on totally real cubic fields the authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant d 500000 thereby extending the existing table of fields with d 100000 by i 0 angell 1 serious defects in angell s table are pointed out for each field running number discriminant coefficients of a generating polynomial integral basis class number and a fundamental pair of units are listed the article contains statistics about the following subjects distribution of class numbers fields in which every norm positive unit is totally positive nonconjugate fields with the same discriminant fields with noncyclic class group the fields are tabulated by means of a method due to davenport and heilbronn 7 8 which leads to a unique normalized generating polynomial the given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced

class groups of number fields numerical heuristics extending previous work of h w lenstra jr and the first author we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four given the signature and the galois group of the galois closure the theoretical justifications for these conjectures will appear elsewhere but the agreement with the existing tables is quite good 1 introduction and notations in 3 h w lenstra jr and the first author developed a method for conjecturing quantitative results on class groups of quadratic fields and cyclic extensions of prime degree in a forthcoming paper 4 we shall show that this technique can be extended to a much wider class of number fields and also to relative extensions the aim of the present paper is to rapidly make available the numerical conjec tures obtained for people not really interested in our heuristic reasoning or not wanting to wait for 4 to appear hence apart from a total lack of justifications for the conjectures that we present this paper is essentially self contained the plan is as follows in the rest of this section we present the notations used in the sequel some of them being nonstandard and in general differing from the notations of 3 we urge the reader to read the notations carefully before applying the conjectures in the next section we present templates for the subsequent conjectures and then the conjectures themselves illustrated by numerical examples first for their own sake and second as a double check for the reader to understand the templates these conjectures are given for all types of fields of degree less than or equal to four in the final section we comment on the consistency of the conjectures with existing tables which is quite good combinatorial notations if x is a set xi denotes its cardinality

an asymptotic analysis of the 1 3 4 hamiltonian normal form abstract the normal form of the hamiltonian 1 3 4 resonance which exhibits two simultaneous resonances of differing orders is studied asymptotically since the two resonances have different strengths the exact solution of the primary single resonance system may be used to construct an action angle transformation the resulting standard form system is solved asymptotically by canonical near identity averaging transformations in addition to the hamiltonian itself and its unperturbed part which are two exact constants of the motion a third independent adiabatic invariant of the original hamiltonian system is constructed the results apply directly to the problem of a free electron laser with weak self fields a specific model problem is studied numerically to verify the asymptotic validity of the results over long times

on class groups of random number fields the main aim of the present paper is to disprove the cohen lenstra martinet heuristics in two different ways and to offer possible corrections we also recast the heuristics in terms of arakelov class groups giving an explanation for the probability weights appearing in the general form of the heuristics we conclude by proposing a rigorously formulated cohen lenstra martinet conjecture

computation of class numbers of quadratic number fields we explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields we therefore end up with a simpler and faster method for computing class numbers of quadratic number fields we also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields

computation of relative class numbers of cm fields let be a nontrivial hecke character on a strict ray class group of a totally real number field l of discriminant d textbf l then l 0 is an algebraic number of some cyclotomic number field we develop an efficient technique for computing the exact values at s 0 of such abelian hecke l functions over totally real number fields l let f denote the norm of the finite part of the conductor of then roughly speaking we can compute l 0 in o d textbf l f x 0 5 epsilon elementary operations we then explain how the computation of relative class numbers of cm fields boils down to the computation of exact values at s 0 of such abelian hecke l functions over totally real number fields l finally we give examples of relative class number computations for cm fields of large degrees based on computations of l 0 over totally real number fields of degree 2 and 6 this paper being an abridged version of lou4 the reader will find there all the details glossed over here

connection between gaussian periods and cyclic units this paper finds that all known parametric families of units in real quadratic cubic quartic and sextic fields with prime conductor are linear combinations of gaussian periods and exhibits these combinations this approach is used to find new units in the real quintic field for prime conductors p n4 5n3 15n2 25n 25

computation of relative class numbers of cm fields by using hecke l functions we develop an efficient technique for computing values at s 1 of hecke l functions we apply this technique to the computation of relative class numbers of non abelian cm fields n which are abelian extensions of some totally real subfield l we note that the smaller the degree of l the more efficient our technique is in particular our technique is very efficient whenever instead of simply choosing l n the maximal totally real subfield of n we can choose l real quadratic we finally give examples of computations of relative class numbers of several dihedral cm fields of large degrees and of several quaternion octic cm fields with large discriminants

geometry and chaos near resonant equilibria of 3 dof hamiltonian systems in this paper we study the dynamics near resonant elliptic equilibria in three degree of freedom hamiltonian systems the resonances we consider have multiplicity two and the corresponding local normal form for the equilibrium is integrable at cubic order we prove the existence of families of 3 tori and whiskered 2 tori with nearby chaotic dynamics in the quartic normal form the whiskers of the 2 tori intersect in a non trivial way giving rise to multi pulse homoclinic and heteroclinic connections these connections survive in the full system as orbits homoclinic to invariant 3 spheres

a gieseker type degeneration of moduli stacks of vector bundles on curves we construct a new degeneration of the moduli stack of vector bundles over a smooth curve when the curve degenerates to a singular curve which is irreducible with one double point we prove that the total space of the degeneration is smooth and its special fibre is a divisor with normal crossings furthermore we give a precise description of how the normalization of the special fibre of the degeneration is related to the moduli space of vector bundles over the desingularized curve

class group frequencies of real quadratic function fields the degree 4 case the distribution of ideal class groups of f q t m t is examined for degree four monic polynomials m e f q t when f q is a finite field of characteristic greater than 3 with q e 20000 100000 or q e 1020000 1100000 and m is irreducible or has an irreducible cubic factor particular attention is paid to the distribution of the p sylow part of the class group and these results agree with those predicted using the cohen lenstra heuristics to within about 1 part in 10000 an alternative set of conjectures specific to the cases under investigation is in even sharper agreement

chaos in the 1 2 3 hamiltonian normal form the normal form of the hamiltonian 1 2 3 resonance to degree 3 contains seven families of periodic solutions of which one can be complex unstable associated with this complex unstable solution is an invariant manifold n on which the dynamics can be characterised completely one of the ingredients of n is a set of homoclinic orbits in the normal form to degree 4 the set of homoclinic orbits breaks up for certain parameter values into one homoclinic orbit this enables us to apply silnikov devaney theory to prove at this stage numerically the existence of a horseshoe map in the system with the implication of non integrability and chaos in the normal form

the expectation of success using a monte carlo factoring method some statistics on quadratic class numbers a method has been proposed for factoring an integer n by using the structure of the class groups of quadratic fields of radicand kn for various small multipliers k we discuss the method and an implementation of the method and various theoretical questions which have an impact on the practical use of the method in factoring some of the theoretical questions relate to the nature of class numbers and class groups we present extensive statistical results on the class numbers and class groups of imaginary quadratic fields 1 the method we deal with binary quadratic forms a b c of discriminant b2 4ac n n 0 the equivalence classes of forms of a fixed discriminant under transformations of the modular group t form a finite abelian group the class group the order of the class group is the class number h h n the crucial fact used either implicitly or explicitly in several factoring methods is that for odd n the classes of order 2 in the class group called ambiguous classes are precisely the classes containing forms p p p2 n 4p for the various divisors p of tv thus finding ambiguous forms leads to finding factors for even tv there are also ambiguous classes represented by forms p 0 n p in addition to the classes mentioned above shanks used the explicit class group structure in his factoring method clasno shan he estimated h with the product formula then fiddled in the class group until the exact value for h was found he then found an ambiguous form by finding a form a b c for which 2 was not the identity the new method which we shall henceforth refer to as the cps method classgroup clasno pollard p l synthesis is somewhat less direct schn we let m be the product of all small odd primesp i raised to large exponents a i we then compute g for forms until we find an for which is not the identity writing h 2mh it is certainly true that if h m then g2 is ambiguous in short we exponentiate forms to huge odd powers that we hope contain all the odd factors of the class number if this comes to pass then we can get ambiguous forms and then factors of n actually as with many factoring methods there are several deeper levels of subtlety first it is not really necessary that h m the class groups being finite received march 8 1983 revised october 11 1983 1980 mathematics subject classification primary 10 04 10a25 12a25 1984 american mathematical society 0025 5718 84 1 00 25 per page

on real cyclic sextic fields a table of units and class numbers of real cyclic sextic fields with conductor 16 1 leopoldt s unit index and the signature rank of the unit group

on the computation of a table of complex cubic fields with discriminant d 10 6 a method for finding all the nonisomorphic complex cubic fields with discriminant d 106 is described three different methods were used to find the class number of each of these fields the speed of these techniques is discussed and several tables illustrating the computational results are presented these include tables of the distribution of the fields and the class numbers and the class group structures of these fields

the cohen lenstra heuristic methodology and results in number theory great efforts have been undertaken to study the cohen lenstra probability measure on the set of all finite abelian p groups on the other hand group theorists have studied a probability measure on the set of all partitions induced by the probability that a randomly chosen n n matrix over fp is contained in a conjugacy class associated with this partitions for n this paper shows that both probability measures are identical as a consequence a multitude of results can be transferred from each theory to the other one the paper contains a survey about the known methods to study the probability measure and about the results that have been obtained so far from both communities

some computational results on a problem concerning powerful numbers let d be a positive square free integer and let x y j d be the fundamental unit in the order with z basis 1 vd an algorithm which is of time complexity 0 d1 4 for any positive e is developed for determining whether or not d y results are presented for a computer run of this algorithm on all d 108 the conjecture of ankeny artin and chowla is verified for all primes 1 mod 4 less than 109

asymptotic analysis of a class of three degree of freedom hamiltonian systems near stable equilibria abstract a conservative near integrable hamiltonian dynamical system is examined which to leading order consists of three uncoupled harmonic oscillators with constant frequencies in the ratio 1 2 for certain rational formally the problem considered can arise by perturbing any three degree of freedom hamiltonian near a stable equilibrium point so that the hamiltonian consists of a power series expansion in a small parameter where successive terms are homogeneous polynomials of increasing degree in the coordinates and the momenta the special case of two exact simultaneous resonances one in the first perturbation term and one in the second is examined and explicit asymptotic solutions are obtained the solution procedure involves reducing the original hamiltonian to two degrees of freedom using one integral of the motion then transforming to standard form to find two additional adiabatic invariants by near identity averaging canonical transformations a specific example is studied numerically to verify the asymptotic validity of the results over long times

efficient computation of root numbers and class numbers of parametrized families of real abelian number fields let k m be a parametrized family of simplest real cyclic cubic quartic quintic or sextic number fields of known regulators e g the so called simplest cubic and quartic fields associated with the polynomials p m x x3 mx2 m 3 x 1 and p m x x 4 mx 3 6x 2 mx 1 we give explicit formulas for powers of the gaussian sums attached to the characters associated with these simplest number fields we deduce a method for computing the exact values of these gaussian sums these values are then used to efficiently compute class numbers of simplest fields finally such class number computations yield many examples of real cyclotomic fields q p of prime conductors p 3 and class numbers h p greater than or equal to p however in accordance with vandiver s conjecture we found no example of p for which p divides h p

on the parallel generation of the residues for the continued fraction factoring algorithm etude des proprietes du developpement en fraction continue d un irrationnel quadratique application au developpement d un algorithme efficace de factorisation

the continued fraction algorithm and regulator for quadratic function fields of characteristic 2 the continued fraction expansion and infrastructure for quadratic congruence function fields of odd characteristic have been well studied recently these ideas have even been used to produce cryptosystems much less is known concerning the continued fraction expansion and infrastructure for quadratic function fields of even characteristic we will explore these ideas and show that the situation is very similar to the odd characteristic case this exploration will result in a method for computing the regulator for quadratic function fields of characteristic 2

on the computation of unit groups and class groups of totally complex quartic fields in this paper we describe the computation of a system of fundamental units and of the class group for each totally real quartic field y of discriminant less than 106 generating equations integral bases and the galois groups for all those fields were recently given by buchmann and ford

a wonderful embedding of the loop group i describe the wonderful compactification of loop groups these compactifications are obtained by adding normal crossing boundary divisors to the group lg of loops in a reductive group g or more accurately to the semi direct product c lg in a manner equivariant for the left and right c lg actions the analogue for a torus group t is the theory of toric varieties for an adjoint group g this is the wonderful compactification of de concini and procesi the loop group analogue is suggested by work of faltings in relation to the compactification of moduli of g bundles over nodal curves

on cyclic cubic fields a table of class numbers and units in cyclic cubic fields with conductor o for3 f 2 a 6 mod9 b 3 or 6 mod9 b o for 3 if introduced by hasse 8 p 12 let 6 6 6 denote the gaussian periods for a generating cubic character of k taken with a suitable sign then k q o and see

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

on unit groups and class groups of quartic fields of signature 2 1 this is the third and last paper of a series now completing the description of the unit group and class group of all quartic number fields f of discriminant df with idfi 106

chebyshev spectral solution of nonlinear volterra hammerstein integral equations abstract in this paper the chebyshev spectral cs method for the approximate solution of nonlinear volterra hammerstein integral equations f o k r g r y r d r 0 t is investigated the method is applied to approximate the solution not to the equation in its original form but rather to an equivalent equation z t g t y t t 1 1 the function z is approximated by the nth degree interpolating polynomial zn with coefficients determined by discretizing g t y t at the chebyshev gauss labatto nodes we then define the approximation to y to be of the form y n t f t 1 1 k t s z n s d s 1 1 and establish that under suitable conditions limn yn t y t uniformly in t finally a numerical experiment for a nonlinear volterra hammerstein integral equation is presented which confirms the convergence demonstrates the applicability and the accuracy of the chebyshev spectral cs method

how many rational points does a random curve have a large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties for an elliptic curve over q the set of rational points forms a finitely generated abelian group the ranks of these groups when ranging over all elliptic curves are conjectured to be evenly distributed between rank 0 and rank 1 with higher ranks being negligible we will describe these conjectures and discuss some results on bounds for average rank highlighting the recent work of bhargava and shankar

local analysis of formal stability and existence of fixed points in 4d symplectic mappings abstract symplectic mappings in a four dimensional phase space are analysed in the neighbourhood of an elliptic fixed point whose eigenvalues are close to satisfy a double resonance condition the perturbative tools of resonant normal forms are outlined the analysis of the interpolating hamiltonian of the normal form allows one to determine the existence and the stability of the fixed points which are due to the double resonance only some combinations of stabilities are allowed due to the asymptotic character of the series the results are formal but supported by numerical computations

l functions and class numbers of imaginary quadratic fields and of quadratic extensions of an imaginary quadratic field starting from the analytic class number formula involving its lfunction we first give an expression for the class number of an imaginary quadratic field which in the case of large discriminants provides us with a much more powerful numerical technique than that of counting the number of reduced definite positive binary quadratic forms as has been used by buell in order to compute his class number tables then using class field theory we will construct a periodic character x defined on the ring of integers of a field k that is a quadratic extension of a principal imaginary quadratic field k such that the zeta function of k is the product of the zeta function of k and of the l function l s x we will then determine an integral representation of this l function that enables us to calculate the class number of k numerically as soon as its regulator is known it will also provide us with an upper bound for these class numbers showing that hua s bound for the class numbers of imaginary and real quadratic fields is not the best that one could expect we give statistical results concerning the class numbers of the first 50000 quadratic extensions of q i with prime relative discriminant and with k q a non galois quartic extension our analytic calculation improves the algebraic calculation used by lakein in the same way as the analytic calculation of the class numbers of real quadratic fields made by williams and broere improved the algebraic calculation consisting in counting the number of cycles of reduced ideals finally we give upper bounds for class numbers of k that is a quadratic extension of an imaginary quadratic field k which is no longer assumed to be of class number one 1 class numbers of imaginary quadratic fields let k be an imaginary quadratic field with discriminant d 0 and character x the analytic class number formula for this field is h k k idl x 2nr knowing the functional equations satisfied by the zeta function of k and the riemann zeta function one can easily deduce the functional equation satisfied by their quotient l s x i e f s f 1 s with f s di f s l 2s 1 x a t ts t received by the editor february 12 1991 and in revised form may 17 1991 1991 mathematics subject classification primary 1 r29 secondary 11 r1 6 i 1992 american mathematical society 0025 5718 92 1 00 25 per page

computation of real quadratic fields with class number one a rapid method for determining whether the real quadratic field si s d has class number one is described the method makes use of the infrastructure idea of shanks to determine the regulator of w and then uses the generalized riemann hypothesis to rapidly estimate l l x to the accuracy needed for determining whether or not the class number of 3 is one the results of running this algorithm on a computer for all prime values of d up to 109 are also presented together with further results on runs on intervals of size 107 starting at 10 t 9 10 16

gromov witten gauge theory we introduce a modular completion of the stack of maps from stable marked curves to the quotient stack pt c x and use this stack to construct some gauge theoretic analogues of the gromov witten invariants we also indicate the generalization of these invariants to the quotient stacks x c x where x is a smooth proper complex algebraic variety

a note on class number one in certain real quadratic and pure cubic fields let p be any odd prime and let h p be the class number of the real quadratic field 2 f p the results of a computer run to determine the density of the field fp with h p 1 and p 108 are presented similar results are given for pure cubic fields 2 fp with p 106 1 introduction let p be any odd prime and let h h p be the class number of the quadratic field 2 p it is well known that h p is odd but the problem of how frequently h p 1 although it goes back to gauss is still unsolved if we let it a b x denote the number of primes of the form a bk less than or equal to x and f a b x denote the number of these primes p for which h p 1 we find see lakein 5 from a large table of kuroda 4 that r l 4 x l 4 x 77 l 4 x 7765 for x 2776817 that is over 77 of all the primes 1 mod4 up to 2776817 have h p 1 indeed according to the recent heuristic results of cohen and lenstra see cohen 1 we would expect that h p 1 with probability 75446 in order to test this heuristic we developed and ran a computer program which determined whether or not h p 1 for all primes p 108 in the next section of this note we give the results of this computer run in the following section we present some data for certain pure cubic fields p with p 106 2 the quadratic case in order to find h p we made use of the well known formula

quintic polynomials and real cyclotomic fields with large class numbers we study a family of quintic polynomials discoverd by emma lehmer we show that the roots are fundamental units for the corresponding quintic fields these fields have large class numbers and several examples are calculated as a consequence we show that for the prime p 641491 the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401 in an appendix we give a characterization of the simplest quadratic cubic and quartic fields

the picard group of the universal moduli space of vector bundles on stable curves abstract we construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its picard group as a consequence we obtain an explicit description of the picard groups of the universal moduli stack of vector bundles on smooth curves and of the schmitt s compactification over the stack of stable curves we prove some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group in particular we give necessary and sufficient conditions for the existence of poincare bundles over the universal curve of an open substack of the rigidification generalizing a result of mestrano ramanan

irregular behaviour arising from quasiperiodic forcing of simple quantum systems insight from perturbation theory the perturbation of simple quantum systems with almost periodic time dependent forces can produce irregular dynamics which is manifested in i broadband fourier spectra and ii rapid decay of autocorrelation functions some classical perturbation methods for differential equations provide insight into this behaviour by showing the existence of a dense spectrum on the real line such irregular behaviour is shown also to occur in some model linear systems of differential equations e g a quasiperiodic mathieu type equation

normal forms and quasi integrals for the hamiltonians of magnetic bottles the well known birkhoff gustavson normal form theory suffers from the restraint that the quadratic part of the hamiltonian must be of the harmonic oscillator type in this paper we describe a generalized normal form concept which can be applied to any polynomial hamiltonian thus rendering the above restriction to harmonic oscillators unnecessary as in the classical theory we can derive an asymptotic expression for a second integral of motion the truncated formal integral the quasi integral exhibits good convergence properties in regions of phase space where the dynamics is regular whereas in chaotic regions the convergence deteriorates in order to exemplify these findings we apply the theory to a hamiltonian describing a particular type of magnetic bottle which cannot be analysed using the birkhoff gustavson normal form we calculate the quasi integrals up to and including the 14th order and analyse their convergence properties

quadratic class numbers and character sums we present an algorithm for computing the class number of the quadratic number field of discriminant d the algorithm terminates unconditionally with the correct answer and under the grh executes in o d 1 4 steps the technique used combines algebraic methods with burgess theorem on character sums to estimate l 1 x d we give an explicit version of burgess theorem valid for prime discriminants and as an application we compute the class number of a 32 digit discriminant