on elliptic curves of prime power conductor over imaginary quadratic fields with class number 1 the main result of this paper is to extend from q to each of the nine imaginary quadratic fields of class number one a result of serre 1987 and mestre oesterle 1989 namely that if e is an elliptic curve of prime conductor then either e or a 2 3 or 5 isogenous curve has prime discriminant for four of the nine fields the theorem holds with no change while for the remaining five fields the discriminant of a curve with prime conductor is either prime or the square of a prime the proof is conditional in two ways first that the curves are modular so are associated to suitable bianchi newforms and second that a certain level lowering conjecture holds for bianchi newforms we also classify all elliptic curves of prime power conductor and non trivial torsion over each of the nine fields in the case of 2 torsion we find that such curves either have cm or with a small finite number of exceptions arise from a family analogous to the setzer neumann family over q

on the ultimate complexity of factorials it has long been observed that certain factorization algorithms provide a way to write the product of many different integers succinctly in this paper we study the problem of representing the product of all integers from 1 to n i e n by straight line programs formally we say that a sequence of integers an is ultimately f n computable if there exists a nonzero integer sequence mn such that for any n anmn can be computed by a straight line program using only additions subtractions and multiplications of length at most f n shub and smale 12 showed that if n is ultimately hard to compute then the algebraic version of np p is true assuming a widely believed number theory conjecture concerning smooth numbers in a short interval a subexponential upper bound exp c log n log log n for the ultimate complexity of n is proved in this paper and a randomized subexponential algorithm constructing such a short straight line program is presented as well

on stationary renewal reward processes where most rewards are zero we consider a stationary version of a renewal reward process i e a renewal process where a random variable called a reward is associated with each renewal the rewards are nonnegative and i i d but each reward may depend on the distance to the next renewal we give an explicit bound for the total variation distance between the distribution of the accumulated reward over the interval 0 l and a compound poisson distribution the bound depends in its simplest form only on the first two joint moments of t and y or i y 0 where t is the distance between successive renewals and y is the reward if t and y are independent and le y or lp y 0 is bounded or y binary valued then the bound is o e y as e y 0 or o p y 0 as p y 0 0 to prove our result we generalize a poisson approximation theorem for point processes by barbour and brown derived using stein s method and palm theory to the case of compound poisson approximation and combine this theorem with suitable couplings

the homological torsion of psl_2 of the imaginary quadratic integers denote by q sqrt m with m a square free positive integer an imaginary quadratic number field and by a its ring of integers the bianchi groups are the groups sl 2 a we reveal a correspondence between the homological torsion of the bianchi groups and new geometric invariants which are effectively computable thanks to their action on hyperbolic space we expose a novel technique the torsion subcomplex reduction to obtain these invariants we use it to explicitly compute the integral group homology of the bianchi groups furthermore this correspondence facilitates the computation of the equivariant k homology of the bianchi groups by the baum connes conjecture which is verified by the bianchi groups we obtain the k theory of their reduced c algebras in terms of isomorphic images of their equivariant k homology

torsion homology growth and cycle complexity of arithmetic manifolds let m be an arithmetic hyperbolic 3 manifold such as a bianchi manifold we conjecture that there is a basis for the second homology of m where each basis element is represented by a surface of low genus and give evidence for this we explain the relationship between this conjecture and the study of torsion homology growth

proving modularity for a given elliptic curve over an imaginary quadratic field we present an algorithm to determine if the l series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree by the work of harris soudry taylor taylor and berger harcos cf hst93 tay94 and bh we can associate to an automorphic representation a family of compatible p adic representations our algorithm is based on faltings serre s method to prove that p adic galois representations are isomorphic modularity for rational elliptic curves was one of the biggest achievements of last century little is known for general number fields in the case of totally real number fields some techniques do apply but the result is far from being proven the case of not totally real fields is more intractable to taylor wiles machinery in this paper we present an algorithm to determine if the l series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree or not the algorithm is based on faltings serre s method to prove isomorphism of p adic galois representation by the work of harris soudry taylor taylor and berger harcos cf hst93 tay94 and bh we can associate to an automorphic representation a family of compatible p adic representations and an elliptic curve has such a family of representations as well in the natural way the paper is organized as follows on the first section we present the algorithms which depend on the residual representations on the second section we review the results of p adic representations attached to automorphic forms on imaginary quadratic fields on the third section we explain falting serre s method on galois representations on the fourth section we prove that the algorithm gives the right answer at last we show some examples and some gp code writen for the examples 2 algorithm

stickelberger elements and modular parametrizations of elliptic curves in the present paper we shall give evidence to support the claim conjecture i below and 1 3 that every elliptic curve a o which can be parametrized by modular functions admits a canonical modular parametrization whose properties can be related to intrinsic properties of a in particular we will see how such a parametrizat ion can be used to prove some rather pleasant integrality properties of stickelberger elements ad p adic l functions attached to a in addition if conjecture i is true then we can give an intrinsic characterization of the isomorphism class of a special elliptic curve in the q isogeny class of a distinguished by modular considerations for most of the paper we have opted for the concrete approach and defined modular parametrizations in terms of x i n definition 1 1 however to justify our view of these parametrizations as being canonical we begin here with a more intrinsic definition recall that shimura 19 chap 6 see w 1 of this paper has defined a compatible system of canonical models of modular curves xs ses where 5 p is a certain collection of open subgroups of the group gl 2 az over the finite adeles a i of q we define the adelic upper half plane to be the pro variety ll m xs and give the q structure induced by the s field of modular functions whose q expansions at the 0 cusp have coefficients in q a modular parametrization of a is a q morphism a which sends

complete classification of the torsion structures of rational elliptic curves over quintic number fields abstract we classify the possible torsion structures of rational elliptic curves over quintic number fields in addition let e be an elliptic curve defined over q and let g e q tors be the associated torsion subgroup we study for a given g which possible groups g h could appear such that h e k tors for k q 5 in particular we prove that at most there is one quintic number field k such that the torsion grows in the extension k q i e e q tors e k tors

compound poisson approximation for markov chains using stein s method let be a stationary harris recurrent markov chain on a polish state space s f with stationary distribution let n n i 1 i i s 1 be the number of visits to s 1 f by where s 1 is rare in the sense that s 1 is small we want to find an approximating compound poisson distribution for l n such that the approximation error measured using the total variation distance can be explicitly bounded with a bound of order not much larger than s 1 this is motivated by the observation that approximating poisson distributions often give larger approximation errors when the visits to s 1 by tend to occur in clumps and also by the compound poisson limit theorems of classical extreme value theory we here propose an approximating compound poisson distribution which in a natural way takes into account the regenerative properties of harris recurrent markov chains a total variation distance error bound for this approximation is derived using the compound poisson stein equation of barbour chen and loh and certain couplings when the chain has an atom s 0 e g a singleton such that s 0 0 the bound depends only on much studied quantities like hitting probabilities and expected hitting times which satisfy poisson s equation as by products we also get upper and lower bounds for the error in the approximation with poisson or normal distributions the above results are illustrated by numerical evaluations of the error bound for some markov chains on finite state spaces

binary hermitian forms over a cyclotomic field abstract let be a primitive fifth root of unity and let f be the cyclotomic field f q let o f be the ring of integers we compute the voronoi polyhedron of binary hermitian forms over f and classify gl 2 o conjugacy classes of perfect forms the combinatorial data of this polyhedron can be used to compute the cohomology of the arithmetic group gl 2 o and hecke eigenforms

invariants de classes exemples de non annulation en dimension superieure the so called class invariant homomorphism measures the galois module structure of torsors under a finite flat group scheme g which lie in the image of a coboundary map associated to an isogeny between neron models of abelian varieties with kernel g when the varieties are elliptic curves with semi stable reduction and the order of g is coprime to 6 it is known that the homomorphism vanishes on torsion points in this paper using weil restrictions of elliptic curves we give the construction for any prime number p 2 of an abelian variety a of dimension p endowed with an isogeny with kernel p whose coboundary map is surjective in the case when a has rank zero and the p part of the picard group of the base is non trivial we obtain examples where does not vanish on torsion points

the cuspidal group and special values of l functions the structure of the cuspidal divisor class group is investigated by relating this structure to arithmetic properties of special values of l functions of weight two eisenstein series a new proof of a theorem of kubert proposition 3 1 concerning the group of modular units is derived as a consequence of the method the key lemma is a nonvanishing result theorem 2 1 for values of the l function attached to a one dimensional cohomology class over the relevant congruence subgroup proposition 4 7 provides data regarding eisenstein series and associated subgroups of the cuspidal divisor class group which the author hopes will simplify future calculations in the cuspidal group let xr q be shimura s canonical model over q of the complete modular curve associated to the congruence group r c sl2 z where r is one of the groups r n r1 n or ro n the cuspidal divisor class group cr c pic xr plays an important role in the theory of the arithmetic of the modular curve xr for example in mazur s proof of ogg s conjecture 13 and in the proof by mazur and wiles of the main conjecture of iwasawa theory 14 subgroups of the cuspidal group lead to the isogenies used in the eisenstein descent theory these subgroups can also be used to give congruence formulas for the universal special values of the l function of xr which are compatible with the descent theoretic results and the conjecture of birch and swinnerton dyer 3 5 12 19 the structure of the cuspidal group has been investigated extensively by kubert and lang 6 7 their approach is based on a study of the group of modular units in the present paper we propose another approach based on a study of integrality and divisibility properties of special values of l functions of weight two eisenstein series typically such an l function is a product of two dirichlet l functions whose special values are closely related to the arithmetic of cyclotomic fields using a theorem of l tllashington 20 on the non p part of the class number in cyclotomic zp extensions we describe a method of computing subgroups of cr which should be useful for the descent theory the corresponding congruence formulas for universal special values of l functions are an immediate consequence of the method since eisenstein series occur in many settings our point of view offers the prospect of generalization such a generalization to hilbert blumenthal varieties may be an received by the editors september 4 1984 1980 mathematics subject classification primary llf67 llg16 secondary llfll llf33 l partially supported by nsf grant mcs 82 01762 t1985 american mathematical society 0002 9947 8s 1 00 25 per page

nonlinear thermoelastic plate equations global existence and decay rates for the cauchy problem abstract we consider the cauchy problem in r n for quasilinear thermoelastic kirchhoff type plate equations where the heat conduction is modeled by either the cattaneo law or by the fourier law additionally we take into account possible inertial effects considering nonlinearities which are of fourth order in the space variable we deal with a quasilinear system which triggers difficulties typical for nonlinear schrodinger equations the different models considered are systems of mixed type comparable to schrodinger parabolic or schrodinger hyperbolic systems the main task consists in proving sophisticated a priori estimates leading to obtaining the global existence of solutions for small data neither known nor expected for the cauchy problem in pure plate theory nor available before for the coupled system under investigation where only special cases bounded domains within with analytic semigroup setting or the cauchy problem with semilinear nonlinearities had been treated before

sequences of rational torsions on abelian varieties we address the question of how fast the available rational torsion on abelian varieties over increases with dimension the emphasis will be on the derivation of sequences of torsion divisors on hyperelliptic curves work of hellegouarch and lozach and klein may be made explicit to provide sequences of curves with rational torsion divisors of orders increasing linearly with respect to genus the main results in 2 are applications of a new technique which provide sequences of hyperelliptic curves for all torsions in an interval a g a g b g wherea g is quadratic ing andb g is linear ing as well as providing an improvement from linear to quadratic these results provide a wide selection of torsion orders for potential use by those involved in computer integration we conclude by considering possible techniques for divisors of non hyperelliptic curves and for general abelian varieties

gonality of the modular curve x1 n in this paper we compute the gonality over q of the modular curve x1 n for all n 40 and give upper bounds for each n 250 this allows us to determine all n for which x1 n has infinitely points of degree 8 we conjecture that the modular units of q x1 n are freely generated by f 2 f n 2 1 where f k is obtained from the equation for x1 k

torsion subgroups of rational elliptic curves over the compositum of all d4 extensions of the rational numbers abstract let e q be an elliptic curve and let q d 4 be the compositum of all extensions of q whose galois closure has galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of d 4 in this article we first show that q d 4 is in fact the compositum of all d 4 extensions of q and then we prove that the torsion subgroup of e q d 4 is finite and determine the 24 possibilities for its structure we also give a complete classification of the elliptic curves that have each possible torsion structure in terms of their j invariants

p adic l functions of bianchi modular forms the theory of overconvergent modular symbols developed by rob pollack and glenn stevens gives a beautiful and effective construction of the p adic l function of a modular form in this paper we give an analogue of their results for bianchi modular forms that is modular forms over imaginary quadratic fields in particular we prove control theorems that say that the canonical specialisation map from overconvergent to classical bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable hecke operators we also give an explicit link between the classical modular symbol attached to a bianchi modular form and critical values of its l function which then allows us to construct p adic l functions of bianchi modular forms

computing a lower bound for the canonical height on elliptic curves over number fields computing a lower bound for the canonical height is a crucial step in determining a mordell weil basis for elliptic curves this paper presents an algorithm for computing such a lower bound for elliptic curves over number fields without searching for points the algorithm is illustrated by some examples

computing heights on elliptic curves we describe how to compute the canonical height of points orn elliptic curves tate has given a rapidly converging series for archimedean local heights over r we describe a modified version of tate s series which also converges over c and give an efficient procedure for calculating local heights at non archimedean places in this way we can calculate heights over number fields having complex embeddings we also give explicit estimates for the tail of our series and present several examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 alxy a3y x3 a2x2 a4x a6 the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 11 viii section 9 or 6 chapter vi the canonical height is an extremely important theoretical tool in the arithmetic theory of elliptic curves being used for such diverse purposes as studying values of l functions 5 numbers of integral points 12 and transcendence theory 9 it is also important as a computational tool such as its use in zagier s algorithm for finding integral points up to large bounds 18 it is thus of interest to have an efficient method for calculating the canonical height of a point the usual definition of h as a limit h p limn 0 4 nh x 2np is not practical for computation instead one uses the fact that the canonical height can be written as a sum of local heights one term for each distinct absolute value on k 2 h p e na p vemk for example if k q then mk can be identified with the set of rational primes together with the usual absolute value on q the multiplicities nv are chosen so that the product formula holds and so that h is independent of the choice of the field k the local height corresponding to a non archimedean absolute value is given by intersection theory in a well known manner see e g 2 4 or 7 chapter 11 section 5 we will describe a quick way to compute non archimedean local heights in section 5 the local height for an archimedean absolute value is given by a transcendental function and so efficient computation is somewhat more difficult j tate 15 1988 american mathematical society 0025 5718 88 1 00 25 per page 339 received august 20 1987 revised october 21 1987 1980 mathematics subject classification 1985 revision primary 11g05 14k07 11d25 this work was partially supported by nsf grant dms 8612393 current address mathematics department brown university providence ri 02912 this content downloaded from 207 46 13 103 on thu 20 oct 2016 04 37 48 utc all use subject to http about jstor org terms 340 joseph h silverman has given an easily computed power series which works for real absolute values precisely for a given curve e and point p x y he gives a sequence of easily computed numbers co cl so that

elliptic curves with abelian division fields let e be an elliptic curve over mathbb q and let n ge 1 the central object of study of this article is the division field mathbb q e n that results by adjoining to mathbb q the coordinates of all n torsion points on e overline mathbb q in particular we classify all curves e mathbb q such that mathbb q e n is as small as possible that is when mathbb q e n mathbb q zeta n and we prove that this is only possible for n 2 3 4 or 5 more generally we classify all curves such that mathbb q e n is contained in a cyclotomic extension of mathbb q or equivalently by the kronecker weber theorem when mathbb q e n mathbb q is an abelian extension in particular we prove that this only happens for n 2 3 4 5 6 or 8 and we classify the possible galois groups that occur for each value of n

discrete models of coupled dynamic thermoelasticity for stress temperature formulations in this article the author studies the properties of discrete approximations for mathematical models of coupled thermoelasticity in the stress temperature formulation since many applied problems deal with steep gradients of thermal fields the main emphasis is given to the investigation of non smooth solutions of non stationary thermoelasticity convergence of operator difference schemes on weak solutions of thermoelasticity is proved and the dispersion analysis of models is performed error estimates and the results of computational experiments are presented

periods of cusp forms and elliptic curves over imaginary quadratic fields in this paper we explore the arithmetic correspondence between on the one hand isogeny classes of elliptic curves e defined over an imaginary quadratic field k of class number one and on the other hand rational newforms f of weight two for the congruence subgroups where n is an ideal in the ring of integers r of k this continues work of the first author and forms part of the ph d thesis of the second author in each case we compute numerically the value of the l series at and compare with the value of which is predicted by the birch swinnerton dyer conjecture finding agreement to several decimal places in particular we find that whenever has a point of infinite order several examples are given in detail from the extensive tables computed by the authors

decay and growth estimates for solutions of second order and third order differential operator equations abstract we obtained decay and growth estimates for solutions of second order and third order differential operator equations in a hilbert space applications to initial boundary value problems for linear and nonlinear non stationary partial differential equations modeling the strongly damped nonlinear improved boussinesq equation the dual phase lag heat conduction equations the equation describing wave propagation in relaxing media and the moore gibson thompson equation are given

large solutions and smoothing properties for nonlinear thermoelastic systems abstract we study the existence of global solutions for large data for a class of systems of thermoelastic type with non local nonlinearities moreover we investigate smoothing properties of such coupled systems in an abstract setting including some models for thermoelastic plates in this setting classical second order thermoelasticity and viscoleasticity of integral type are recognized as limiting cases decay rates of sobolev norms of the solutions as time tends to infinity are also investigated

sur la nature non cyclotomique des points d ordre fini des courbes elliptiques appendice par e kowalski et p michel

the solution of nonstrictly hyperbolic conservation laws may be hard to compute some simple examples of nonstrictly hyperbolic conservation laws are presented that admit certain instability properties with respect to l 1 perturbations of the initial data one of the examples shows that a piecewise constant approximation generates an approximate solution with erroneous qualitative behaviour another example shows that the approximate solution is not well defined for both of these examples a classical stable entropy solution can be given by an explicit formula these properties may cause difficulties for numerical methods based on the solution of riemann problems

darmon points on elliptic curves over number fields of arbitrary signature we present new constructions of complex and pp adic darmon points on elliptic curves over base fields of arbitrary signature we conjecture that these points are global and present numerical evidence to support our conjecture

on the level of modular curves that give rise to isolated j invariants abstract we say a closed point x on a curve c is sporadic if c has only finitely many closed points of degree at most deg x and that x is isolated if it is not in a family of effective degree d divisors parametrized by p 1 or a positive rank abelian variety see section 4 for more precise definitions and a proof that sporadic points are isolated motivated by well known classification problems concerning rational torsion of elliptic curves we study sporadic and isolated points on the modular curves x 1 n in particular we show that any non cuspidal non cm sporadic respectively isolated point x x 1 n maps down to a sporadic respectively isolated point on a modular curve x 1 d where d is bounded by a constant depending only on j x conditionally we show that d is bounded by a constant depending only on the degree of q j x so in particular there are only finitely many j invariants of bounded degree that give rise to sporadic or isolated points

long time behavior for a nonlinear plate equation with thermal memory abstract we consider a nonlinear plate equation with thermal memory effects due to non fourier heat flux laws first we prove the existence and uniqueness of global solutions as well as the existence of a global attractor then we use a suitable ojasiewicz simon type inequality to show the convergence of global solutions to single steady states as time goes to infinity under the assumption that the nonlinear term f is real analytic moreover we provide an estimate on the convergence rate

compound poisson process approximation compound poisson processes are often useful as approximate models when describing the occurrence of rare events in this paper we develop a method for showing how close such approximations are our approach is to use stein s method directly rather than by way of declumping and a marked poisson process this has conceptual advantages but entails technical difficulties several applications are given to illustrate the procedure

level structures on abelian varieties kodaira dimensions and lang s conjecture abstract assuming lang s conjecture we prove that for a prime p number field k and positive integer g there is an integer r such that no principally polarized abelian variety a k has full level p r structure to this end we use a result of zuo to prove that for each closed subvariety x in the moduli space a g of principally polarized abelian varieties of dimension g there exists a level m x such that the irreducible components of the preimage of x in a g m are of general type for m m x

torsion of elliptic curves over cyclic cubic fields we determine all the possible torsion groups of elliptic curves over cyclic cubic fields over non cyclic totally real cubic fields and over complex cubic fields