courbes algebriques et equations multiplicatives we study the intersection of an algebraic curve c lying in a multiplicative torus over bar mathbb q with the union of all algebraic subgroups of codimension 2 finiteness of this set has already been proved by bombieri masser and zannier under the assumption that c is not contained in a translate of a proper subtorus following this result the question of the minimal hypothesis implying finiteness has been raised by these authors giving rise to the conjecture finiteness holds precisely when c is not contained in a proper subgroup we prove here this statement which is also a special case of more general conjectures stated independently by zilber and pink our proof takes its inspiration from an article by remond and viada concerning the zilber pink conjecture for curves lying in a power of an elliptic curve hence it relies on a uniform version of the vojta inequality proven via the generalized vojta inequality of remond the main task is to establish a lower bound for some intersection numbers here on a whole family of surfaces obtained by blowing up a compactification of c c

ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

torsion points on families of squares of elliptic curves in a recent paper we proved that there are at most finitely many complex numbers 0 1 such that the points 2 sqrt 2 2 lambda and 3 sqrt 6 3 lambda are both torsion on the elliptic curve defined by y 2 x x 1 x here we give a generalization to any two points with coordinates algebraic over the field q and even over c this implies a special case of a variant of pink s conjecture for a variety inside a semiabelian scheme namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes

height uniformity for integral points on elliptic curves we recall the result of d abramovich and its generalization by p pacelli on the uniformity for stably integral points on elliptic curves it says that the lang vojta conjecture on the distribution of integral points on a variety of logarithmic general type implies the uniformity for the numbers of stably integral points on elliptic curves in this paper we will investigate its analogue for their heights under the assumption of the vojta conjecture basically we will show that the vojta conjecture gives a naturally expected simple uniformity for their heights

current algebra of classical non linear sigma models the current algebra of classical non linear sigma models on arbitrary riemannian manifolds is analyzed it is found that introducing in addition to the noether currentj associated with the global symmetry of the theory a composite scalar fieldj the algebra closes under poisson brackets

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

the difference between the weil height and the canonical height on elliptic curves estimates for the difference of the weil height and the canonical height of points on elliptic curves are used for many purposes both theoretical and computational in this note we give an explicit estimate for this difference in terms of the j invariant and discriminant of the elliptic curve the method of proof suggested by serge lang is to use the decomposition of the canonical height into a sum of local heights we illustrate one use for our estimate by computing generators for the mordell weil group in three examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 x ax b with a and b in the ring of integers of k the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 10 chapter viii 9 or 6 chapter vi the canonical height is determined by this property together with the fact that the difference

hilbert s tenth problem for function fields of varieties over number fields and p adic fields abstract let k be a subfield of a p adic field of odd residue characteristic and let l be the function field of a variety of dimension n 1 over k then hilbert s tenth problem for l is undecidable in particular hilbert s tenth problem for function fields of varieties over number fields of dimension 1 is undecidable

higher local conservation laws for nonlinear sigma models on symmetric spaces we show that for the classical two dimensional nonlinear model on a riemannian symmetric space of dimensionm and rankp there existp independent series of higher local conservation laws and we reduce the field equations of the model to a system of nonlinear partial differential equations possessing an associated lax pair and involvingm p independent variables

quadratic pseudopotentials for gl n c principal sigma models abstract matrix pseudopotentials for the gl n c principal sigma model in two dimensional minkowski space are derived through the method of closed differential ideals the structure of the pseudopotential equations is chosen to contain only quadratic nonlinearities of the matrix riccati type a set of further constraints is deduced which leads to inner backlund transformations depending on two complex parameters through a geometrical interpretation of such matrix riccati equations and the associated constraints a linearization procedure is derived and the equations are reduced to those of zakharov mikhailov and shabat 7 10 an inductive procedure is applied to explicitly solve an iterated sequence of such backlund transformations through a purely algebraic transformation of solutions to the first step the resulting nonlinear superposition formula is used to demonstrate a permutability theorem and to analyse the appearance of singularities fr nous deduisons les pseudo potentiels matriciels pour le modele sigma principal a valeurs dans gl n c defini sur l espace de minkowski bi dimensionnel en utilisant la methode des ideaux differentiels fermes on restreint la structure des equations des pseudopotentiels de facon a ce que n apparaissent que des non linearites quadratiques du type riccati matriciel nous deduisons un ensemble de contraintes supplementaires menant a des transformations de backlund internes dependantes de deux parametres complexes gr ce a l interpretation geometrique de ces equations de riccati matricielles et des contraintes associees nous deduisons un procede de linearisation et les equations se reduisent a celles de zakharov mikhailov et shabat 7 10 un procede inductif est utilise pour resoudre explicitement une succession de transformations de backlund par une transformation purement algebrique des solutions de la premiere etape la formule de superposition non lineaire qui en resulte permet de demontrer un theoreme de permutabilite et d analyser l apparition de singularites

linear integral transformations and hierarchies of integrable nonlinear evolution equations abstract integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations these are riemann hilbert type of integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an indefinite parameter k the potential matrices in these equations obey a set of nonlinear recursion relations leading to a heirarchy of nonlinear partial differential equations in decreasing order the same equations give rise to a reciprocal hierarchy associated with heisenberg ferromagnet type of equations central in the treatment is an embedding of the hierarchy into an infinite matrix structure which is constructed on the basis of the integral equations in terms of this infinite matrix structure the equations governing the hierarchies become quite simple furthermore it leads in a straightforward way to various generalizations such as to other types of linear spectral problems multicomponent system and lattice equations generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment finally some results on conserved densities and the hamiltonian structure are briefly discussed

special points on fibered powers of elliptic surfaces consider a fibered power of an elliptic surface we characterize its subvarieties that contain a zariski dense set of points that are torsion points in fibers with complex multiplication this result can be viewed as a mix of the manin mumford and andre oort conjecture and is related to a conjecture of pink the main technical tool is a new height inequality we also use it to give another proof of a case of gubler s result on the bogomolov conjecture over function fields

fluctuation dissipation theorems in the generic formalism the viscoelastic fluids abstract fluctuation dissipation relations fdr for a maxwell fluid are computed via the generic formalism this formalism is determined by four building blocks two potentials total energy and entropy and two matrices which determine the dynamics the fdr exhibits specific features arising from the type of entropy used in the formalism but the understanding of fluctuations in a given non equilibrium system is reduced to determining a single friction matrix we show that consistent results with fundamental laws of irreversible thermodynamics for maxwell fluid are obtained if it is used the corresponding entropy in lit if structural properties of the fluid are used as thermodynamic variables giving rise to extended irreversible thermodynamics the resulting fdr also shows the desired properties we discuss these results under the scope of the relation among levels of description and statistical properties of fluctuating thermodynamic fields

the algebraic curve of classical superstrings on ads5 s5 we investigate the monodromy of the lax connection for classical iib superstrings on ads5 s5 for any solution of the equations of motion we derive a spectral curve of degree 4 4 the curve consists purely of conserved quantities all gauge degrees of freedom have been eliminated in this form the most relevant quantities of the solution such as its energy can be expressed through certain holomorphic integrals on the curve this allows for a classification of finite gap solutions analogous to the general solution of strings in flat space the role of fermions in the context of the algebraic curve is clarified finally we derive a set of integral equations which reformulates the algebraic curve as a riemann hilbert problem they agree with the planar one loop open image in new window supersymmetric gauge theory proving the complete agreement of spectra in this approximation

existence of simple jacobian varieties of genus g with rank at least 4g 5 we prove the existence of infinitely many curves of genus g over the rational number field which have absolutely simple jacobian varieties with mordell weil rank at least 4 g 5

time structure invariance criteria for closure approximations in many areas of physics time evolution equations for moments of distributions are expressed in terms of higher order moments closure approximations are then introduced in an ad hoc fashion to reduce the higherorder moments to functions of the lower order ones herein the time structure invariance of the poisson bracket as manifested through the jacobi identity is used to derive constraint relationships on these approximations these constraints severely limit the allowable functionality of general closures and help to define the boundaries within which future investigations should concentrate s1063 651x 97 10410 x

density of rational points on del pezzo surfaces of degree one we state conditions under which the set s k of k rational points on a del pezzo surface s of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is zariski dense for example it suffices to require that the elliptic fibration s p 1 induced by the anticanonical map has a nodal fiber over a k rational point of p 1 it also suffices to require the existence of a point in s k that does not lie on six exceptional curves of s and that has order 3 on its fiber of the elliptic fibration this allows us to show that within a parameter space for del pezzo surfaces of degree 1 over r the set of surfaces s defined over q for which the set s q is zariski dense is dense with respect to the real analytic topology we also include conditions that may be satisfied for every del pezzo surface s and that can be verified with a finite computation for any del pezzo surface s that does satisfy them c 2014 elsevier inc all rights reserved

integrable extension of nonlinear sigma model we propose an integrable extension of nonlinear sigma model on the target space of hermitian symmetric space hss starting from a discussion of soliton solutions of o 3 model and an integrally extended version of it we construct general theory defined on arbitrary hss by using the coadjoint orbit method it is based on the exploitation of a covariantized canonical structure on hss this term results in an additional first order derivative term in the equation of motion which accommodates the zero curvature representation infinite conservation laws of nonlocal charges in this model are derived

derivative nonlinear schrodinger equations and hermitian symmetric spaces it is shown that to each hermitian symmetric space there corresponds an integrable system of generalised derivative nonlinear schrodinger equations nls the nonlinear terms are related to the curvature tensor of the associated symmetric space the hamiltonian form of the equations is presented the results are an extension of those presented in an earlier paper 1983 on generalised nls equations associated with symmetric and reductive homogeneous spaces

torsion anomalous points and families of elliptic curves we prove that there are at most finitely many complex lambda neq 0 1 such that two points on the legendre elliptic curve y 2 x x 1 x lambda with coordinates x 2 3 both have finite order this is a very special case of some conjectures on unlikely intersections in semiabelian schemes

a simple example for comparing generic with rational non equilibrium thermodynamics during the last 15 years a bracket formalism of dissipative continuum physics has been developed which resulted in a formulation which is shortly denoted as generic generic has been applied to different problems of continuum thermodynamics often in this way that a well known problem was reformulated in generic formalism to learn some more about the generic procedure we consider a gas which is contained in a cylinder closed by a piston moving with friction we treat this simple discrete system with rational non equilibrium thermodynamics by using liu s procedure and for comparison also with the generic formalism both different procedures yield in the same results especially in the same entropy production differences similarities and fundamental presuppositions of both formalisms are discussed and compared

torsion points on families of products of elliptic curves abstract in a recent paper we proved a special case of a variant of pink s conjecture for a variety inside a semiabelian scheme namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes here we go ahead with the programme of settling the conjecture for general abelian surface schemes by completing the proof for all non simple surfaces this involves some entirely new and crucial issues

darboux transformation and exact multisolitons of cpn nonlinear sigma model abstract exact multisolitons of c p n sigma model are obtained by using darboux transformation in terms of quasideterminant darboux matrix the multisoliton solutions are shown to be expressed in terms of quasideterminants of gelfand and retakh 1991 10 the method of quasideterminant darboux matrix has been shown to be related with the dressing method that has already been used to compute iteratively exact multisolitons of the c p n sigma model by means of nonlinear superposition principle of backlund transformations explicit expressions of one two and three soliton solutions of c p 1 model have been computed by using properties of quasideterminants

torsion points pell s equation and integration in elementary terms the main results of this paper involve general algebraic differentials omega on a general pencil of algebraic curves we show how to determine if omega is integrable in elementary terms for infinitely many members of the pencil in particular this corrects an assertion of james davenport from 1981 and provides the first proof even in rather strengthened form we also indicate analogies with work of andre and hrushovski and with the grothendieck katz conjecture to reach this goal we first provide proofs of independent results which extend conclusions of relative manin mumford type allied to the zilber pink conjectures we characterize torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2 in turn we present yet another application of the latter results to a rather general pencil of pell equations a 2 db 2 1 over a polynomial ring we determine whether the pell equation with squarefree d is solvable for infinitely many members of the pencil

on ranks of twists of elliptic curves and power free values of binary forms let e be an elliptic curve over q by the rank of e we shall mean the rank of the group of rational points of e mestre 31 improving on the work of neron 34 cf 13 39 and 46 has shown that there is an infinite family of elliptic curves over q with rank at least 12 however computational work see 3 4 7 and 47 suggests that a typical elliptic curve will have much smaller rank with curves of rank 0 or 1 being predominant indeed brumer 6 has proved subject to the birch swinnerton dyer conjecture the shimura taniyama weil conjecture and the generalized riemann hypothesis that the average rank of an elliptic curve ordered according to its faltings height is at most 2 3 in this article we shall study the behaviour of the rank as we run over twists of a given elliptic curve over q that is we shall restrict our attention to families of elliptic curves defined over q which are isomorphic over c there are families of quadratic cubic quartic and sextic twists see for example proposition 5 4 of chapter x of 42 let e be an elliptic curve over q with weierstrass equation y2 x ax b and for any non zero integer d let ed 2 3 denote a quadratic twist of e given by the equation dy2 x ax b let r d denote the rank of ed note that if d1 and d2 are non zero integers then ed is isomorphic to ed over q if and only if d1 d2 is the square of a rational number subject to the conjectures of birch and swinnerton dyer and of shimura taniyama and weil goldfeld 14 conjectured in 1979 that

homogeneous kahler manifolds paving the way towards new supersymmetric sigma models homogeneous kahler manifolds give rise to a broad class of supersymmetric sigma models containing as a rather special subclass the more familiar supersymmetric sigma models based on hermitian symmetric spaces in this article all homogeneous kahler manifolds with semisimple symmetry groupg are constructed and are classified in terms of dynkin diagrams explicit expressions for the complex structure and the kahler structure are given in terms of the lie algebra g ofg it is shown that for compactg one can always find an einstein kahler structure which is unique up to a constant multiple and for which the kahler potential takes a simple form

some series of infinitely many symmetry generators in symmetric space chiral models this paper shows that starting from any conserved current generated by some given infinitesimal symmetry generator one may use finite dual transformations to induce infinitely many infinitesimal symmetry generators thus besides starting from ordinary isotopic and space time translation this paper also discovers the infinitesimal generators for backlund transformation for dual symmetry itself and other general cases and then uses them to generate infinitely many local or nonlocal currents respectively

neron models of jacobians over base schemes of dimension greater than 1 we investigate to what extent the theory of neron models of jacobians and of abel jacobi maps extends to relative curves over base schemes of dimension greater than 1 we give a necessary and sufficient criterion for the existence of a neron model we use this to show that in general neron models do not exist even after making a modification or even alteration of the base on the other hand we show that neron models do exist outside some codimension 2 locus

bounded height in pencils of finitely generated subgroups in this paper we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric mordell lang theorem by replacing finiteness by emptyness for the intersection of varieties and subgroups all moving in a pencil except for bounded height values of the parameters and excluding identical relations more precisely an instance of the results is as follows consider the torus scheme g r m c over a curve c defined over q and let be a subgroup scheme generated by finitely many sections satisfying some necessary conditions further let v be any subscheme then there is a bound for the height of the points p c q such that for some which does not generically lie in v p lies in the fiber vp we further offer some direct diophantine applications to illustrate once again that the results implicitly contain information absent from the previous bounds in this context

preservation of thermodynamic structure in model reduction based on the availability of an invariant manifold we develop a model reduction procedure that preserves thermodynamic structure more concretely we construct the poisson and irreversible brackets of the general equation for the nonequilibrium reversible irreversible coupling of nonequilibrium thermodynamics by means of the ideas originally introduced for handling constraints the general ideas are then applied to the kramers problem that is the description of transitions between two potential wells separated by a high barrier this example reveals how a fortuitous cancellation mechanism that allows a logarithmic entropy to generate a linear diffusion equation is inherited by a master equation resulting from model reduction

log fano varieties over function fields of curves consider a smooth log fano variety over the function field of a curve suppose that the boundary has positive normal bundle choose an integral model over the curve then integral points are zariski dense after removing an explicit finite set of points on the base curve