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

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

finding regular simple paths in graph databases we consider the following problem given a labelled directed graph g and a regular expression r find all pairs of nodes connected by a simple path such that the concatenation of the labels along the path satisfies r the problem is motivated by the observation that many recursive queries in relational databases can be expressed in this form and by the implementation of a query language bf g based on this observation we show that the problem is in general intractable but present an algorithm than runs in polynomial time in the size of the graph when the regular expression and the graph are free of conflicts we also present a class of languages whose expressions can always be evaluated in time polynomial in the size of both the graph and the expression and characterize syntactically the expressions for such languages

on arithmetic fundamental lemmas we present a relative trace formula approach to the gross zagier formula and its generalization to higher dimensional unitary shimura varieties as a crucial ingredient we formulate a conjectural arithmetic fundamental lemma for unitary rapoport zink spaces we prove the conjecture when the rapoport zink space is associated to a unitary group in two or three variables

a probabilistic model for quadrature rules the approximate formula of quadrature rules is usually indicated as a bf x email protected i 1nw if x i where x i are the integration nodes and w i are the corresponding coefficients during last decades various cases of above type formula such as gauss quadrature rules newton cotes rules and so on have been introduced to somehow estimate x i and w i in this paper we propose a quadrature model based on a probabilistic approach to estimate the related nodes and coefficients for this purpose we apply the normal distribution as a special case and present an error analysis for the obtained probabilistic formulas some numerical examples are given in this way to show the efficiency of the proposed model

algorithms for chow heegner points via iterated integrals let e q be an elliptic curve of conductor n and let f be the weight 2 newform on g0 n associated to it by modularity building on an idea of s zhang an article by darmon rotger and sols describes the construction of so called chow heegner points pt f e q indexed by algebraic correspondences t x0 n x0 n it also gives an analytic formula depending only on the image of t in cohomology under the complex cycle class map for calculating pt f numerically via chen s theory of iterated integrals the present work describes an algorithm based on this formula for computing the chow heegner points to arbitrarily high complex accuracy carries out the computation for all elliptic curves of rank 1 and conductor n 100 when the cycles t arise from hecke correspondences and discusses several important variants of the basic construction

a basic class of symmetric orthogonal polynomials using the extended sturm liouville theorem for symmetric functions abstract in this research by applying the extended sturm liouville theorem for symmetric functions a basic class of symmetric orthogonal polynomials bcsop with four free parameters is introduced and all its standard properties such as a generic second order differential equation along with its explicit polynomial solution a generic orthogonality relation a generic three term recurrence relation and so on are presented then it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class they are respectively the generalized ultraspherical polynomials generalized hermite polynomials and two other sequences of symmetric polynomials which are finitely orthogonal on and can be expressed in terms of the mentioned class directly in this way two half trigonometric sequences of orthogonal polynomials as special sub cases of bcsop are also introduced

labeled cuts in graphs in this paper we study labeled extensions of the classical s t mincut problem in which we are given a graph g v e two specific vertices s t v a set l of labels and a labeling e l of the edges the goal is to choose a subset l l of labels so that s and t become disconnected when deleting the edges with labels in l we give an algorithm with an o n 2 3 approximation factor guarantee which improves the o m approximation guarantee of zhang et al 2009 16 we also consider variants in which selected subsets of paths between s and t have to be removed instead of all paths these labeled cut problems are much harder than the classical mincut problem

formal language constrained path problems given an alphabet sigma a directed graph g whose edges are weighted and sigma labeled and a formal language l subseteq sigma the formal language constrained shortest simple path problem consists of finding a shortest simple path p in g complying with the additional constraint that l p in l here l p denotes the unique word obtained by concatenating the sigma labels of the edges along the path p the main contributions of this paper include the following we show that the formal language constrained shortest path problem is solvable efficiently in polynomial time when l is restricted to be a context free language cfl when l is specified as a regular language we provide algorithms with improved space and time bounds in contrast we show that the problem of finding a simple path between a source and a given destination is np hard even when l is restricted to fixed simple regular languages and to very simple classes of graphs e g complete grids for the class of treewidth bounded graphs we show that i the problem of finding a regular language constrained simple path between source and destination is solvable in polynomial time and ii the extension to finding cfl constrained simple paths is np complete our results extend the previous results in siam j comput 24 1995 pp 1235 1258 proceedings of the 76th annual meeting of the transportation research board 1997 and proceedings of the 9th acm sigact sigmod sigart symposium on database systems 1990 pp 230 242 several additional extensions and applications of our results in the context of transportation problems are presented for instance as a corollary of our results we obtain a polynomial time algorithm for the best k similar path problem studied in proceedings of the 76th annual meeting of the transportation reasearch board 1997 the previous best algorithm was given by proceedings of the 76th annual meeting of the transportation research board 1997 and takes exponential time in the worst case

examples of abelian surfaces with everywhere good reduction we describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction these examples provide evidence for the eichler shimura conjecture for hilbert modular forms over a real quadratic field several of the examples also support a conjecture of brumer and kramer on abelian varieties associated to siegel modular forms with paramodular level structures

characterizations of δ volterra lattice a symmetric orthogonal polynomials interpretation abstract in this paper we introduce the volterra lattice which is interpreted in terms of symmetric orthogonal polynomials it is shown that the measure of orthogonality associated with these systems of orthogonal polynomials evolves in t like 1 x 2 1 t x where is a given positive borel measure moreover the volterra lattice is related to the toda lattice from miura or backlund transformations the main ingredients are orthogonal polynomials which satisfy an appell condition with respect to the forward difference operator and the characterization of the point spectrum of a jacobian operator that satisfies a volterra equation lax type theorem we also provide an explicit example of solutions of volterra and toda lattices and connect this example with the results presented in the paper

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

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

a technique for the numerical solution of initial value problems based on a class of birkhoff type interpolation method in this paper a class of birkhoff type interpolation problems on arbitrary nodal points is studied the explicit representation characterization the uniqueness and the error function are explicitly given furthermore we apply the obtained birkhoff type interpolation method to find i the numerical solution of high order initial value problems ivps and the corresponding error of this approximation ii the approximation of some special functions with their explicit error functions and iii new interpolatory type quadrature formulae of precision degree at least m n 1 and m kn 1 m n email protected n n k 2 numerical examples are included to demonstrate the validity and applicability of the technique proposed in this paper and a comparison is made with the existing results the results reveal that the new method is effective simple and accurate

incomplete symmetric orthogonal polynomials of finite type generated by a generalized sturm liouville theorem in a classical case orthogonal polynomial sequences are in such a way that the nth polynomial has the exact degree n such sequences are complete and form a basis of the space for any arbitrary polynomial in this paper we introduce some incomplete sets of finite orthogonal polynomials that do not contain all degrees but they are solutions of some symmetric generalized sturm liouville problems although such polynomials do not possess all properties as in classical cases they can be applied to functions approximation theory as we will compute their explicit norm square values

hida families and rational points on elliptic curves let e be an elliptic curve over q of conductor n mp having a prime p n of multiplicative reduction because e is modular it corresponds to a normalised weight two eigenform on 0 n whose q expansion is denoted f n anq letx hom z p z p z p 1 z zp which contains z as a dense subset by associating to k z the character x xk 2 denote by a u the ring of cp valued p adic analytic functions on a compact open subset u of x hida s theory associates to f a neighborhood u of 2 x which can be assumed for simplicity to be contained in the residue class of 2 modulo p 1 and a formal q expansion

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

approximate strong separation with application in fractional graph coloring and preemptive scheduling in this paper we show that approximation algorithms for the weighted independent set and s dimensional knapsack problem with ratio a can be turned into approximation algorithms with the same ratio for fractional weighted graph coloring and preemptive resource constrained scheduling in order to obtain these results we generalize known results by grotschel lovasz and schrijver on separation non emptiness test optimization and violation in the direction of approximability

characterizations of generalized hermite and sieved ultraspherical polynomials a new characterization of the generalized hermite polynomials and of the orthogonal polynomials with respect to the measure ixy 1 x2 1 2dx is derived which is based on a reversing property of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of laguerre and jacobi polynomials a similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind these results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order

twists of hilbert modular forms the theory of newforms for hilbert modular forms is summarized including a statement of a strong multiplicity one theorem and a characterization of newforms as eigenfunctions for a certain involution whose dirichlet series has a prescribed euler product the general question of twisting hilbert modular newforms by arbitrary fecke characters is considered and the exact level of a character twist of a hilbert modular form is determined conditions under which the twist of a newform is a newform are given applications include a strengthening in the elliptic modular case of a theorem of atkin and li s regarding the characterization of imprimitive newforms as well as its generalization to the hilbert modular case and a decomposition theorem for certain spaces of newforms as the direct sum of twists of spaces of newforms of lower level

orders in quaternion algebras on etudie certains ordres dans des algebres de quaternions et on developpe une formule de trace calculable pour les matrices de brandt associees a ces ordres speciaux

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

sieved ultraspherical polynomials the continuous q ultraspherical polynomials contain a number of important examples as limiting or special cases one of these arose in allaway s ph d thesis in a previous paper we solved a characterization problem essentially equivalent to allaway s and showed that these polynomials arose from the q ultraspherical polynomials when q approached a root of unity a second class of such polynomials is found and the recurrence relation and orthogonality relation are found for each of these polynomials the orthogonality is interesting because the weight function has a finite number of zeros in 1 1 generating functions and other formulas are also found

a basic class of symmetric orthogonal functions with six free parameters in a previous paper we introduced a basic class of symmetric orthogonal functions bcsof by an extended theorem for sturm liouville problems with symmetric solutions we showed that the foresaid class satisfies the differential equation x 2 px 2 q f n x x rx 2 s f n x l nx 2 1 1 n c 2 f n x 0 where l n n q 1 1 1 n 2 r n 1 q 1 1 1 n 2 p email protected s q 1 q and contains four important sub classes of symmetric orthogonal functions moreover for q 1 it is reduced to a basic class of symmetric orthogonal polynomials bcsop which respectively generates the generalized ultraspherical polynomials generalized hermite polynomials and two other sequences of finite symmetric orthogonal polynomials in this paper again by using the extended theorem we introduce a further basic class of symmetric orthogonal functions with six parameters and obtain its standard properties we show that the new class satisfies the equation x 2 px 2 q f n x x rx 2 s f n x a nx 2 1 nc d f n x 0 in which c d are two free parameters and a n denotes eigenvalues corresponding to the defined class we then introduce four orthogonal sub classes of the foresaid class and study their properties in detail since the introduced class is a generalization of bcsof for email protected 2 the four mentioned sub classes naturally generalize the generalized ultraspherical polynomials generalized hermite polynomials and two sequences of finite classical symmetric orthogonal polynomials again

a new type of weighted quadrature rules and its relation with orthogonal polynomials in this research we introduce a new type of weighted quadrature rules as x x f x p m 1 x f dx i 1ai m f m b i m r m n f in which p m 1 x f m 1 j 0 f j x j jl x is a positive function f m x denotes the mth derivative of the function f x and r m n f is the error function we determine the error function analytically and obtain the unknowns a i m b i m n l 1 explicitly so that the above formula is exact for all polynomials of degree at most 2n m 1 in particular we emphasize on the sub case x f x f dx i 1 a i 1 f b i 1 r 1 n f with the precision 2n one degree higher than gauss quadrature precision degree and show that under some specific conditions the two foresaid formulas can be connected to the current weighted quadrature rules the best application of the case m 1 in the second formula is when is a known root of the function f x for instance x x g t dt and x x g x dx are two samples in which f 0 finally we present various analytic examples of above rules and introduce a more general form of the mentioned formulas as x f x p m 1 x f dx n i 1 k j 0 di m j r i r m k n f

un analogue du calcul de heegner on demontre les deux resultats suivants a si p est un nombre premier impair congru a 2 modulo 9 la courbe elliptique x 3 y 3 2pz 3 possede une infinite de points rationnels b si p est un nombre premier congru a 5 modulo 9 la courbe elliptique x 3 y 3 2p 2 z 3 possede une infinite de points rationnels

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

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

on sieved orthogonal polynomials iii orthogonality on several intervals on introduit deux generalisations des polynomes de chelryshev dont le spectre continu est x 2 c 1 c t k x 2 c 1 c ou c est un parametre positif on definit les fonctions poids on obtient les fonctions generatrices et les formules explicites

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

a basic class of symmetric orthogonal polynomials of a discrete variable abstract by using a generalization of sturm liouville problems in discrete spaces a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters which generalizes all classical discrete symmetric orthogonal polynomials is introduced the standard properties of these polynomials such as a second order difference equation an explicit form for the polynomials a three term recurrence relation and an orthogonality relation are presented it is shown that two hypergeometric orthogonal sequences with 20 different weight functions can be extracted from this class moreover moments corresponding to these weight functions can be explicitly computed finally a particular example containing all classical discrete symmetric orthogonal polynomials is studied in detail

fast algorithms for polynomial interpolation integration and differentiation for functions tabulated at chebyshev nodes on an interval spectral interpolation and indefinite integration can be performed stably and efficiently via the fast fourier transform for many other sets of nodes such as the gaussian nodes on an interval the classical interpolation and indefinite integration schemes are stable but slow requiring o n 2 arithmetic operations with n the number of nodes in the discretization of the interval in this paper a group of algorithms is presented for the efficient evaluation of lagrange polynomial interpolants at multiple points on the line and for the rapid indefinite integration and differentiation of functions tabulated at nodes other than chebyshev the interpolation scheme requires o n cdot log 1 varepsilon arithmetic operations and o n cdot log n n cdot log 1 varepsilon operations are required for the integration and differentiation schemes where varepsilon is the precision of computations and n is the number of nodes t

the p adic gross zagier formula for elliptic curves at supersingular primes let p be a prime number and let e be an elliptic curve defined over of conductor n let k be an imaginary quadratic field with discriminant prime to pn such that all prime factors of n split in k b perrin riou established the p adic gross zagier formula that relates the first derivative of the p adic l function of e over k to the p adic height of the heegner point for k when e has good ordinary reduction at p in this article we prove the p adic gross zagier formula of e for the cyclotomic p extension at good supersingular prime p our result has an application for the full birch and swinnerton dyer conjecture suppose that the analytic rank of e over is 1 and assume that the iwasawa main conjecture is true for all good primes and the p adic height pairing is not identically equal to zero for all good ordinary primes then our result implies the full birch and swinnerton dyer conjecture up to bad primes in particular if e has complex multiplication and of analytic rank 1 the full birch and swinnerton dyer conjecture is true up to a power of bad primes and 2

a generalization of fourier trigonometric series in this paper by using the extended sturm liouville theorem for symmetric functions we introduce the differential equation f n t n a 1 1 n 2 2 a a 1 cos 2t 1 1 n 2 f n t 0 as a generalization of the differential equation of trigonometric sequences sinnt n 1 and cosnt n 0 for a 0 and obtain its explicit solution in a simple trigonometric form we then prove that the obtained sequence of solutions is orthogonal with respect to the constant weight function on 0 p and compute its norm square value explicitly one of the important advantages of this generalization is to find some new infinite series a practical example is given in this sense

nonvanishing of l series associated to cubic twists of elliptic curves has a rational solution an extensive compilation of the older history of the problem is given in dickson 6 within the past century researchers have tried to exploit either explicitly or unknowingly the fact that the curve 0 1 is in fact an elliptic curve during the nineteenth century lucas and later sylvester used a descent argument to prove that 0 1 had no solution for infinitely many d in certain congruence classes mod 9 and 18 see 6 ch xxi zagier and kramarz 19 have produced a great deal of numerical evidence about the l series of the curves based on these computations they have argued heuristically that for

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

stark heegner points and the cohomology of quaternionic shimura varieties let f be a totally real field of narrow class number one and let e f be a modular semistable elliptic curve of conductor n 1 l etk f be a non cm quadratic extension with disc k n 1 such that the sign in the functional equation of l e k s is 1 suppose further that there is a prime p n that is inert in k w e describe a p adic construction of points on e which we conjecture to be rational over ring class fields of k f and satisfy a shimura reciprocity law these points are expected to behave like classical heegner points and can be viewed as new instances of the stark heegner point construction of 5 the key idea in our construction is a reinterpretation of darmon s theory of modular symbols and mixed period integrals in terms of group cohomology

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

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

canonical models of arithmetic 1 e curves the fuchsian groups of signature 1 e are the simplest class of fuchsian groups for which the calculation of the corresponding quotient of the upper half plane presents a challenge this paper considers the finite list of arithmetic 1 e groups we define canonical models for the associated quotients by relating these to genus 1 shimura curves these models are then calculated by applying results on the mathfrak p adic uniformization of shimura curves and hilbert modular forms

a basic class of symmetric orthogonal functions using the extended sturm liouville theorem for symmetric functions by using the extended sturm liouville theorem for symmetric functions we introduced a basic class of symmetric orthogonal polynomials bcsop in a previous paper the mentioned class satisfies a differential equation of the formx 2 px 2 q f n x x rx 2 s f n x n r n 1 p x 2 1 1 n s 2 f n x 0and contains four main sequences of symmetric orthogonal polynomials in this paper again by using the mentioned theorem we introduce a basic class of symmetric orthogonal functions bcsof as a generalization of bcsop and obtain its standard properties we show that the latter class satisfies the equationx 2 px 2 q f n x x rx 2 s f n x a nx 2 1 1 n b 2 f n x 0 in which b is a free parameter and email protected n denotes eigenvalues corresponding to bcsof we then consider four sub classes of defined orthogonal functions class and study their properties in detail since bcsof is a generalization of bcsop for b s the four mentioned sub classes respectively generalize the generalized ultraspherical polynomials generalized hermite polynomials and two other finite sequences of symmetric polynomials which were introduced in the previous work

numerical solution of variable order fractional nonlinear quadratic integro differential equations based on the sixth kind chebyshev collocation method abstract in this paper a sixth kind chebyshev collocation method will be considered for solving a class of variable order fractional nonlinear quadratic integro differential equations v ofnqides the operational matrix of variable order fractional derivative for sixth kind chebyshev polynomials is derived and then a collocation approach is employed to reduce the v ofnqide to a system of nonlinear algebraic equations convergence analysis of the proposed method is evaluated and the rate of convergence is established finally some numerical test examples are investigated to validate the accuracy and robustness of the proposed approach

on constructing new expansions of functions using linear operators let t u be two linear operators mapped onto the function f such that u t f f but t u f f in this paper we first obtain the expansion of functions t u f and u t f in a general case then we introduce four special examples of the derived expansions first example is a combination of the fourier trigonometric expansion with the taylor expansion and the second example is a mixed combination of orthogonal polynomial expansions with respect to the defined linear operators t and u in the third example we apply the basic expansion u t f f x to explicitly compute some inverse integral transforms particularly the inverse laplace transform and in the last example a mixed combination of taylor expansions is presented a separate section is also allocated to discuss the convergence of the basic expansions t u f and u t f

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

triple product formula and the subconvexity bound of triple product l function in level aspect in this paper we derive a nice general formula for the local integrals of triple product formula whenever one of the representations has sufficiently higher level than the other two as an application we generalize venkatesh and woodbury s work on the subconvexity bound of triple product l function in level aspect allowing joint ramifications higher ramifications general unitary central characters and general special values of local epsilon factors

hilbert modular forms and the gross stark conjecture let f be a totally real eld and an abelian totally odd character of f in 1988 gross stated a p adic analogue of stark s conjecture that relates the value of the derivative of the p adic l function associated to and the p adic logarithm of a p unit in the extension of f cut out by in this paper we prove gross s conjecture when f is a real quadratic eld and is a narrow ring class character the main result also applies to general totally real elds for which leopoldt s conjecture holds assuming that either there are at least two primes above p in f or that a certain condition relating the l invariants of and 1 holds this condition on l invariants is always satised when is quadratic

on constructing new interpolation formulas using linear operators and an operator type of quadrature rules let t u be two linear operators mapped onto the function f such that u t f f but t u f f in this paper interpolating the functions of type t u f is presented in a general case as a special case the linear operators t f l xf t dt and u f df x dx are considered to interpolate the family of incomplete special functions three new examples of interpolation formulas together with their analytic error are also given as the special samples of the mentioned operator method finally by using the foresaid method a basic class of operator type quadrature rules is defined and its properties are investigated

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