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

zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

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

behaviors of solutions for a singular diffusion equation abstract this paper is devoted to some behaviors of solutions of the initial boundary problem for a singular diffusion equation namely localization and large time behavior after given some special explicit solutions it is proved that solutions of the problem possess the localization property next l 2 decay estimate as t is proved by a rather standard energy method finally by comparison with a special solution the expected l decay estimate is derived

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

positivity properties of viscosity solutions of a degenerate parabolic equation we consider the problem i u t u u u 2 in r n r u x 0 u 0 x in r n where is a nonnegative constant and u 0 is a bounded continuous and nonnegative function on r n

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

self similar intermediate asymptotics for a degenerate parabolic filtration absorption equation the equation tu u xx2u c 1 xu 2 is known in literature as a qualitative mathematical model of some biological phenomena here this equation is derived as a model of the groundwater flow in a water absorbing fissurized porous rock therefore we refer to this equation as a filtration absorption equation a family of self similar solutions to this equation is constructed numerical investigation of the evolution of non self similar solutions to the cauchy problems having compactly supported initial conditions is performed numerical experiments indicate that the self similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state identical zero an open problem caused by the nonuniqueness of the solution of the cauchy problem is discussed

computing discrete logarithms in real quadratic congruence function fields of large genus the discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems recently real quadratic congruence function fields were used to construct a public key distribution system the security of this public key system is based on the difficulty of a discrete logarithm problem in these fields in this paper we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus this algorithm is a generalization of similar algorithms for real quadratic number fields

on the one dimensional p laplacian with a singular nonlinearity abstract in this paper we are concerned with the existence and nonexistence of positive solutions of boundary value problems for the one dimensional p laplacian with a singular nonlinearity in the case of the model equation we give the necessary and sufficient conditions of the existence of positive solutions for both the dirichlet problem and the periodic problem

smooth ideals in hyperelliptic function fields recently several algorithms have been suggested for solving the discrete logarithm problem in the jacobians of high genus hyperelliptic curves over finite fields some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense we explicitly show how these density results can be derived all proofs are purely combinatorial and do not exploit analytic properties of generating functions

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

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

epicomplete archimedean l groups and vector lattices an object g in a category is epicomplete provided that the only morphisms out of g which are simultaneously epi and mono are the isomorphisms we characterize the epicomplete objects in the category arch whose objects are the archimedean lattice ordered groups archimedean groups and whose morphisms are the maps preserving both group and lattice structure fhomomorphisms recall that a space is basically disconnected if the closure of each cozero subset is open theorem the following are equivalent for g arch a g is arch epicomplete b g is an arch extremal suboject of d y for some basically disconnected compact hausdorffspace y here d y denotes the continuous extended realvalued functions on y which are finite on a dense subset c g is conditionally and laterally a complete meaning each countable subset of positive elements of g which is either bounded or pairwise disjoint has a supremum and g is divisible the analysis of arch rests on an analysis of the closely related category w whose objects are of the form g u where g e arch and u is a weak unit meaning g au 0 implies g 0 for all g 6 g and whose morphisms are the homomorphism preserving the weak unit theorem the following are equivalent for g w ew a g u is w epicomplete b g u is xv isomorphic to d y 1 c g u is conditionally and laterally o complete and g is divisible

explict bounds and heuristics on class numbers in hyperelliptic function fields in this paper we provide tight estimates for the divisor class number of hyperelliptic function fields we extend the existing methods to any hyperelliptic function field and improve the previous bounds by a factor proportional to g with the help of new results we thus obtain a faster method of computing regulators and class numbers furthermore we provide experimental data and heuristics on the distribution of the class number within the bounds on the class number these heuristics are based on recent results by katz and sarnak our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest

discontinuous viscosity solutions of a degenerate parabolic equation we study a nonlinear degenerate parabolic equation of the second order regularizing the equation by adding some artificial viscosity we construct a generalized solution we show that this solution is not necessarily continuous at all points

the parallelized pollard kangaroo method in real quadratic function fields we show how to use the parallelized kangaroo method for computing invariants in real quadratic function fields specifically we show how to apply the kangaroo method to the infrastructure in these fields we also show how to speed up the computation by using heuristics on the distribution of the divisor class number and by using the relatively inexpensive baby steps in the real quadratic model of a hyperelliptic function field furthermore we provide examples for regulators and class numbers of hyperelliptic function fields of genus 3 that are larger than those ever reported before

homogeneous diffusion in ℝ with power like nonlinear diffusivity we study the nonnegative solutions of the initial value problem ut ur ux p 1ux x u x 0 l1 where p 0 r p 0 the local velocity of propagation of the solutions is identified as v vx vx p 1 where v cu with r p 1 p and c r p r p 1 is the nonlinear potential our main result is the a priori estimate vx vx p 1 x frac 1 2p r t

existence of the maximal weak solution for a class of singular parabolic equations abstract in this paper we study the initial boundary value problem for a class of singular parabolic equations under some additional conditions we prove that there exists a weak solution for the problem by parabolic regularization moreover we show that the solution obtained by parabolic regularization is the maximal solution among all weak solutions

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

free α extensions of an archimedean vector lattice and their topological duals arch denotes the category of archimedean vector lattices with vector lattice homomorphisms and denotes an uncountable cardinal number or the symbol arch denotes the category of arch objects with complete arch morphisms

on some computational problems in finite abelian groups we present new algorithms for computing orders of elements discrete logarithms and structures of finite abelian groups we estimate the computational complexity and storage requirements and we explicitly determine the o constants and constants we implemented the algorithms for class groups of imaginary quadratic orders and present a selection of our experimental results our algorithms are based on a modification of shanks baby step giant step strategy and have the advantage that their computational complexity and storage requirements are relative to the actual order discrete logarithm or size of the group rather than relative to an upper bound on the group order

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

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

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

real and imaginary quadratic representations of hyperelliptic function fields a hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field if at least one of the rational prime divisors is rational over the field of constants then it also can be represented as an imaginary quadratic extension of the rational function field the arithmetic in the divisor class group can be realized in the second case by cantor s algorithm we show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved furthermore we show how the two representations are connected and compare the computational complexity

computation of the class number and class group of a complex cubic field let h and g be respectively the class number and the class group of a complex cubic field of discriminant a a method is described which makes use of recent ideas of lenstra and schoof to develop fast algorithms for finding h and g under certain riemann hypotheses it is shown that these algorithms will compute h in o 1a 1 5 elementary operations and g in o 1a 1 4 elementary operations finally the results of running some computer programs to determine h and g for all pure cubic fields 9 vdi with 2 s d 30 000 are summarized

quasi f covers of tychonoff spaces a tychonoff topological space is called a quasi f space if each dense coz ero set of x is c embedded in x in canad j math 32 1980 657 685 dashiell hager and henriksen construct the minimal quasi f cover qff x of a compact space x as an inverse limit space and identify the ring c qf x as the order cauchy completion of the ring c x in on perfect irreducible preimuges topology proc 9 1984 173 189 vermeer constructed the minimal quasi f cover of an arbitrary tychonoff space in this paper the minimal quasi f cover of a compact space x is constructed as the space of ultrafilters on a certain sublattice of the boolean algebra of regular closed subsets of x the relationship between qf x and qf bx is studied in detail and broad conditions under which b qf x qf bx are obtained together with examples of spaces for which the relationship fails here bx denotes

nonuniqueness of solutions for a singular diffusion problem in this paper we consider a singular diffusion problem and show by constructing a counterexample that the weak solution to the problem is not unique the proof consists of several steps first we prove that there exists a maximal weak solution to the problem we show that the support of the continuous maximal weak solution cannot decrease in time then we cite an example of a nonnegative continuous function with shrinking support that also solves the problem and therefore the problem possesses at least two weak solutions for some continuous nonnegative initial data