643 adic algorithm

harmonic maass forms of weight 1 the object of this paper is to initiate a study of the fourier coefficients of a weight one harmonic maass form and relate them to the complex galois representation associated to a weight one newform which is the form s image under a certain differential operator in this paper our focus will be on weight one dihedral newforms of prime level p 3 mod 4 in this case we give properties of the fourier coefficients that are similar to and sometimes reduce to cases of stark s conjectures on derivatives of l functions we also give a new modular interpretation of certain products of differences of singular moduli studied by gross and zagier finally we provide some numerical evidence that the fourier coefficients of a mock modular form whose shadow is exotic are similarly related to the associated complex galois representation

time domain boundary element analysis of elastic wave interaction with a crack the mathematical formulation of the problem of transient wave interaction with a crack in a homogeneous isotropic linearly elastic solid has been reduced to the solution of an integral equation over the insonified crack face the integral equation relates the unknown crack opening displacement which depends on time and position to the incident wave field the integral equation has been solved numerically by a time stepping method in conjunction with a boundary element discretization of the crack surface for normal incidence of a longitudinal step stress wave on a penny shaped crack results as functions of time have been obtained for the crack opening displacement the elastodynamic mode i stress intensity factor and the scattered far field

comments on the grad procedure for the fokker planck equation if the kinetic equation of a macroscopic system is expanded with respect to the velocity in terms of orthogonal functions e g in terms of hermite functions one obtains an infinite hierarchy of equations for the expansion coefficients grad s method consists in truncating this hierarchy and investigating the remaining finite system in this paper we set up conditions under which this procedure is rigorously justified in case of the fokker planck equation

transcendental lattices and supersingular reduction lattices of a singular k3 surface a k3 surface x defined over a field k of characteristic 0 is called singular if the neron severi lattice ns x of x k is of rank 20 let x be a singular k3 surface defined over a number field f for each embedding f c we denote by t x the transcendental lattice of the complex k3 surface x obtained from x by for each prime p of f at which x has a supersingular reduction xp we define l x p to be the orthogonal complement of ns x in ns xp we investigate the relation between these lattices t x and l x p as an application we give a lower bound for the degree of a number field over which a singular k3 surface with a given transcendental lattice can be defined

restriction of coherent hilbert eisenstein series in pure appl math q 1 305 340 2005 yang constructed a family of coherent hilbert eisenstein series and asked several questions concerning the span of their diagonal restrictions in this paper we will answer one of them where the construction essentially comes from a biquadratic number field k surprisingly the span depends on the non vanishing of the central critical value of the l function associated to a weight 2 eigenform

hilbert class polynomials and traces of singular moduli where q e the values of j z at imaginary quadratic arguments in the upper half of the complex plane are known as singular moduli singular moduli are algebraic integers which play prominent roles in classical and modern number theory see c bcsh for example hilbert class fields of imaginary quadratic fields are generated by singular moduli furthermore isomorphism classes of elliptic curves with complex multiplication are distinguished by singular moduli throughout let d 0 3 mod 4 be a positive integer so that d is the discriminant of an order in an imaginary quadratic field and let h d be the hurwitz kronecker class number for the discriminant d let qd be the set of positive definite integral binary quadratic forms note including imprimitive forms if there are any

a novel derivation of non hypersingular time domain bies for transient elastodynamic crack analysis abstract a novel application of conservation integrals in transient elastodynamic crack analysis is presented to derive non hypersingular time domain boundary integral equations bies the new derivation is based on an elastodynamic conservation integral which is employed to express the displacement gradients in terms of an integral over the surface of the crack bies are obtained by substituting this representation integral into hooke s law and by taking a limit process the bies obtained in this manner are valid for arbitrary crack configurations and they are immediately nonhypersingular the laplace and the fourier transforms of these bies agree with those obtained by other authors by using the conventional derivation in conjunction with regularization techniques numerical examples show that the time domain bies presented here can yield highly accurate results for elastodynamic stress intensity factors

a nonsplit sum of coefficients of modular forms we introduce and study certain truncated sums of hecke eigenvalues of gl2 automorphic forms along quadratic polynomials a power saving estimate is established and new applications to moments of critical l values associated to quadratic fields are derived an application to the asymptotic behavior of the height of heegner points and singular moduli is discussed in detail

algebraic formulas for the coefficients of half integral weight harmonic weak maass forms abstract we prove that the coefficients of certain weight 1 2 harmonic maass forms are traces of singular moduli for weak maass forms to prove this theorem we construct a theta lift from spaces of weight 2 harmonic weak maass forms to spaces of weight 1 2 vector valued harmonic weak maass forms on mp 2 z a result which is of independent interest we then prove a general theorem which guarantees with bounded denominator when such maass singular moduli are algebraic as an example of these results we derive a formula for the partition function p n as a finite sum of algebraic numbers which lie in the usual discriminant 24 n 1 ring class field

central values of hecke l functions of cm number fields 0 introduction it is well known that the zeta function of cm complex multiplication abelian varieties can be given in terms of l functions of associated hecke characters in this paper we prove a formula expressing the central special value of the l function of certain hecke characters in terms of theta functions the formula easily implies that the central value is nonnegative and yields a criterion for its positivity combining this criterion with the work of arthaud and rubin we show that certain cm elliptic curves have mordell weil rank zero over their field of definition let f be a totally real number field of degree t and let be a quadratic hecke character of f of conductor f such that 2f f 1 given a cm extension e of f we consider the twist can of by a canonical hecke character can of

equidistribution of heegner points and ternary quadratic forms we prove new equidistribution results for galois orbits of heegner points with respect to single reduction maps at inert primes the arguments are based on two different techniques primitive representations of integers by quadratic forms and distribution relations for heegner points our results generalize an equidistribution result with respect to a single reduction map established by cornut and vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow moreover for fixed fundamental discriminant and variable conductor we deduce an effective surjectivity theorem for the reduction map from heegner points to supersingular points at a fixed inert prime our results are applicable to the setting considered by kolyvagin in the construction of the heegner points euler system

rankin selberg l functions in the level aspect keywords moments rankin selberg convolution level aspect convexity breaking reference tan article 2002 003doi 10 1215 s0012 7094 02 11416 1 record created on 2008 11 14 modified on 2017 05 12

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

a spectral proof of class number one we continue our previous work on the subject of re proving the heegner baker stark theorem giving another effective resolution of this conjecture of gauss namely there are exactly 9 imaginary quadratic fields mathbf q sqrt q with class number 1 specifically the list is q in 3 4 7 8 11 19 43 67 163 we again follow an ansatz of goldfeld considering a modular form l function of analytic rank 2 and reducing the situation to showing sufficient cancellation of the dirichlet series coefficients when restricted to the principal form previously we then chose a specific rank 2 elliptic curve with complex multiplication by mathbf q sqrt 1 and deduced the desired cancellation from an equi distribution result of hooley s for roots of a quadratic polynomial to varying moduli herein we instead use spectral techniques filling in details of work of templier and tsimerman to complete the proof relying on the duke iwaniec bound for fourier coefficients of half integral weight maass forms unlike the previous method we do not require complex multiplication and any analytic rank 2 modular form whose level has an even number of prime factors suffices

fricke s two valued modular equations the modular equation of order b is a polynomial relation between j z and j z b which has astronomically large coefficients even for small values of b fricke showed that a two valued relation exists for 37 small values of b this relation would have much smaller coefficients and would also be convenient for finding singular moduli although fricke produced no two valued relations explicitly no doubt because of the tedious amount of algebraic manipulation they are now found by use of macsyma for 31 cases ranging from b 2 to 49 fricke provided the equations necessary to generate the relations with two corrections required the remaining six cases of order 39 41 47 50 59 71 require an extension of fricke s methods using the discriminant function theta functions and power series approximations

mixed variational principles for dynamic response of thermoelastic and poroelastic continua abstract the primary objective of the present work is to make further connections between variational methods on the one hand and reversible and irreversible thermodynamics on the other this begins with the development of a new stationary principle involving mixed field variables for continuum problems in infinitesimal dynamic thermoelasticity by defining lagrangian and dissipation functions in terms of physically relevant contributions and invoking the rayleigh formalism for damped systems we are able to recover the governing equations of thermoelasticity as the euler lagrange equations this includes the balance laws of linear momentum and entropy energy the constitutive models for elastic response and heat conduction and the natural boundary conditions by including energy contributions associated with second sound phenomena one eliminates the paradox of infinite thermal propagation speeds and the resulting set of governing equations has an elegant symmetry which is most easily seen in the fourier wave number domain a related formulation for dynamic poroelasticity yields two new stationary mixed variational principles depending upon the selection of primary field variables these governing equations can also exhibit an elegant structure which can deepen our understanding of the underlying phenomena and the thermoelastic poroelastic analogy in addition to the theoretical significance the variational formulations developed here can provide the basis for a class of optimization based methods for computational mechanics

uniform distribution of heegner points let e be a modular elliptic curve over q of conductor n let k denote an imaginary quadratic field of discriminant d with n d 1 if p is a prime then there exists a unique zp extension k k such that gal k q acts nontrivially on gal k k the field k is called the anticyclotomic zp extension of k let e k denote the mordell weil group of e over k then a fundamental conjecture of mazur maz predicts that the size of e k is controlled by the prime factorization of n in k equivalently mazur s conjecture relates the size of the mordell weil group to the sign in the functional equation of certain l series the conjecture was verified by greenberg rohrlich and rubin in what mazur calls the exceptional case when e has complex multiplication by k more generally they settled the conjecture for certain abelian varieties with complex multiplication for a discussion of this cm case we refer the reader to gre roh and rub our goal in this paper is to treat the generic case which occurs either when e has no cm or when the field of complexmultiplications is distinct from k under certain conditions on e and k mazur s conjecture predicts that the group e k is finitely generated our main result asserts that this is in fact the case at least when p is an ordinary prime for e or when the class number of k is prime to p the main new ingredient we introduce is that of equidistribution following ideas used by ferrero andwashington to study the cyclotomic invariant more precisely we show that theheegner points associated to definite quaternion algebras are uniformly distributed on the components of a certain curve x and that the elements of a certain galois group act independently in a suitable sense this combined with a special value formula due to gross allows us to conclude that the special values of anticyclotomic l functions are almost always nonzero so that the statement about the mordell weil groups follows from the machinery of euler systems as developed by bertolini and darmon bd

trilinear forms and the central values of triple product l functions we give an explicit formula for certain global trilinear forms that appear in jacquet s conjecture in terms of local trilinear forms and the central values of triple product l functions

twisted borcherds products on hilbert modular surfaces and their cm values we construct a family of rational functions m on a hilbert modular surface from the classical j invariant and its hecke translates these functions are obtained by means of a multiplica tive analogue of the doi naganuma lifting and can be viewed as twisted borcherds products we then study when the value of m at a cm point associated to a nonbiquadratic quartic cm field generates the cm class field of the reflex field for the real quadratic field q 5 we factorize the norm of some of these cm values to q 5 numerically

cm values of regularized theta lifts and harmonic weak maas forms of weight 1 we study special values of regularized theta lifts at complex multiplication cm points in particular we show that cm values of borcherds products can be expressed in terms of finitely many fourier coefficients of certain harmonic weak maas forms of weight 1 as it turns out these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve our results imply a conjecture of duke and li and give a new proof of the modularity of a certain arithmetic generating series of weight 1 studied by kudla rapoport and yang

choosing the correct elliptic curve in the cm method we give an elementary way to distinguish between the twists of an ordinary elliptic curve e over f p in order to identify the one with p 1 2u points when p u 2 dv 2 with 2u 2v and e is constructed using the cm method for finding elliptic curves with a prescribed number of points our algorithms consist in most cases of reading off simple congruence conditions on u and v modulo 4

an arithmetic intersection formula on hilbert modular surfaces in this paper we obtain an explicit arithmetic intersection formula on a hilbert modular surface between the diagonal embedding of the modular curve and a cm cycle associated to a non biquadratic cm quartic field this confirms a special case of the author s conjecture with j bruinier in by and is a generalization of the beautiful factorization formula of gross and zagier on singular moduli as an application we proved the first non trivial non abelian chowla selberg formula a special case of colmez conjecture

faltings heights of cm cycles and derivatives of l functions we study the faltings height pairing of arithmetic special divisors and cm cycles on shimura varieties associated to orthogonal groups we compute the archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a rankin l function we prove the conjecture in certain cases where the shimura variety has dimension 0 1 or 2 in particular we obtain a new proof of the gross zagier formula

transient elastic wave analysis of 3 d large scale cavities by fast multipole bem using implicit runge kutta convolution quadrature abstract existing boundary element methods bems have been proven to be efficient numerical techniques particularly in modeling wave propagation problems but still remain limitations in solving large scale problems the main disadvantages may be caused by several problems including high cost and large amount of computer memory for the computation a time domain bem in which a collocation method is used for the time discretization also shows numerical instability for small cfl courant friedrichs lewy number in this work we propose a new three dimensional 3 d approach for solving transient elastic wave propagation problems of large scale spherical and spheroidal cavities by the convolution quadrature bem accelerated by fast multipole method fmm using an implicit runge kutta scheme in abbreviation we name it as irk based cq fmbem two special techniques pertaining to the fmm including the scaling of modified spherical bessel functions and the truncation method are used to enhance the efficiency and stability of our proposed method this approach is found to be particularly suitable for 3 d large scale problems as its high accuracy stability of time marching process and computational efficiency these properties are subsequently illustrated through numerical examples dealing with large scale wave problems of 3 d single spherical and spheroidal cavities and multiple equally and irregularly spherical cavities the accuracy of the present formulation is verified by comparing the obtained results with available reference solutions in the literature some numerical aspects of time increments sizes and topological shapes of cavities and their influences on the deformations and the waveforms are investigated also detailed investigation of the computational efficiency of our proposed method such as cpu time required memory etc is presented all the implementation tasks are carried out using the tokyo tech supercomputer which is called tsubame 2 5 42

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

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

a single region time domain bem for dynamic crack problems abstract a time domain boundary element method which allows the analysis of dynamic crack problems by using a single region formulation is presented the present method generates the distinct set of boundary integral equations by applying the displacement equation to one of the crack surfaces and the traction equation to the other the boundary of the structure is divided into continuous semi discontinuous and discontinuous quadratic elements the temporal variation of boundary displacements and tractions is approximated by piecewise linear and constant functions respectively the dynamic stress intensity factors are calculated using the crack opening displacements and the path independent integral this method is used to study dynamic behaviour of stationary cracks in finite and infinite domains in two dimensional analysis the results for two examples are compared with other reported solutions showing good agreement

complex multiplication cycles and kudla rapoport divisors ii we study the intersections of special cycles on a unitary shimura variety of signature n 1 1 and show that the intersection multiplicities of these cycles agree with fourier coecients of eisenstein series the results are new cases of conjectures of kudla and suggest a gross zagier theorem for unitary shimura varieties

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