construction of hyperelliptic function fields of high three rank we present several explicit constructions of hyperelliptic function fields whose jacobian or ideal class group has large 3 rank our focus is on finding examples for which the genus and the base field are as small as possible most of our methods are adapted from analogous techniques used for generating quadratic number fields whose ideal class groups have high 3 rank but one method applicable to finding large l ranks for odd primes l 3 is new and unique to function fields algorithms examples and numerical data are included

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

on the 3 sylow subgroup of the class group of quadratic fields in this paper we give a large number of quadratic fields q vd whose class group h d has 3 sylow subgroup h3 d with rank r3 d 1 they include 20 discriminants d 0 with r3 d 5 the associate real fields have r3 d 4 the distribution of the h3 d is studied and its relative frequency is compared with the heuristic conjectures of cohen and lenstra some of the h d are recorded since they are of special interest some related topics are also considered

secondary terms in counting functions for cubic fields we prove the existence of secondary terms of order x5 6 in the davenport heilbronn theorems on cubic fields and 3 torsion in class groups of quadratic fields for cubic fields this confirms a conjecture of datskovsky wright and roberts we also prove a variety of generalizations including to arithmetic progressions where we discover a curious bias in the secondary term roberts s conjecture has also been proved independently by bhargava shankar and tsimerman in contrast to their work our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms developed by shintani and datskovsky wright

quadratic fields with special class groups for every prime number p 5 it is shown that under certain hypotheses on x e q the imaginary quadratic fields q vx2p 6xp 1 have ideal class groups with noncyclic p parts several numerical examples with p 5 and 7 are presented these include the field q v 4805446 123032518648268510536 the 7 part of its class group is isomorphic to c 7 x c 7 x c 7 where c n denotes a cyclic group of order n

error estimates for the davenport heilbronn theorems we obtain the first known power saving remainder terms for the theorems of davenport and heilbronn on the density of discriminants of cubic fields and the mean number of 3 torsion elements in the class groups of quadratic fields in addition we prove analogous error terms for the density of discriminants of quartic fields and the mean number of 2 torsion elements in the class groups of cubic fields these results prove analytic continuation of the related dirichlet series to the left of the line r s 1

extremal segments in random sequences we investigate the probability for the largest segment with total displacement q in an n step random walk to have length l using analytical exact enumeration and monte carlo methods we reveal the complex structure of the probability distribution in the large n limit in particular the size of the longest loop has a distribution with square root singularity at l identical to l n 1 an essential singularity at l 0 and a discontinuous derivative at l 1 2

trace class operators and commutators abstract we identify the linear span of commutators ab ba where a is a trace class operator and b is any bounded operator on a separable hilbert space or where a b are both hilbert schmidt operators thus answering questions raised by work of anderson vaserstein and weiss

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

tabulation of cubic function fields via polynomial binary cubic forms we present a method for tabulating all cubic function fields over mathbb f q t whose discriminant d has either odd degree or even degree and the leading coefficient of 3d is a non square in mathbb q up to a given bound x on d q deg d our method is based on a generalization of belabas method for tabulating cubic number fields the main theoretical ingredient is a generalization of a theorem of davenport and heilbronn to cubic function fields along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms the algorithm requires o q 4 x 1 epsilon field operations when d has odd degree and o q 5 x 1 epsilon when d has even degree it performs quite well in practice the algorithm examples and numerical data for q 5 7 11 13 are included

traces of compact operators and the noncommutative residue abstract we extend the noncommutative residue of m wodzicki on compactly supported classical pseudo differential operators of order d and generalise a connes trace theorem which states that the residue can be calculated using a singular trace on compact operators contrary to the role of the noncommutative residue for the classical pseudo differential operators a corollary is that the pseudo differential operators of order d do not have a unique trace pseudo differential operators can be non measurable in connes sense other corollaries are given clarifying the role of dixmier traces in noncommutative geometry including the definitive statement of connes original theorem

a fast algorithm to compute cubic fields we present a very fast algorithm to build up tables of cubic fields real cubic fields with discriminant up to 10 11 and complex cubic fields down to 10 11 have been computed

class groups of complex quadratic fields we present 75 new examples of complex quadratic fields that have 5 rank of their class groups 5 3 only one of these fields has 5 rank of its class group 3 the field q 25855935i511807 has a class group isomorphic to c 5 x c 5 x c 5 x c 5 x 7 2 x 7 11828 the fields were obtained by applying ideas of j f mestre to the 5 isogeny 11 xn 1

multiplicities of dihedral discriminants given the discriminant dk of a quadratic field k the number of cyclic relative extensions n k of fixed odd prime degree p with dihedral absolute galois group of order 2p which share a common conductor is called the multiplicity of the dihedral discriminant d f2 p d in this paper general formulas for multiplicities of dihedral discriminants are derived by analyzing the p rank of the ring class group mod of k for the special case p 3 c4 3 an elementary proof is given additionally the theory is illustrated by a discussion of all known discriminants of multiplicity 5 of totally real and complex cubic fields introduction let p be an odd prime and at q a cyclic extension of degree p then it is well known 9 15 7 that the conductor of k must have the form pe qi it where e 0 or e 2 0 and the qi are pairwise distinct rational primes satisfying qi x mod p for i x t the discriminant of k is just a power of the conductor di fp x if for any positive integer the number of cyclic extensions k q of degree p which share the same conductor is denoted by m f then i p where p dimf qx q qx f p dimr syl u z fz zp p t w with x f r x r f x q reqx r x mod and w e moebius inversion yields an explicit formula for m f m pe qx qt p l w 1 it is the aim of the present paper to establish similar formulas for multiplicities of discriminants in the case of non galois extensions l q of degree p with dihedral normal closure yv of degree 2p for the sake of illustration the formulas are applied to discriminants with multiplicities up to 16 of non galois received december 10 1990 revised april 5 1991 1991 mathematics subject classification primary 11r20 11r11 11r16

on quadratic fields with large 3 rank davenport and heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields which has been used to tabulate the latter we give a simpler proof of their theorem then analyze and improve the table building algorithm it computes the multiplicities of the o x general cubic discriminants real or imaginary up to x in time o x and space o x or more generally in time o x x m and space o m x for a freely chosen positive m a variant computes the 3 ranks of all quadratic fields of discriminant up to x with the same time complexity but using only m o 1 units of storage as an application we obtain the first 1618 real quadratic fields with r3 4 and prove that q 5393946914743 is the smallest imaginary quadratic field with 3 rank equal to 5

sums of commutators in non commutative banach function spaces let m be a ii factor and denote by its normal faithful semi finite trace for any rearrangement invariant kothe function space x on 0 let x m be the associated non commutative banach function space this paper is concerned with ideals in m of the form ix m m x m that are contained in lp m for some p 0 it is proved that an element t in ix m is a finite sum of commutators of the form a b with a ix m and b m if and only if the function t 1t t t d t belongs to x where t is the brown spectral measure of t and t t t is the non increasing rearrangement of the function with respect to t this extends to general banach function spaces a result obtained by kalton for quasi banach ideals of compact operators and implies that the dixmier s trace of a quasi nilpotent element in l1 m is always zero

density of cubic field discriminants in this paper we give a conjectural refinement of the davenport heilbronn theorem on the density of cubic field discriminants we explain how this refinement is plausible theoretically and agrees very well with computa tional data let an be the number of isomorphism classes of abelian cubic fields with discrim inant n let bn be the number of isomorphism classes of non abelian cubic fields with discriminant n the numbers an are very well understood the numbers bn have been the subject of extensive theoretical and computational study for at least sixty years but are less well understood the object of this note is to contribute to the study of these bn by bringing together the theoretical and computational literature for define g x

commutator structure of operator ideals commutator structure of operator ideals ken dykema 1 tadeusz figiel gary weiss and mariusz wodzicki department of mathematics texas a m university college station tx 77843 3368 usa b instytut matematyczny polskiej akademii nauk 81 825 sopot poland department of mathematical sciences university of cincinnati cincinnati oh 45221 0025 usa department of mathematics university of california berkeley ca 94720 3840 usa

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p