on the lattice of cotorsion theories abstract we discuss the lattice of cotorsion theories for abelian groups first we show that the sublattice of the well studied rational cotorsion theories can be identified with the well known lattice of types using a recently developed method for making ext vanish we also prove that any power set together with the ordinary set inclusion and thus any poset can be embedded into the lattice of all cotorsion theories

a survey of public key cryptosystems we give an overview of the most important public key cryptosystems and discuss the difficult task of evaluating the merit of such systems

absolute e rings abstract a ring r with 1 is called an e ring if end z r is ring isomorphic to r under the canonical homomorphism taking the value 1 for any end z r moreover r is an absolute e ring if it remains an e ring in every generic extension of the universe e rings are an important tool for algebraic topology as explained in the introduction the existence of an e ring r of each cardinality of the form 0 was shown by dugas mader and vinsonhaler 1987 9 we want to show the existence of absolute e rings it turns out that there is a precise cardinal barrier for this problem the cardinal is the first erd s cardinal defined in the introduction it is a relative of measurable cardinals we will construct absolute e rings of any size but there are no absolute e rings of cardinality the non existence of huge absolute e rings follows from a recent paper by herden and shelah 2009 24 and the construction of absolute e rings r is based on an old result by shelah 1982 31 where families of absolute rigid colored trees with no automorphism between any distinct members are constructed we plant these trees into our potential e rings with the aim to prevent unwanted endomorphisms of their additive group to survive endomorphisms will recognize the trees which will have branches infinitely often divisible by primes our main result provides the existence of absolute e rings for all infinite cardinals i e these e rings remain e rings in all generic extensions of the universe e g using forcing arguments indeed all previously known e rings dugas mader and vinsonhaler 1987 9 gobel and trlifaj 2006 23 of cardinality 2 0 have a free additive group r in some extended universe thus are no longer e rings as explained in the introduction our construction also fills all cardinal gaps of the earlier constructions which have only sizes 0 these e rings are domains and as a by product we obtain the existence of absolutely indecomposable abelian groups compare gobel and shelah 2007 22

gauss class number problem for imaginary quadratic fields a result first proved by heilbronn h in 1934 the disquisitiones also contains tables of binary quadratic forms with small class numbers actually tables of imaginary quadratic fields of small class number with even discriminant which is a much easier problem to deal with and gauss conjectured that his tables were complete in modern parlance we can rewrite gauss tables we are including both even and odd discriminants in the following form

applying sieving to the computation of quadratic class groups we present a new algorithm for computing the ideal class group of an imaginary quadratic order which is based on the multiple polynomial version of the quadratic steve factoring algorithm although no formal analysis is given we conjecture that our algorithm has sub exponential complexity and computational experience shows that it is significantly faster in practice than existing algorithms

integral similarity and commutators of integral matrices abstract let f be a field m n f the algebra of n n matrices over f and a m n f with trace a 0 the following facts are well known i if a is not a scalar then a is similar over f to a matrix with zero diagonal ii a p q pq qp for some p q m n f we consider the situation when f is replaced by the ring of integers z we show that ii holds in this case for every n 1 this result has been proved for n 2 by lissner and vaserstein independently we show also that i holds if f is replaced by z for n 2 provided a ai mod p for all integers a and primes p

small generators of the ideal class group assuming the generalized riemann hypothesis bach has shown that the ideal class group c k of a number eld k can be generated by the prime ideals of k having norm smaller than 12 logjdiscriminant k j 2 this result is essential for the computation of the class group and units of k by buchmann s algorithm currently the fastest known however once c k has been computed one notices that this bound could have been replaced by a much smaller value and so much work could have been saved we introduce here a short algorithm which allows us to reduce bach s bound substantially usually by a factor 20 or so the bound produced by the algorithm is asymptotically worse than bach s but favorable constants make it useful in practice

an almost full embedding of the category of graphs into the category of groups abstract we construct a functor f g raphs g roups which is faithful and almost full in the sense that every nontrivial group homomorphism f x f y is a composition of an inner automorphism of fy and a homomorphism of the form ff for a unique map of graphs f x y when f is composed with the eilenberg mac lane space construction k f x 1 we obtain an embedding of the category of graphs into the unpointed homotopy category which is full up to null homotopic maps we provide several applications of this construction to localizations i e idempotent functors we show that the questions 1 is every orthogonality class reflective 2 is every orthogonality class a small orthogonality class have the same answers in the category of groups as in the category of graphs in other words they depend on set theory 1 is equivalent to weak vop nka s principle and 2 to vop nka s principle additionally the second question considered in the homotopy category is also equivalent to vop nka s principle

solving sparse linear equations over finite fields a coordinate recurrence method for solving sparse systems of linear equations over finite fields is described the algorithms discussed all require o n 1 omega n 1 log k n 1 field operations where n 1 is the maximum dimension of the coefficient matrix omega is approximately the number of field operations required to apply the matrix to a test vector and the value of k depends on the algorithm a probabilistic algorithm is shown to exist for finding the determinant of a square matrix also probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error

large e rings exist a ring r with 1 is called an e ring provided hom r r z r under the map cp 1 the class of e rings was defined and studied by schultz s in 1973 and further investigated by bowshell and schultz bs in 1977 examples of e rings include z n subrings of q and pure subrings of the ring of p adic integers more interesting examples are the torsion free e rings of finite rank these were characterized in bs as those rings quasi isomorphic to r x r x x r where each ri is a strongly indecomposable subring of an algebraic number field and hom r rj 0 for i j in spite of their seemingly specialized nature such rings have played an important role in the theory of torsion free groups of finite rank dating back to beaumont pierce bpl bp2 and pierce p in 1960 1961 see also aprvw nr pv ri relatively little has been published on infinite rank torsion free e rings indeed until recently the only examples of these were provided by the pure subrings of the p adic integers in this paper the black box of shelah is used to construct a host of large e rings we show that any torsion free p reduced p cotorsion free commutative ring s may be embed88 0021 8693 87 3 00

large localizations of finite groups abstract we construct examples of localizations in the category of groups which take the mathieu group m 11 to groups of arbitrarily large cardinality which are abelian up to finitely many generators the paper is part of a broader study on the group theoretic properties which are or are not preserved by localizations

an l 1 3 algorithm for ideal class group and regulator computation in certain number fields we analyse the complexity of the computation of the class group structure regulator and a system of fundamental units of a certain class of number fields our approach differs from buchmann s who proved a complexity bound of l 1 2 o 1 when the discriminant tends to infinity with fixed degree we achieve a subexponential complexity in o l 1 3 o 1 when both the discriminant and the degree of the extension tend to infinity by using techniques due to enge and gaudry in the context of algebraic curves over finite fields

algorithms in algebraic number theory in this paper we discuss the basic problems of algorithmic algebraic number theory the emphasis is on aspects that are of interest from a purely mathematical point of view and practical issues are largely disregarded we describe what has been done and more importantly what remains to be done in the urea we hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them

a fast rigorous technique for computing the regulator of a real quadratic field we present a new algorithm for computing the regulator of a real quadratic field q d based on an algorithm for unconditionally verifying the correctness of the regulator produced by a subexponential algorithm that runs in expected time o d 1 6 e under the generalized riemann hypothesis the correctness of our algorithm relies on no unproven hypotheses and is currently the fastest known unconditional algorithm for computing the regulator a number of implementation issues and performance enhancements are discussed and we present the results of computations demonstrating the efficiency of the new algorithm

outer automorphisms of locally finite p groups abstract every group is an outer automorphism group of a locally finite p group this extends an earlier result m dugas r gobel on locally finite p groups and a problem of philip hall s j algebra 159 1 1993 115 138 about countable outer automorphism groups it is also in sharp contrast to results concerning the existence of outer automorphisms of nilpotent groups in w gaschutz nichtabelsche p gruppen besitzen aussere p automorphismen j algebra 4 1966 1 2 o puglisi a note on the automorphism group of a locally finite p group bull london math soc 24 5 1992 437 441 a e zalesskii a nilpotent p group has an outer automorphism dokl akad nauk sssr 196 1971 751 754

a rigorous time bound for factoring integers in this paper a probabilistic algorithm is exhibited that factors any positive integer n into prime factors in expected time at most ln 2 1 o for n oo where l a b exp b logx a loglogx l a many practical factoring algorithms including the quadratic sieve and the elliptic curve method are conjectured to have an expected running time that satisfies the same bound but this is the first algorithm for which the bound can be rigorously proved nevertheless this does not close the gap between rigorously established time bounds and merely conjectural ones for factoring algorithms this is due to the advent of a new factoring algorithm the number field sieve which is conjectured to factor any positive integer n in time ln i 0 1 the algorithm analyzed in this paper is a variant of the class group relations method which makes use of class groups of binary quadratic forms of negative discriminant this algorithm was first suggested by seysen and later improved by a k lenstra who showed that the algorithm runs in expected time at most ln 12 1 o 1 if one assumes the generalized riemann hypothesis the main device for removing the use of the generalized riemann hypothesis from the proof is the use of multipliers in addition a character sum estimate for algebraic number fields is used with an explicit dependence on possible exceptional zeros of the corresponding l functions another factoring algorithm using class groups that has been proposed is the random class groups method it is shown that there is a fairly large set of numbers that this algorithm cannot be expected to factor as efficiently as had previously been thought department of mathematics university of california berkeley california 94720 e mail address hwl math berkeley edu department of mathematics university of georgia athens georgia 30602 e mail address carl ada math uga edu this content downloaded from 157 55 39 186 on tue 12 apr 2016 08 54 17 utc all use subject to http about jstor org terms

classifying e algebras over dedekind domains an r algebra a is said to be a generalized e algebra if a is isomorphic to the algebra endr a generalized e algebras have been extensively investigated in this work they are classified modulo cotorsion free modules when the underlying ring r is a dedekind domain

are all localizing subcategories of stable homotopy categories coreflective abstract we prove that in a triangulated category with combinatorial models every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large cardinal axiom vop nka s principle is assumed true it follows that under the same assumptions orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories the existence of such a bijection was left as an open problem by hovey palmieri and strickland in their axiomatic study of stable homotopy categories and also by neeman in the context of well generated triangulated categories

algebraic entropy for abelian groups the theory of endomorphism rings of algebraic structures allows in a natural way a systematic approach based on the notion of entropy borrowed from dynamical systems here we study the algebraic entropy of the endomorphisms of abelian groups introduced in 1965 by adler konheim and mcandrew the so called addition theorem is proved this expresses the algebraic entropy of an endomorphism of a torsion group as the sum of the algebraic entropies of the restriction to a invariant subgroup and of the endomorphism induced on the quotient group particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all of whose endomorphisms have zero algebraic entropy the significance of this class arises from the fact that any group not in this class can be shown to have endomorphisms of infinite algebraic entropy and we also investigate such groups a uniqueness theorem for the algebraic entropy of endomorphisms of torsion abelian groups is proved

geometric and symmetry properties of a nondegenerate diffusion process a diffusion process with smooth and nondegenerate elliptic infinitesimal generator on a manifold m induces a riemannian metric g on m this paper discusses in detail different symmetry properties of such a diffusion by geometric methods partial differential equations associated with the generator are studied likewise with an eye to modelling and applications to filtering relationships between symmetries of deterministic systems and symmetries of diffusion processes are delineated the incidence of a stochastic framework on the properties of an original deterministic system are then illustrated in different examples the construction of a diffusion process with given symmetries is also addressed and resulting geometric problems are raised

automorphisms of torsion free nilpotent groups of class two we construct 2 divisible torsion free abelian groups g admitting an alternating bilinear map we use these groups g to find nilpotent groups n of class 2 such that aut n modulo a natural normal subgroup is a prescribed group

localizations of torsion free abelian groups ii abstract a homomorphism a b between abelian groups a b is called a localization of a if for each hom a b there is a unique end b such that it is well known that if a z then b is an e ring and 1 is the identity of b we investigate localizations of rank 1 groups a l q of type it turns out that localizations of l can be surprisingly complicated if l m is a localization and l is a subring of q then m is simply an e ring that is also an l module if l is not a subring things get more complicated if m m then tensor products of l and e rings come into play it is possible that m m and we can say very little in this case another topic under consideration are localizations of e rings frequently localizations of e rings are e rings again but we find examples where this is not the case

hermite normal form computation using modulo determinant arithmetic this paper describes a new class of hermite normal form solution procedures which perform modulo determinant arithmetic throughout the computation this class of procedures is shown to possess a polynomial time complexity bound which is a function of the length of the input string computational results are also given

symmetry groups and translation invariant representations of markov processes the symmetry groups of the potential theory of a markov process xt are used to introduce new algebraic and topological structures on the state space and the process for example let g be the collection of bijections sp on e which preserve the collection of excessive functions assume there is a transitive subgroup h of the symmetry group g such that the only map 9 e h fixing a point e e e is the identity map on e there is a bijection p e h so that the algebraic structure of h can be carried to e making e into a group if there is a left quasi invariant measure on e then there is a topology on e making e into a locally compact second countable metric group there is also a time change r t of xt such that xt t is a translation invariant process on e and xt t is right continuous with left limits in the new topology

cotilting and a hierarchy of almost cotorsion groups abstract we describe the structure of all cotilting and partial cotilting groups and of all cotilting torsion free classes of groups under v l we also describe all tilting and partial tilting groups as a by product we introduce new classes of groups called almost cotorsion groups unlike the well known cotorsion groups the almost cotorsion ones have a rather complex structure we prove that any cotorsion free ring can be realized as an endomorphism ring of an almost cotorsion group

curves arising from kronecker modules abstract let k be an algebraically closed field a kronecker module m is a pair of k vector spaces s t together with a k bilinear map k 2 s t the space s is called the domain space of m while t is called the range space of m to each power series in k x we attach a kronecker module p whose domain and range spaces are denoted by v and v respectively both v and v are modules over the endomorphism algebra end p of p we show that if end p is non trivial then the sequence of coefficients of is defined by a linear or a quadratic recursion in the quadratic case end p is the coordinate ring of an affine curve an affine curve is called realizable when its coordinate ring is isomorphic to some end p we show that the realizable curves can be constructed up to birational equivalence by pairs of non zero polynomials p q with deg q p the curves realized herein are embedded in k d where d deg p the planarity of these curves when d 3 remains open our main result on realization of curves is that a cubic curve is realizable if and only if its coordinate ring is either k x or the coordinate ring of a cubic in weierstrass form

non simple localizations of finite simple groups abstract often a localization functor in the category of groups sends a finite simple group to another finite simple group we study when such a localization also induces a localization between the automorphism groups and between the universal central extensions as a consequence we exhibit many examples of localizations of finite simple groups which are not simple

e r algebras that are sharply transitive modules abstract we will consider modules over principal ideal domains that are neither fields nor complete discrete valuation rings and show their wild behavior in two instances which we explain for simplicity in the category of abelian groups in the first case we consider constructions of large e rings this question was answered in 1987 by dugas mader vinsonhaler m dugas a mader c vinsonhaler large e rings exist j algebra 108 1 1987 88 101 recall that a ring a is an e ring if the canonical map end z a a 1 is an isomorphism the second question was raised in 1997 by emmanuel dror farjoun and answered recently in r gobel s shelah uniquely transitive torsion free abelian groups in rings modules algebras and abelian groups in pure appl math vol 236 marcel dekker 2004 pp 271 290 there is a non trivial torsion free abelian group g other than z and with pure elements such that its automorphism group autg acts sharply transitive on the pure elements of g these abelian groups g are called ut groups or ut modules for more general rings recall that pure elements of a torsion free abelian group are those divisible only by 1 and note that automorphisms leave the family p g of pure elements of g invariant the two apparently distinct questions can be unified and strengthened the link between them is the notion of a pure invertible ring a which simply means that all pure elements of a z seen as abelian group are units of the commutative ring a for technical reasons to have non trivial pure elements we also require that a is 1 free as abelian group in the first part of this paper we transform farjoun s problem into a ring theoretic question on a it turns out that a must be a pure invertible ring and an e ring at the same time in the second part of the paper we construct such rings assuming zfc ch the special continuum hypothesis thus the existence of ut groups follows and as a byproduct we obtain a new class of e rings which are at the same time principal ideal domains

new quadratic polynomials with high densities of prime values hardy and littlewood s conjecture f implies that the asymptotic density of prime values of the polynomials fa x x2 x a a z is related to the discriminant 1 4a of fa x via a quantity c the larger c is the higher the asymptotic density of prime values for any quadratic polynomial of discriminant a technique of bach allows one to estimate c accurately for any 0 given the class number and regulator of the real quadratic order with discriminant the manitoba scalable sieve unit mssu has shown us how to rapidly generate many discriminants for which c is potentially large and new methods for evaluating class numbers and regulators of quadratic orders allow us co compute accurate estimates of c efficiently even for values of with as many as 70 decimal digits using these methods we were able to find a number of discriminants for which under the assumption of the extended riemann hypothesis c is larger than any previously known examples

every endomorphism of a local warfield module of finite torsion free rank is the sum of two automorphisms abstract a well known result of p hill s 1969 trans amer math soc 141 99 105 says that any endomorphism of a totally projective abelian p group for any odd prime is the sum of two automorphisms we will extend this result to local warfield modules of finite torsion free rank over odd primes

on endomorphism rings of non separable abelian p groups there have been spectacular advances in recent years in the so called realization problem for certain classes of abelian groups and modules the advances have derived mainly from powerful combinatorial set theoretic tools pioneered by shelah in the case of separable abelian p groups these results appear e g in 1 4 51 for non separable abelian p groups the most significant contribution has been 2 our objective in this paper is to extend the results in 2 and derive analogous results to those obtained in 4 51 it is perhaps worth pointing out that our methods are entirely algebraic the necessary set theoretical work has been carried out in the separable case and no further set theoretical arguments are needed in this the work is reminiscent of s all our set theory with one exception is in zfc and this is not acknowledged in the statement of individual results in the exceptional case where we work in godel s constructible universe v l this is acknowledged by appending v l after the statement number the principal realization result can be stated as

cotorsion theories and splitters let r be a subring of the rationals we want to investigate self splitting r modules g that is ext r g g 0 holds and follow schultz 22 to call such modules splitters free modules and torsion free cotorsion modules are classical examples for splitters are there others answering an open problem by schultz 22 we will show that there are more splitters in fact we are able to prescribe their endomorphism r algebras with a free r module structure as a byproduct we are able to answer a problem of salce 21 showing that all rational cotorsion theories have enough injectives and enough projectives

a rigorous subexponential algorithm for computation of class groups let c d denote the gauss class group of quadratic forms of a negative discriminant d or equivalently the class group of the imaginary quadratic field q a we give a rigorous proof that there exists a las ve gas algorithm that will compute the structure of c d with an expected running time of l d vf2 1 bit operations where l d exp vlog d log log d thus of course also includes the computation of the class number h d the cardinality of c d ibm research division almaden research center k53 802 650 harry road san jose california 95 120 6099 current address k mccurley organization 1423 sandia national laboratories albuquerque new mexico 87185 this content downloaded from 207 46 13 180 on thu 08 sep 2016 05 47 29 utc all use subject to http about jstor org terms

a family of commutative endomorphism algebras abstract let k x be the quotient field of the polynomial ring k x over an algebraically closed field k given an integer n 2 a finite sequence m m 2 m n of n 1 positive integers a sequence 2 n of k linear functionals on k x and a height function h there is attached a kronecker module e e m h in this paper we show that when e has no finite dimensional direct summand then end e is a commutative k subalgebra of the algebra of n n matrices over k x consequently when end e is an integral domain its transcendence degree over k is at most one we give sufficient conditions for end e to be isomorphic to k and also show that k is the only field that occurs as end e examples where k x k x x 1 and some subalgebras of k x n occur as end e are given the determination of all the commutative k subalgebras of m n k x realizable as end e remains open yet surprisingly it can be shown that the coordinate ring of a cubic equation in weierstrass normal form y 2 x 3 ax 2 bx c is realizable

homotopical localizations of module spectra we prove that stable f localizations where f is any map of spectra preserve ring spectrum structures and module spectrum structures under suitable hypotheses and we use this fact to describe all possible localizations of the integral eilenberg mac lane spectrum hz as a consequence of this study we infer that localizations of stable gems are stable gems and it also follows that there is a proper class of nonequivalent stable localizations

implications of large cardinal principles in homotopical localization the existence of arbitrary cohomological localizations on the homotopy category of spaces has remained unproved since bousfield settled the same problem for homology theories in the decade of 1970 this is related with another open question namely whether or not every homotopy idempotent functor on spaces is an f localization for some map f we prove that both questions have an affirmative answer assuming the validity of a suitable large cardinal axiom from set theory vop nka s principle we also show that it is impossible to prove that all homotopy idempotent functors are f localizations using the ordinary zfc axioms of set theory zermelo fraenkel axioms with the axiom of choice since a counterexample can be displayed under the assumption that all cardinals are nonmeasurable which is consistent with zfc

a probabilistic factorization algorithm with quadratic forms of negative discriminant we propose a probabilistic algorithm for factorization of an integer n with run time expvlog n log log n v54 1 asymptotically our algorithm will be as fast as the wellknown factorization algorithm of morrison and brillhart the latter algorithm will fail in several cases and heuristic assumptions are needed for its run time analysis our new algorithm will be analyzed under the assumption of the extended riemann hypothesis and it will be of las vegas type on input n the new algorithm will factor n with probability 12 in case of prime n the algorithm will prove the primality of n with probability 2 introduction until the last decade the centuries old problem of factoring integers was mainly a problem for specialists worldwide interest in factoring integers increased dramatically in 1978 when rivest shamir and adleman 32 published their public key cryptosystem whose security relies on the fact that some large integers are hard to factor gauss 9 already discovered a close connection between the factorization of a natural number n and the theory of quadratic forms of discriminant 4n now quadratic forms are one of the most important tools for factoring integers examples for efficient factorization algorithms are among others the algorithms of morrison and brillhart 26 and lenstra and schnorr 22 the former works with the continued fraction expansion of fn which is closely related to the theory of quadratic forms of discriminant 4n see 20 while the latter works with quadratic forms of discriminant 4n at present the most efficient factorization algorithm is the quadratic sieve algorithm see 29 which can also be expressed in terms of quadratic forms for an overview of modem factorization algorithms we refer to the papers of guy 10 monier 25 and pomerance 29 a deeper understanding of the theory of quadratic forms is of great importance for the analysis of modem factorization algorithms in this paper we shall only deal with the theory of quadratic forms of negative discriminant which is considerably more simple than the theory of forms of positive discriminants using this theory we obtain a probabilistic factorization algorithm with run time exp vlog n log log n v4 in this paper we denote by log x the natural logarithm of x we have 5 4 1 118 received may 17 1985 revised june 12 1986 1980 mathematics subject classification primary 10a30 68c25 1987 american mathematical society 0025 5718 87 1 00 25 per page

symmetries of stochastic differential equations a geometric approach a new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations sdes the correspondence between an algebra of weak symmetries for a given sde and an algebra of strong symmetries for a modified sde is proved under suitable regularity assumptions this general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional brownian motion

statistical evidence for small generating sets for an integer n let g n denote the smallest x such that the primes x generate the multiplicative group modulo n we offer heuristic arguments and numerical data supporting the idea that g n log 2 1 log n log log n asymptotically we believe that the coefficient 1 log 2 is optimal finally we show the average value of g n for n n is at least 1 o l loglognlogloglogn and give a heuristic argument that this is also an upper bound this work gives additional evidence independent of the erh that primality testing can be done in deterministic polynomial time if our bound on g n is correct there is a deterministic primality test using o log n 2 multiplications modulo n

computing hilbert class polynomials with the chinese remainder theorem we present a space efficient algorithm to compute the hilbert class polynomial h d x modulo a positive integer p based on an explicit form of the chinese remainder theorem under the generalized riemann hypothesis the algorithm uses o d 1 2 e log p space and has an expected running time of o d 1 e we describe practical optimizations that allow us to handle larger discriminants than other methods with d as large as 10 13 and h d up to 10 6 we apply these results to construct pairing friendly elliptic curves of prime order using the cm method

bilinear forms in weyl sums for modular square roots and applications abstract let q be a prime p 1 and let n q p denote the number of rational primes p p that split in the imaginary quadratic field q q the first part of this paper establishes various unconditional and conditional under existence of a siegel zero lower bounds for n q p in the range q 1 4 e p q for any fixed e 0 this improves upon what is implied by work of pollack and benli pollack the second part of this paper is dedicated to proving an estimate for a bilinear form involving weyl sums for modular square roots equivalently salie sums our estimate has a power saving in the so called polya vinogradov range and our methods involve studying an additive energy coming from quadratic residues in f q this bilinear form is inspired by the recent automorphic motivation the second moment for twisted l functions attached to kohnen newforms has recently been computed by the first and fourth authors so the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form these include the equidistribution of quadratic roots of primes products of primes and relaxations of a conjecture of erd s odlyzko sarkozy

solving thue equations without the full unit group the main problem when solving a thue equation is the computation of the unit group of a certain number field it is shown that the knowledge of a subgroup of finite index is in fact sufficient two examples linked with the primitive divisor problem for lucas and lehmer sequences are given la principale difficulte lors de la resolution d une equation de thue reside dans le calcul du groupe des unites du corps de nombres associe on montre qu il suffit en fait de connaitre un sous groupe d indice fini de ce groupe on donne deux exemples li

localization of algebras over coloured operads we give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories our approach encompasses a number of previous results about preservation of structures under localizations such as loop spaces or infinite loop spaces and provides new results of the same kind for instance under suitable assumptions homotopical localizations preserve ring spectra in the strict sense not only up to homotopy modules over ring spectra and algebras over commutative ring spectra as well as maps between ring spectra module maps and algebra maps it is principally the treatment of module spectra and their maps that led us to the use of coloured operads also called enriched multicategories in this context

conditional bounds for the least quadratic non residue and related problems this paper studies explicit and theoretical bounds for several inter esting quantities in number theory conditionally on the generalized riemann hypothesis specically we improve the existing explicit bounds for the least qua dratic non residue and the least prime in an arithmetic progression we also rene the classical conditional bounds of littlewood for l functions at s 1 in par ticular we derive explicit upper and lower bounds for l 1 and 1 it and deduce explicit bounds for the class number of imaginary quadratic elds finally we improve the best known theoretical bounds for the least quadratic non residue and more generally the least k th power non residue

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