mazur tate elements of nonordinary modular forms we establish formulae for the iwasawa invariants of mazur tate elements of cuspidal eigenforms generalizing known results in weight 2 our first theorem deals with forms of medium weight and our second deals with forms of small slope we give examples illustrating the strange behavior which can occur in the high weight high slope case

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

a formula for the derivative of the p adic l function of the symmetric square of a finite slope modular form let f be a modular form of weight k and nebentypus psi by generalizing a construction of dabrowski and delbourgo we construct a p adic l function interpolating the special values of the l function l s rm sym 2 f otimes xi where xi is a dirichlet character when s k 1 and xi psi 1 this p adic l function vanishes due to the presence of a so called trivial zero we give a formula for the derivative at s k 1 of this p adic l function when the form f is steinberg at p if the weight of f is even the conductor is even and squarefree and the nebentypus is trivial this formula implies a conjecture of benois

the decay mode solutions for the cylindrical kp equation abstract the decay mode solutions for the cylindrical kadomtsev petviashvili equation can be obtained by the backlund transformation and hirota method

linear superposition principle of hyperbolic and trigonometric function solutions to generalized bilinear equations linear subspaces of hyperbolic and trigonometric function solutions to generalized bilinear equations are analyzed necessary and sufficient conditions are presented to apply the linear superposition principles applications of an algorithm using weights are made together with a few concrete examples

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

on the eigencurve at classical weight 1 points we show that the p adic eigencurve is smooth at classical weight one points which are regular at p and give a precise criterion for etaleness over the weight space at those points our approach uses deformations of galois representations

rational solutions of a differential difference kdv equation the toda equation and the discrete kdv equation in this paper a series of rational solutions are presented for a differential difference analogue of the kdv equation the toda equation and the discrete kdv equation these rational solutions are obtained using hirota s bilinear formalism and backlund transformations the crucial step is the use of nonlinear superposition formulae

a rigid analytic gross zagier formula and arithmetic applications let f be a newform of weight 2 and squarefree level n its fourier coefficients generate a ring of whose fraction field kf has finite degree over q fix an imaginary quadratic field k of discriminant prime to n corresponding to a dirichlet character e the l series l f k s l f s l f 0 e s of f over k has an analytic continuation to the whole complex plane and a functional equation relating l f k s to l f k 2 s assume that the sign of this functional equation is 1 so that l f k s vanishes to even order at s 1 this is equivalent to saying that the number of prime factors of n which are inert in k is odd fix any such prime say p the field k determines a factorization n n nof n by taking n resp nto be the product of all the prime factors of n which are split resp

coupling integrable couplings and bi hamiltonian structure associated with the boiti pempinelli tu hierarchy a kind of new lie algebra is introduced whose corresponding loop algebra is followed to define from which two isospectral problems are established whose compatibility condition gives rise to a coupling integrable coupling of the boiti pempinelli tu bpt hierarchy by making use of the quadratic form identity its bi hamiltonian structure is obtained finally we reduce the coupling integrable coupling to obtain two integrable couplings of the bpt hierarchy furthermore the well known mkdv equation and a new coupled equation with two dependent variables are produced respectively

the main conjecture for cm elliptic curves at supersingular primes at a prime of ordinary reduction the iwasawa main conjecture for elliptic curves relates a selmer group to a p adic l function in the supersingular case the statement of the main conjecture is more complicated as neither the selmer group nor the p adic l function is well behaved recently kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case namely kobayashi s conjecture relates modified selmer groups which he defined with modified padic l functions defined by the first author in this paper we prove kobayashi s conjecture for elliptic curves with complex multiplication

p adic l functions and p adic periods of modular forms let e be an elliptic curve which is defined over q and has stable reduction modulo a given prime p assuming that e is modular one can associate to e a p adic l function lp e s see mz swd a v vi mz t t for its construction in various cases this function is defined by a certain interpolation property and is analytic for sezp in this paper we will assume that e has split multiplicative reduction at p under this assumption the interpolation property implies that lp e 1 0 we will prove a formula for ep e 1 which was discovered experimentally by mazur tate and teitelbaum mz t t by tate s p adic urfiformization theory there is a p adic integer q epzp which we refer to as the tate period for e and a p adic analytic isomorphism

quasi periodic waves and an asymptotic property for the asymmetrical nizhnik novikov veselov equation based on a multi dimensional riemann theta function the hirota bilinear method is extended to explicitly construct multi periodic quasi periodic wave solutions for the asymmetrical nizhnik novikov veselov equation among these periodic waves two periodic waves are a direct generalization of well known cnoidal waves their surface pattern is two dimensional the main physical result is the description of the behavior of nonlinear waves in shallow water a limiting procedure is presented to analyze asymptotic properties of the two periodic waves in details relations between the periodic wave solutions and the well known soliton solutions are established it is rigorously shown that the periodic wave solutions tend to the soliton solutions under a small amplitude limit

universal eigenvarieties trianguline galois representations and p adic langlands functoriality using the overconvergent cohomology modules introduced by ash and stevens we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces building on work of ash stevens urban and others we also formulate a precise modularity conjecture linking trianguline galois representations with overconvergent cohomology classes in the course of giving evidence for this conjecture we establish several new instances of p adic langlands functoriality our main technical innovations are a family of universal coefficients spectral sequences for overconvergent cohomology and a generalization of chenevier s interpolation theorem

on bilinear representations and infinite conservation laws of a nonlinear variable coefficient equation the kdv equation for non uniform media with relaxation effects has extensive applications in mathematics and physics satsuma 1976 20 and hirota 1979 21 once investigated its algebraic properties in the paper we shall generalize it a more general integrable equation with variable coefficients and further discuss the corresponding bilinear representation backlund transformation lax pair and infinite conservation laws by making use of the bell polynomials

quantitative non vanishing of dirichlet l values modulo p let p be an odd prime and k a non negative integer let n be a positive integer such that p not mid n and lambda a dirichlet character modulo n we obtain quantitative lower bounds for the number of dirichlet character chi modulo f with the critical dirichlet l value l k lambda chi being p indivisible here f rightarrow infty with n f 1 and p not mid f phi f we explore the indivisibility via an algebraic and a homological approach the latter leads to a bound of the form f 1 2 the p indivisibility yields results on the distribution of the associated p selmer ranks we also consider an iwasawa variant it leads to an explicit upper bound on the lowest conductor of the characters factoring through the iwasawa mathbb z ell extension of mathbb q for an odd prime ell ne p with the corresponding critical l value twists being p indivisible

the periodic wave solutions for a 2 1 dimensional akns equation in this paper a 2 1 dimensional akns equation is considered bilinear backlund transformation is presented to explicitly construct periodic wave solutions based on a multidimensional riemann theta function then the one and two periodic wave solutions are presented and it is also shown that the soliton solutions can be reduced from the periodic wave solutions

an integrable symmetric 2 1 dimensional lotka volterra equation and a family of its solutions a symmetric 2 1 dimensional lotka volterra equation is proposed by means of a dependent variable transformation the equation is firstly transformed into multilinear form and further decoupled into bilinear form by introducing auxiliary independent variables a bilinear backlund transformation is found and then the corresponding lax pair is derived explicit solutions expressed in terms of pfaffian solutions of the bilinear form of the symmetric 2 1 dimensional lotka volterra equation are given as a special case of the pfaffian solutions we obtain soliton solutions and dromions

on the birch and swinnerton dyer conjecture let f be a finite extension of the rational field q if e is an elliptic curve defined over f then the mordell weil group e f of points on e with coordinates in f is a finitely generated abelian group let l e f s be the hasseweil zeta function of e over f which at least in the case where e has complex multiplication is known to be an analytic function on the entire complex plane birch and swinnerton dyer have conjectured that l e f s has a zero at s l of order precisely equal to the rank of e f over z the strongest result known in support of this conjecture is the following theorem of coates and wiles 4

geometric deformations of modular galois representations let f be a newform on 1 n and v f the 2 dimensional p adic galois representation attached to f let s be a finite set of primes containing the primes divisors of np and denote by adv f the adjoint of v f under some mild conditions on f we show that h1 g g s adv f 0 using this result we show that the universal deformation space of the residual representation attached to f is smooth and 3 dimensional at the point corresponding to f when f has finite slope one can also use this result to give a deformation theoretic description of the eigencurve of coleman mazur at the point corresponding to f

hirota bilinear equations with linear subspaces of hyperbolic and trigonometric function solutions linear superposition principles of hyperbolic and trigonometric function solutions are analyzed for hirota bilinear equations with an aim to construct a specific sub class of n soliton solutions formulated by linear combinations of hyperbolic and trigonometric functions an algorithm using weights is discussed and a few illustrative application examples are presented

une correspondance de jacquet langlands p adique we extend the jacquet langlands correspondence between the hecke modules of usual and quaternionic modular forms to overconvergent p adic forms of finite slope we show that this correspondence respects p adic families and is induced by an isomorphism between some associated eigencurves

some remarks on the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication we establish several results towards the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields namely i the existence of an appropriate p adic l function building on works of hida and perrin riou ii the basic structure theory of the dual selmer group following works of coates hachimori venjakob et al and iii the implications of dihedral or anticyclotomic main conjectures with basechange the result of i is deduced from the construction of hida and perrin riou which in particular is seen to give a bounded distribution the result of ii allows us to deduce a corank formula for the p primary part of the tate shafarevich group of an elliptic curve in the z p 2 extension of an imaginary quadratic field finally iii allows us to deduce a criterion for one divisibility of the two variable main conjecture in terms of specializations to cyclotomic characters following a suggestion of greenberg as well as a refinement via basechange c 2011 elsevier inc all rights reserved

negative order korteweg de vries equations in this paper based on the regular korteweg de vries kdv system we study negative order kdv nkdv equations particularly their hamiltonian structures lax pairs conservation laws and explicit multisoliton and multikink wave solutions thorough bilinear backlund transformations the nkdv equations studied in our paper are differential and actually derived from the first member in the negative order kdv hierarchy the nkdv equations are not only gauge equivalent to the camassa holm equation through reciprocal transformations but also closely related to the ermakov pinney systems and the kupershmidt deformation the bi hamiltonian structures and a darboux transformation of the nkdv equations are constructed with the aid of trace identity and their lax pairs respectively the single and double kink wave and bell soliton solutions are given in an explicit formula through the darboux transformation the one kink wave solution is expressed in the form of tanh while the one bell soliton is in the form of sech and both forms are very standard the collisions of two kink wave and two bell soliton solutions are analyzed in detail and this singular interaction differs from the regular kdv equation multidimensional binary bell polynomials are employed to find bilinear formulation and backlund transformations which produce n soliton solutions a direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the nkdv equations furthermore the relations between quasiperiodic wave solutions and soliton solutions are clearly described finally we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions

critical p adic l functions we attach p adic l functions to critical modular forms and study them we prove that those l functions fit in a two variables p adic l function defined locally everywhere on the eigencurve

p adic interpolation of square roots of central l values of modular forms we construct a meromorphic function on part of the eigencurve that interpolates a square root of a ratio of quadratic twists of the central modular l value

two variable p adic l functions attached to eigenfamilies of positive slope consider a classical cusp eigenform f n 1 a n f q n of weight k 2 for 0 n with a dirichlet character mod n and let l f s n 1 n a n f n s denote the l function of f twisted with an arbitrary dirichlet character for a prime number p 5 consider a family of cusp eigenforms f k of weight k k f k n 1 a n f k q n containing f f k such that the fourier coefficients a n f k are given by certain p adic analytic functions k a n f k the purpose of this paper is to construct a two variable p adic l function attached to coleman s family f k of cusp eigenforms of a fixed positive slope v p p 0 where p p k is an eigenvalue which depends on k of the atkin operator u u p our p adic l function interpolates the special values l f k s at points s k with s 1 2 k 1 we give a construction using the rankin selberg method and the theory of p adic integration on a profinite group y with values in an affinoid k algebra a where k is a fixed finite extension of q p our p adic l functions are p adic mellin transforms of certain a valued measures in their turn such measures come from eisenstein distributions with values in certain banach a modules m m n a of families of overconvergent forms over a

quasi periodic waves of the n 1 supersymmetric modified korteweg de vries equation abstract based on the hirota bilinear method and the riemann theta function a straightforward way is shown to construct quasi periodic wave solutions of supersymmetric equations the resulting theory is applied to the supersymmetric modified korteweg de vries equation further we analyze the asymptotic properties of the solutions and give their asymptotic relations between the quasi periodic wave solutions and the soliton solutions

algorithms for the arithmetic of elliptic curves using iwasawa theory we explain how to use results from iwasawa theory to obtain information about p parts of tate shafarevich groups of specific elliptic curves over q our method provides a practical way to compute x e q p in many cases when traditional p descent methods are completely impractical and also in situations where results of kolyvagin do not apply e g when the rank of the mordell weil group is greater than 1 we apply our results along with a computer calculation to show that x e q p 0 for the 1 534 422 pairs e p consisting of a non cm elliptic curve e over q with conductor 30 000 rank 2 and good ordinary primes p with 5 p 1000 and surjective mod p representation

on l functions of elliptic curves and cyclotomic towers in this paper we shall strengthen a recent theorem of ralph greenberg 1 concerning l functions of elliptic curves and anticyclotomic towers let k be an imaginary quadratic field and m an abelian extension of k possibly of infinite degree we say that the extension m k is anticyclotomic if m is galois over q and if the nontrivial element of gal k q acts on gal m k by inversion now fix a finite set p of rational primes and consider the compositum in an algebraic closure of k of all anticyclotomic extensions of k which are unramified outside p we denote this compositum by l and refer to it as the maximal anticyclotomic extension of k unramified outside p the field l may also be described as the union of all ring class fields of k with conductor divisible only by primes in p class field theory shows that gal l k is isomorphic to the product of a finite group and the group

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

overconvergent modular forms and the fontaine mazur conjecture we prove a conjecture of fontaine and mazur on modularity of representations of g which are potentially semi stable at p for representations coming from finite slope overconvergent eigenforms we also give an application of our technique to a question of gouvea on overconvergence of certain modular forms if is a suitable representation of g s on a two dimensional vector space over a finite field product of m and the s a finite set of primes we construct a rigid analytic subspace of the universal deformation space of which is defined by a purely representation theoretic condition and contains the eigencurve of coleman mazur conjecturally it is equal to this curve

integrability of the modified generalised vakhnenko equation integrability of the modified generalised vakhnenko equation is investigated systematically based on binary bell polynomials its bilinear representation n soliton solutions bilinear backlund transformation and lax pair are succinctly constructed moreover the conservation laws of the modified generalised vakhnenko equation are discussed by using corresponding lax pair furthermore the quasiperiodic solution of the modified generalised vakhnenko equation is presented by applying hirota direct method and riemann theta function the asymptotic behavior of the one periodic wave is analyzed in details it is shown that the one periodic wave solution tends to the one soliton solution under a small amplitude limit 0 finally the new n soliton solutions of the standard vakhnenko equation are presented it would be specially mentioned that all the results of modified generalised vakhnenko equation can be reduced to the generalised vakhnenko equation and standard vakhnenko equation under the special case of 1 and 1 0 respectively

p adic banach spaces and families of modular forms let p be a prime cp the completion of an algebraic closure of the p adic numbers qp and k a nite extension of qp contained in cp let v be the valuation on cp such that v p 1 and let be the absolute value on cp such that x p v x for x cp suppose n is a positive integer prime to p let x1 np denote the modular curve over k which represents elliptic curves with 1 np structure and let up be the hecke operator on modular forms on x1 np which takes a form with q expansion n anq n to the modular form with q expansion n anpq n a modular form f is said to have slope q if there is a polynomial r t over cp such that r up f 0 and such that the newton polygon of r t has only one side and its slope is

exceptional zero formulae and a conjecture of perrin riou let a mathbf q be an elliptic curve with split multiplicative reduction at a prime p we prove an analogue of a conjecture of perrin riou relating p adic beilinson kato elements to heegner points in a mathbf q and a large part of the rank one case of the mazur tate teitelbaum exceptional zero conjecture for the cyclotomic p adic l function of a more generally let f be the weight two newform associated with a let f infty be the hida family of f and let l p f infty k s be the mazur kitagawa two variable p adic l function attached to f infty we prove a p adic gross zagier formula expressing the quadratic term of the taylor expansion of l p f infty k s at k s 2 1 as a non zero rational multiple of the extended height weight of a heegner point in a mathbf q

on the p adic l function of a modular form at a supersingular prime in this paper we study the two p adic l functions attached to a modular form f sum a sb nq sp n at a supersingular prime p when a sb p 0 we are able to decompose both the sum and the difference of the two unbounded distributions attached to f into a bounded measure and a distribution that accounts for all of the growth moreover this distribution depends only upon the weight of f and the fact that a sb p vanishes from this description we explain how the p adic l function is controlled by two iwasawa functions and by two power series with growth which have a fixed infinite set of zeros theorem 5 1 asymptotic formulas for the p part of the analytic size of the tate shafarevich group of an elliptic curve in the cyclotomic direction are computed using this result these formulas compare favorably with results established by m kurihara in 11 and b perrin riou in 23 on the algebraic side moreover we interpret kurihara s conjectures on the galois structure of the tate shafarevich group in terms of these two iwasawa functions

iwasawa theory for elliptic curves at supersingular primes we give a new formulation in iwasawa theory for elliptic curves at good supersingular primes this formulation is similar to mazur s at good ordinary primes namely we define a new selmer group and show that it is of cotorsion then we formulate the iwasawa main conjecture as that the characteristic ideal is generated by pollack s p adic l function we show that this main conjecture is equivalent to kato s and perrin riou s main conjectures we also prove an inequality in the main conjecture by using kato s euler system in terms of the and the invariants of our selmer group we specify the numbers and in the asymptotic formula for the order of the tate shafarevich group by kurihara and perrin riou

quasi periodic wave solutions for the 2 1 dimensional generalized calogero bogoyavlenskii schiff cbs equation abstract the hirota bilinear method is implemented to directly construct quasi periodic wave solutions in terms of theta functions for a hirota bilinear equation the result is applied to the 2 1 dimensional generalized calogero bogoyavlenskii schiff cbs equation thereby yielding their quasi periodic wave solutions under the backlund transformation u y 3 l n f x y t x

two periodic waves and asymptotic property for generalized 2d toda lattice equation abstract in this paper two periodic wave solutions are constructed for the 2 1 dimensional generalized toda lattice equation by using hirota bilinear method and riemann theta function at the same time we analyze in details asymptotic properties of the two periodic wave solutions and give their asymptotic relations between the periodic wave solutions and the soliton solutions

a vector asymmetrical nnv equation soliton solutions bilinear backlund transformation and lax pair a vector asymmetrical nizhnik novikov veselov nnv equation is proposed based on its bilinear form soliton solutions expressed by pfaffians are obtained bilinear backlund transformation and the corresponding lax pair for the vector annv equation are derived

elliptic curves with complex multiplication and the conjecture of birch and swinnerton dyer 1 quick review of elliptic curves 2 2 elliptic curves over c 4 3 elliptic curves over local fields 6 4 elliptic curves over number fields 12 5 elliptic curves with complex multiplication 15 6 descent 22 7 elliptic units 27 8 euler systems 37 9 bounding ideal class groups 43 10 the theorem of coates and wiles 47 11 iwasawa theory and the main conjecture 50 12 computing the selmer group 61

an integrable hierarchy and darboux transformations bilinear backlund transformations of a reduced equation a new integrable hierarchy of evolution equations is obtained by making use of a lie algebra and tu ma scheme from which a new generalized broer kaup gbk equation is produced then two kinds of darboux transformations the bilinear presentation the bilinear backlund transformation and the new lax pair of the gbk equation are generated respectively by employing the bell polynomials

hirota bilinear equations with linear subspaces of solutions we explore when hirota bilinear equations possess linear subspaces of solutions first we establish a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions to hirota bilinear equations second we show that multivariate polynomials whose zeros form a vector space can generate the desired hirota bilinear equations with given linear subspaces of solutions and formulate such multivariate polynomials by using multivariate polynomials which have one and only one zero third applying an algorithm using weights we present parameterizations of wave numbers and frequencies achieved by using one parameter to compute the desired hirota bilinear equations