the quantum superalgebra uq osp 1 2n deformed para bose operators and root of unity representations we recall the relation between the lie superalgebra osp 1 2n and para bose operators the quantum superalgebra uq osp 1 2n defined as usual in terms of its chevalley generators is shown to be isomorphic to an associative algebra generated by so called pre oscillator operators satisfying a number of relations from these relations and the analogue with the non deformed case one can interpret these pre oscillator operators as deformed para bose operators some consequences for uq osp 1 2n cartan weyl basis poincare birkhoff win basis and its hopf subalgebra uq gl n are pointed out finally using a realization in terms of q commuting q bosons we construct an irreducible finite dimensional unitary fock representation of uq osp 1 2n and its decomposition in terms of uq gl n representations when q is a root of unity

computing the p selmer group of an elliptic curve in this paper we explain how to bound the p selmer group of an elliptic curve over a number field k our method is an algorithm which is relatively simple to implement although it requires data such as units and class groups from number fields of degree at most p 2 1 our method is practical for p 3 but for larger values of p it becomes impractical with current computing power in the examples we have calculated our method produces exactly the p selmer group of the curve and so one can use the method to find the mordell weil rank of the curve when the usual method of 2 descent fails

a laplace dunkl equation on s 2 and the bannai ito algebra the dirac dunkl operator on the two sphere associated to the mathbb z 2 3 reflection group is considered its symmetries are found and are shown to generate the bannai ito algebra representations of the bannai ito algebra are constructed using ladder operators eigenfunctions of the spherical dirac dunkl operator are obtained using a cauchy kovalevskaia extension theorem these eigenfunctions which correspond to dunkl monogenics are seen to support finite dimensional irreducible representations of the bannai ito algebra

multifractality of random walks and localized excitations on linear random fractals abstract random walks and localized excitations on topologically one dimensional random fractals such as paths generated by simple random walks in d dimensional lattices are studied these simple systems display a reach scenario of multifractal behavior in both the probability distribution of the random walks and the localization properties of the wave functions

hopf structure and green ansatz of deformed parastatistics algebras deformed parabose and parafermi algebras are revised and endowed with hopf structure in a natural way the noncocommutative coproduct allows for construction of parastatistics fock like representations built out of the simplest deformed bose and fermi representations the construction gives rise to quadratic algebras of deformed anomalous commutation relations which define the generalized green ansatz

second p descents on elliptic curves let p be a prime and let c be a genus one curve over a number field k representing an element of order dividing p in the shafarevich tate group of its jacobian we describe an algorithm which computes the set of d in the shafarevich tate group such that pd c and obtains explicit models for these d as curves in projective space this leads to a practical algorithm for performing explicit 9 descents on elliptic curves over q

on the equation y2 x x2 p generators are found for the group of rational points on the title curve for all primes p 5 mod 8 less than 1 000 the rank is always i in accordance with conjectures of selmer and mordell some of the generators are rather large

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

explicit n descent on elliptic curves iii algorithms this is the third in a series of papers in which we study the n selmer group of an elliptic curve with the aim of representing its elements as genus one normal curves of degree n the methods we describe are practical in the case n 3 for elliptic curves over the rationals and have been implemented in magma one important ingredient of our work is an algorithm for trivialising central simple algebras this is of independent interest for example it could be used for parametrising brauer severi surfaces

the algebra of dual 1 hahn polynomials and the clebsch gordan problem of sl 1 2 the algebra h of the dual 1 hahn polynomials is derived and shown to arise in the clebsch gordan problem of sl 1 2 the dual 1 hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q 1 limit of the dual q hahn polynomials the hopf algebra sl 1 2 has four generators including an involution it is also a q 1 limit of the quantum algebra slq 2 and furthermore the dynamical algebra of the parabose oscillator the algebra h a two parameter generalization of u 2 with an involution as additional generator is first derived from the recurrence relation of the 1 hahn polynomials it is then shown that h can be realized in terms of the generators of two added sl 1 2 algebras so that the clebsch gordan coefficients of sl 1 2 are dual 1 hahn polynomials an irreducible representation of h involving five diagonal matrices and connected to the difference equation of the dual 1 hahn polynomials is constructed

root numbers and ranks in positive characteristic abstract for a global field k and an elliptic curve e over k t silverman s specialization theorem implies rank e k t rank e t k for all but finitely many t p 1 k if this inequality is strict for all but finitely many t the elliptic curve e is said to have elevated rank all known examples of elevated rank for k q rest on the parity conjecture for elliptic curves over q and the examples are all isotrivial some additional standard conjectures over q imply that there does not exist a non isotrivial elliptic curve over q t with elevated rank in positive characteristic an analogue of one of these additional conjectures is false inspired by this for the rational function field k u over any finite field with characteristic 2 we construct an explicit 2 parameter family e c d of non isotrivial elliptic curves over k t depending on arbitrary c d such that under the parity conjecture each e c d has elevated rank

supersymmetry parastatistics and operator realizations of a lie algebra the algebraic structure of parastatistics has been generalized and it is found to be consistent with supersymmetric quantum mechanics with supercharges constructed out of the generalized para bose and para fermi operators it is further shown that the operator algebra of generalized parastatistics offers a realization of the graded orthosymplectic group similar to that of orthogonal and symplectic groups using conventional parastatistics

topological properties of percolation clusters percolation properties including the total cluster mass s the shell mass b and the linear geometrical size r are studied as a function of the topological chemical distance parameter l all critical exponents are shown to be related to an apparently new exponent nu defined by r approximately lv critical exponents are calculated exactly for percolation clusters on the cayley tree a model for 6d percolation for which s approximately l2 b approximately l and r2 approximately l for diffusion on such clusters one finds that l3 approximately t numerical estimates of the exponents are obtained for other dimensions a conjecture which relates nu to beta and nu is discussed

variants of bosonization in parabosonic algebra the hopf and super hopf structures in parabosonic algebra parabosonic algebra in finite or infinite degrees of freedom is considered as a 22 graded associative algebra and is shown to be a 22 graded or super hopf algebra the super hopf algebraic structure of the parabosonic algebra is established directly without appealing to its relation to the osp 1 2n lie superalgebraic structure the notion of super hopf algebra is equivalently described as a hopf algebra in the braided monoidal category cz 2 m the bosonization technique for switching a hopf algebra in the braided monoidal category h m where his a quasitriangular hopf algebra into an ordinary hopf algebra is reviewed in this paper we prove that for the parabosonic algebra p b beyond the application of the bosonization technique to the original super hopf algebra a bosonization like construction is also achieved using two operators related to the parabosonic total number operator both techniques switch the same super hopf algebra p b to an ordinary hopf algebra thus producing two different variants of p b with an ordinary hopf structure

generalized exclusion and hopf algebras we propose a generalized oscillator algebra at the roots of unity with generalized exclusion and we investigate the braided hopf structure we find that there are two solutions these are the generalized exclusions of the bosonic and fermionic types we also discuss the covariance properties of these oscillators

on boson algebras as hopf algebras certain types of generalized undeformed and deformed boson algebras which admit a hopf algebra structure are introduced together with their fock type representations and their corresponding r matrices it is also shown that a class of generalized heisenberg algebras including those underlying physical models such as that of calogero sutherland is isomorphic with one of the types of boson algebra proposed and can be formulated as a hopf algebra

visualising sha 2 in abelian surfaces given an elliptic curve e 1 over a number field k and an element s in its 2 selmer group we give two different ways to construct infinitely many abelian surfaces a such that the homogeneous space representing s occurs as a fibre of a over another elliptic curve e 2 we show that by comparing the 2 selmer groups of e 1 e 2 and a we can obtain information about iii e 1 k 2 and we give examples where we use this to obtain a sharp bound on the mordell weil rank of an elliptic curve as a tool we give a precise description of the m selmer group of an abelian surface a that is m isogenous to a product of elliptic curves e 1 x e 2 one of the constructions can be applied iteratively to obtain information about iii e 1 k 2 n we give an example where we use this iterated application to exhibit an element of order 4 in iii e 1 q

algebraic structure of parabose fock space i the green s ansatz revisited a description of the fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed a new statement of the well known green s ansatz is introduced to help understand the significance of the bilinear commutation relations it is also applied to show how all parabose oscillators of even order respectively odd order can be obtained as irreducible constituents of the fock space associated with a green s ansatz of two three terms analogs of the form in which the parabose green s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order methods used in the paper allow various new results for q deformed parabose oscillators to be derived

two cover descent on hyperelliptic curves we describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve with the property that any rational point will lift to one of the covers in particular if the algorithm returns an empty set then the hyperelliptic curve has no rational points this provides a relatively efficiently tested criterion for solvability of hyperelliptic curves we also discuss applications of this algorithm to curves of genus 1 and to curves with rational points

how to do a p descent on an elliptic curve in this paper we describe an algorithm that reduces the computation of the full p selmer group of an elliptic curve e over a number field to standard number field computations such as determining the p torsion of the s class group and a basis of the s units modulo pth powers for a suitable set s of primes in particular we give a result reducing this set s of bad primes to a very small set which in many cases only contains the primes above p as of today this provides a feasible algorithm for performing a full 3 descent on an elliptic curve over q but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory when the galois module structure of e p is favorable simplifications are possible and p descents for larger p are accessible even today to demonstrate how the method works several worked examples are included

superlocalization correlations and random walks on fractals abstract upper and lower bounds are found for the anomalous decay exponent a of localized wave functions and of various two point correlation functions on typical fractal configurations the decay of the superlocalized wave functions is used to evaluate the decay of the probability distribution of a random walk which is then used to obtain a flory like expression for self avoiding walks emphasis is placed on the differences between typical and average quantities

recursive fermion system in cuntz algebra i embeddings of the car canonical anticommutation relations algebra of fermions into the cuntz algebra 2 or 2 d more generally are presented by using recursive constructions as a typical example an embedding of car onto the u 1 invariant subalgebra of 2 is constructed explicitly generalizing this construction to the case of 2 p an embedding of car onto the u 1 invariant subalgebra of 2 p is obtained restricting a permutation representation of the cuntz algebra we obtain the fock representation of car we apply the results to embed the algebra of parafermions of order p into 2 p according to green s ansatz

second isogeny descents and the birch and swinnerton dyer conjectural formula abstract let e e be an isogeny of prime degree l between elliptic curves defined over a number field we describe how to perform descents on the nontrivial elements in the shafarevich tate group of e which are killed by the dual isogeny this makes computation of l selmer groups of elliptic curves admitting an l isogeny over q feasible for l 5 7 in cases where a descent on e is insufficient and a full l descent would be infeasible as an application we complete the verification of the full birch and swinnerton dyer conjectural formula for all elliptic curves over q of rank zero or one and conductor less than 5000

a description of the superalgebra osp 2 n 1 2 m via green generators an alternative description of the lie superalgebra osp 2n 1 2m in terms of generators and relations is given the generators called green generators are the root vectors of osp 2n 1 2m corresponding to the orthogonal roots

sieving for rational points on hyperelliptic curves we give a new and efficient method of sieving for rational points on hyperelliptic curves this method is often successful in proving that a given hyperelliptic curve suspected to have no rational points does in fact have no rational points we have often found this to be the case even when our curve has points over all localizations qp we illustrate the practicality of the method with some examples of hyperelliptic curves of genus 1

hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the jordan schwinger map the aim of this paper is to show that there is a hopf structure of the parabosonic and parafermionic algebras and this hopf structure can generate the well known hopf algebraic structure of the lie algebras through a realization of lie algebras using the parabosonic and parafermionic extension of the jordan schwinger map the differences between the hopf algebraic and the graded hopf superalgebraic structure on the parabosonic algebra are discussed

explicit isogeny descent on elliptic curves in this note we consider an isogeny descent on a pair of elliptic curves over q we assume that 3 is a prime the main result expresses the relevant selmer groups as kernels of simple explicit maps between finitedimensional f vector spaces defined in terms of the splitting fields of the kernels of the two isogenies we give examples of proving the part of the birch and swinnerton dyer conjectural formula for certain curves of small conductor