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

regression analysis for a functional response functional responses are encountered when units are observed over time although the whole function itself is not observed a sufficiently large number of evaluations as is common with modern recording equipment are assumed to be available functional regression analysis relates the smooth functional response y t to known covariates x by a linear combination of parameter functions t which are to be estimated the model takes the standard form y t x t t t this approach provides an alternative to standard longitudinal data methods used in the biological sciences where less and noisier data necessitate parametric modeling the methodology is illustrated by an application in ergonomics

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

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

numerical grid generation in arbitrary surfaces through a second order differential geometric model abstract in this paper a set of second order partial differential equations for the generation of coordinates in a given surface has been developed and then solved numerically to demonstrate its potential for surface coordinate generation the proposed equations are not some arbitrarily chosen equations but are a consequence of the formulae of gauss and thus carry with them an explicit dependence on the geometric properties of the given surface furthermore these equations are easy to solve and require only the specification of the bounding curves to provide the dirichlet boundary conditions for their solution results of coordinate generation both in simply and doubly connected regions on some known surfaces with the option of coordinate redistribution have been presented extension of this technique to arbitrary surfaces seems to be straightforward

a computational model for organism growth based on surface mesh generation the generation of curvilinear coordinate meshes has been a powerful tool in computational fluid dynamics calculation in the computational modelling of the fluid flows around complex bodies such as an airfoil or a complete aircraft this same technique may be applied to many other computational models in this work the approach is used as part of a computational model to generate simple geometries associated with biological forms or organisms the model adopted was first proposed by cummings and simulates morphogenesis in terms of the geometrical changes occurring during the growth and development of simple organisms this model depends on the generation of a curvilinear coordinate mesh on the surface of an organism previous work has concentrated on the model and its use in generating axisymmetric shapes that are simple models of elementary organisms in this work we describe how the model may be extended to geometrical symmetry breaking this paper describes the methodology of this extension and demonstrates it in the simulation of tentacle growth the resulting computational technique makes it possible to link models of cell bio chemistry and surface deformation

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

new tests for equality of several covariance functions for functional data in this article we propose two new tests for the equality of the covariance functions of several functional populations namely a quasi gpf test and a quasi fmax test whose test statistics are o

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

distances and inference for covariance operators a framework is developed for inference concerning the covariance operator of a functional random process where the covariance operator itself is an object of interest for statistical analysis distances for comparing positive definite covariance matrices are either extended or shown to be inapplicable to functional data in particular an infinite dimensional analogue of the procrustes size and shape distance is developed convergence of finite dimensional approximations to the infinite dimensional distance metrics is also shown for inference a frechet estimator of both the covariance operator itself and the average covariance operator is introduced a permutation procedure to test the equality of the covariance operators between two groups is also considered additionally the use of such distances for extrapolation to make predictions is explored as an example of the proposed methodology the use of covariance operators has been suggested in a philological study of cross linguistic dependence as a way to incorporate quantitative phonetic information it is shown that distances between languages derived from phonetic covariance functions can provide insight into the relationships between the romance languages

bootstrap based testing of equality of mean functions or equality of covariance operators for functional data we investigate the properties of a simple bootstrap method for testing the equality of mean functions or of covariance operators in functional data theoretical size and power results are derived for certain test statistics whose limiting distributions depend on unknown infinite dimensional parameters simulations demonstrate good size and power of the bootstrap based functional tests

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

a studentized permutation test for the comparison of spatial point patterns in this study a new test is proposed for the hypothesis that two or more observed point patterns are realizations of the same spatial point process model to this end the point patterns are divided into disjoint quadrats on each of which an estimate of ripley s k function is calculated the two groups of empirical k functions are compared by a permutation test using a studentized test statistic the proposed test performs convincingly in terms of empirical level and power in a simulation study even for point patterns where the k function estimates on neighboring subsamples are not strictly exchangeable it also shows improved behavior compared with a test suggested by diggle et al for the comparison of groups of independently replicated point patterns in an application to two point patterns from pathology that represent capillary positions in sections of healthy and cancerous tissue our studentized permutation test indicates statistical significance although the patterns cannot be clearly distin

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

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 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

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

second order comparison of gaussian random functions and the geometry of dna minicircles given two samples of continuous zero mean iid gaussian processes on 0 1 we consider the problem of testing whether they share the same covariance structure our study is motivated by the problem of determining whether the mechanical properties of short strands of dna are significantly affected by their base pair sequence though expected to be true had so far not been observed in three dimensional electron microscopy data the testing problem is seen to involve aspects of ill posed inverse problems and a test based on a karhunen loeve approximation of the hilbert schmidt distance of the empirical covariance operators is proposed and investigated when applied to a dataset of dna minicircles obtained through the electron microscope our test seems to suggest potential sequence effects on dna shape supplemental material available online

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

a synopsis of elliptic pde models for grid generation this paper is devoted to an analytical comparison of the various elliptic partial differential equation pde models which are in current use for grid generation these comparisons particularly between the equations from the laplace poisson system and the equations from a gaussian approach have yielded useful expressions connecting the 3d laplacians and the surface beltramians this effort specifically been successful when the transverse coordinate leaving the surface is orthogonal to the surface equations which are derivable from cartesian type poisson and those obtained by using the variational priciple in surface coordinates have also been considered

coordinate system control adaptive meshes the discussion of the formulation and properties of the adaptive mesh algorithm is as complete as space permits as should be clear from the discussion adaptive gridding is most useful for singular perturbation problems where localized regions with large gradients develop spontaneously often choosing the correct weight function is not trivial and sometimes it is appropriate to cause the mesh to respond to structure in several dependent variables simultaneously

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

theoretical foundation of the equations for the generation of surface coordinates references 1 alien h j and perkins e w characteristics of flow over inclined bodies of revolution naca rm a50l07 march 1951 fiechter m wirbelsysteme schlanker rotationskorper und die aerodynamischen kra fte dglr jahrbuch 1969 pp 77 85 hunt b l asymmetric vortex forces and wakes on slender bodies aiaa paper 82 1336 aug 1982 kompenhans j and hartmann k strftmungssichtbarmachung an einem hochangestellten ogivkreiszylinderrumpf mit hilfe der laser lichtschnittmethode dfvlr fb 86 45 1986 prandtl l and tietjens o applied hydroand aeromechanics mcgraw hill new york and london 1934 foeppl l wirbelbewegung hinter einem kreiszylinder sitz ber kdnigl bay akademie d wissenschaften math phys klasse jg 1913 munchen jan 1913 sun y c shen s f and zhu z a numerical simulation of the flow development behind a circular cylinder started from rest and a criterion for investigating unsteady separating flows ada mechanica vol 71 jan 1988 pp 1 20 dyer d e fiddes s p and smith j h b asymmetric vortex formation from cones at incidence a simple inviscid model aeronautical quarterly vol 33 nov 1982 pp 293 312 ng t on leading edge vortex and its control aiaa paper 89 3346 aug 1989 yi8 gao 0 gpo a sai3 a surface christoffel symbols of the second kind

determination of surfaces using beltramians at the separation of two fluid media this paper describes a model formed of a set of elliptic partial differential equations involving beltramians of coordinates for generating the curved surfaces at the separation of two different media the main point is that the differential equations are formed in such a way that the sum of the local principal curvatures appears explicitly which in turn depends on the physical and chemical properties of the two media this dependence is quite obvious as far as the problem of surface tension is concerned the developed partial differential equations have been solved numerically for the simple case of a fluid in contact with a vertical wall and for generation of the surface of revolution due to the capillary effect in a tube of small radius these examples demonstrate the application of the method when the surface tension is important the method may be of value in tackling other problems such as the generation of synclastic and anticlastic surfaces provided that the local principal curvatures can be related to the coordinates by using some physical principle

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