the perron generalized eigenspace and the spectral cone of a cone preserving map a unified treatment is offered to reprove known results on the following four highlights of the combinatorial spectral theory of nonnegative matrices or to extend or partly extend the results to the setting of a linear map preserving a polyhedral proper or proper cone the preferred basis theorem equivalent conditions for equality of the graph theoretic level characteristic and the spectral height characteristic the majorization relation between the two characteristics and the relation between the combinatorial properties of a nonnegative matrix and the positivity of the individual entries in its principal components this is achieved by employing the new concept of spectral cone of a cone preserving map and combining the cone theoretic methods developed in our previous papers on the geometric spectral theory of cone preserving maps with the algebraic analytic method introduced by hartwig neumann and rose and further exploited by neumann and schneider for nonnegative matrices

prime power terms in elliptic divisibility sequences we study a problem on specializations of multiples of rational points on elliptic curves analogous to the mersenne problem we solve this problem when descent via isogeny is possible by giving explicit bounds on the indices of prime power terms in elliptic divisibility sequences associated to points in the image of a nontrivial isogeny we also discuss the uniformity of these bounds assuming the hall lang conjecture

on invariant subspaces of matrices a new proof of a theorem of halmos abstract lmos proved that if a is a matrix and if e is an a invariant subspace then there exist matrices b and c such that ba ab ca ac e is the kernel of b and e is the range of c we present an elementary proof of this result and show that there exist b and c that additionally satisfy bc cb o

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

characteristic invariant subspaces generated by a single vector abstract if f is an endomorphism of a finite dimensional vector space v over a field k then an invariant subspace x v is called hyperinvariant respectively characteristic if x is invariant under all endomorphisms respectively automorphisms that commute with f the characteristic hull of a subset w of v is defined to be the smallest characteristic subspace in v that contains w it is known that characteristic subspaces that are not hyperinvariant can only exist when k 2 in this paper we study subspaces x which are the characteristic hull of a single element in the case where k 2 we derive a necessary and sufficient condition such that x is hyperinvariant

congruences between derivatives of abelian l functions at s 0 let k k be a finite abelian extension of global fields we prove that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s 0 of derivatives of the dirichlet l functions associated to k k we also show that these congruences provide a universal approach to the refined abelian stark conjectures formulated by inter alia stark gross rubin popescu and tate we thereby obtain the first proofs of amongst other things the rubin stark conjecture and the refined class number formulas of both gross and tate for all extensions k k in which k is either an abelian extension of or is a function field

using the theory of cyclotomy to factor cyclotomic polynomials over finite fields we examine the problem of factoring the rth cyclotomic polynomial r x over fp r and p distinct primes given the traces of the roots of r x we construct the coefficients of r x in time o r 4 we demonstrate a deterministic algorithm for factoring r x in time o r 1 2 log p 9 when r x has precisely two irreducible factors finally we present a deterministic algorithm for computing the sum of the irreducible factors of r x in time o r 6

remarks on the schoof elkies atkin algorithm schoof s algorithm computes the number m of points on an elliptic curve e defined over a finite field f q schoof determines m modulo small primes using the characteristic equation of the frobenius of e and polynomials of degree o 2 with the works of elkies and atkin we have just to compute when is a good prime an eigenvalue of the frobenius using polynomials of degree o in this article we compute the complexity of muller s algorithm which is the best known method for determining one eigenvalue and we improve the final step in some cases finally when is bad we describe how to have polynomials of small degree and how to perform computations in schoof s algorithm on x values only

frobenius maps of abelian varieties and finding roots of unity in finite fields if twere done when tis done then twere well it were done quickly macbeth abstract we give a generalization to abelian varieties over finite fields of the algorithm of schoof for elliptic curves schoof showed that for an elliptic curve e over f given by a weierstrass equation one can compute the number of q f rational points of e in time 0 log our result is the following let a be an abelian variety over f then one can compute the characteristic polynomial of the frobenius endomorphism of a in time 0 o q where a and the implied constant depend only on the dimension of the embedding space of a the number of equations defining a and the addition law and their degrees the method generalizing that of schoof is to use the machinery developed by weil to prove the riemann hypothesis for abelian varieties by means of this theory the calculation is reduced to ideal theoretic computations in a ring of polynomials in several variables over f as applications we show how to count the rational points on the reductions modulo primes p of a fixed curve over q in time polynomial in logp we show also that for a fixed prime we can compute the th roots of unity mod p when they exist in polynomial time in logp this generalizes schoof s application of his algorithm to find square roots of a fixed integer x mod p in this paper we generalize to abelian varieties over finite fields the algorithm of schoof 19 for elliptic curves over finite fields and the application given by schoof for his algorithm schoof showed that for an elliptic curve e over a finite field f given by a weierstrass equation one can compute the number of f rational points of f in time polynomial in log q the algorithm computes the characteristic polynomial p t g z t of the frobenius endomorphism of f for an elliptic curve p t is a monic quadratic polynomial with constant term q and the number of fq rational points of e is f l our result is the following theorem a let a be an abelian variety over a finite field f given explicitly as a projective variety with an explicit addition law then one can compute

the witten genus and equivariant elliptic cohomology we construct a thom class in complex equivariant elliptic cohomology extending the equivariant witten genus this gives a new proof of the rigidity of the witten genus which exhibits a close relationship to recent work on non equivariant orientations of elliptic spectra

inflations of self affine tilings are integral algebraic perron we prove that any expanding linear map phi mathbb r mathtt d rightarrow mathbb r mathtt d that is the inflation map in an inflation substitution process generating a self affine tiling of mathbb r mathtt d is integral algebraic and perron this means that phi is linearly conjugate to a restriction of an integer matrix to a subspace mathcal e satisfying a maximal growth condition that generalizes the characterization of perron numbers as numbers that are larger than the moduli of their algebraic conjugates the case of diagonalizable phi has been previously resolved by richard kenyon and boris solomyak and it is rooted in thurston s idea of lifting the tiling from the physical space mathbb r mathtt d to a higher dimensional mathematical space where the tiles their control points sit on a lattice the main novelty of our approach is in lifting the inflation substitution process to the mathematical space and constructing a certain vector valued cocycle defined over the translation induced mathbb r mathtt d action on the tiling space the subspace mathcal e is obtained then by ergodic averaging of the cocycle more broadly we assemble a powerful framework for studying self affine tiling spaces

p adic properties of division polynomials and elliptic divisibility sequences for a fixed rational point p e k on an elliptic curve we consider the sequence of values f n p n 1 of the division polynomials of e at p for a finite field open image in new window we prove that the sequence is periodic for a local field open image in new window we prove under certain hypotheses that there is a power q p e so that for all m 1 the limit of open image in new window exists in k and is algebraic over open image in new window we apply this result to prove an analogous p adic limit and algebraicity result for elliptic divisibility sequences

geometric characterization and classification of marked subspaces abstract given an endomorphism of a finite dimensional c vector space we obtain geometric conditions in order to characterize and to classify the marked invariant subspaces w e that is to say the invariant subspaces having some jordan basis which can be extended to a jordan basis of e both conditions are expressed in terms of the double family of subspaces ker f h im f d the starting point is a new criterion for extensibility of jordan bases

an algorithm to compute volcanoes of 2 isogenies of elliptic curves over finite fields the goal of this paper is presenting an algorithm to determine the structure of the volcano of 2 isogenies of a given elliptic curve over a finite field the core of the algorithm relies on the relationship between the 2 torsion structure of the curves and its level in the volcano as well as on those results that determine the direction of the different outgoing isogenies from each vertex the algorithm is specially efficient for the so called regular volcanoes where the 2 torsion structure is different at every level

super golden gates for pu 2 abstract to each of the symmetry groups of the platonic solids we adjoin a carefully designed involution yielding topological generators of pu 2 which have optimal covering properties as well as efficient navigation these are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups the generators give super efficient 1 qubit quantum gates and are natural building blocks for the design of universal quantum gates

on the equivariant tamagawa number conjecture for tate motives let l be a finite abelian extension of and let k be any subfield of l for each integer r with r 0 we prove the equivariant tamagawa number conjecture for the pair h0 spec l r gal l k

brief paper a class of marked invariant subspaces with an application to algebraic riccati equations invariant subspaces of a matrix a are considered which are obtained by truncation of a jordan basis of a generalized eigenspace of a we characterize those subspaces which are independent of the choice of the jordan basis an application to hamilton matrices and algebraic riccati equations is given

a generic approach to searching for jacobians we consider the problem of finding cryptographically suitable jacobians by applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family we can search for jacobians containing a large subgroup of prime order for a suitable distribution of curves the complexity is subexponential in genus 2 and o n1 12 in genus 3 we give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size improving previous results our approach is particularly effective over low degree extension fields where in genus 2 we find jacobians over fp2 and trace zero varieties over fp3 with near prime orders up to 372 bits in size for p 261 1 the average time to find a group with 244 bit near prime order is under an hour on a pc

eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations abstract udy the perturbation theory of structured matrices under structured rank one perturbations and then focus on several classes of complex matrices generic jordan structures of perturbed matrices are identified it is shown that the perturbation behavior of the jordan structures in the case of singular j hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in 18 28 30 31 thus a generic structured perturbation would not be generic if considered as an unstructured perturbation in other settings of structured matrices the generic perturbation behavior of the jordan structures within the confines imposed by the structure follows the pattern of that of unstructured perturbations

nonlinear neumann boundary conditions for quasilinear degenerate elliptic equations and applications abstract we prove comparison results between viscosity sub and supersolutions of degenerate elliptic and parabolic equations associated to possibly nonlinear neumann boundary conditions these results are obtained under more general assumptions on the equation in particular the dependence in the gradient of the solution and they allow applications to quasilinear possibly singular elliptic or parabolic equations one of the main applications is the extension of the so called level set approach for equations set in bounded domains with nonlinear neumann boundary conditions in such a framework the level set approach provides a weak notion for the motion of hypersurfaces with curvature dependent velocities and a prescribed contact angle at the boundary

on a b t invariant subspaces having extendible brunovsky bases we consider a by invariant subspaces having a brunovsky basis which can be extended to a brunovsky basis of the whole space we obtain a geometrical characterization of this class of a by invariant subspaces and a complete family of numerical invariants to classify them 0 elsevier science inc 1997

local c 0 α estimates for viscosity solutions of neumann type boundary value problems abstract in this article we prove the local c 0 regularity and provide c 0 estimates for viscosity solutions of fully nonlinear possibly degenerate elliptic equations associated to linear or nonlinear neumann type boundary conditions the interest of these results comes from the fact that they are indeed regularity results and not only a priori estimates from the generality of the equations and boundary conditions we are able to handle and the possible degeneracy of the equations we are able to take into account in the case of linear boundary conditions

sigma function solution of the initial value problem for somos 5 sequences the somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients for particular choices of coefficients and initial data integer sequences arise by making the connection with a second order nonlinear mapping with a first integral we prove that the two subsequences of odd even index terms each satisfy a somos 4 fourth order recurrence this leads directly to the explicit solution of the initial value problem for the somos 5 sequences in terms of the weierstrass sigma function for an associated elliptic curve

on the discrete logarithm problem in class groups of curves we study the discrete logarithm problem in degree 0 class groups of curves over finite fields with particular emphasis on curves of small genus we prove that for every fixed g 2 the discrete logarithm problem in degree 0 class groups of curves of genus g can be solved in an expected time of o q 2 2 g where f q is the ground field this result generalizes a corresponding result for hyperelliptic curves given in imaginary quadratic representation with cyclic degree 0 class group and just as this previous result it is obtained via an index calculus algorithm with double large prime variation generalizing this result we prove that for fixed g 0 2 the discrete logarithm problem in class groups of all curves c f q of genus g g 0 can be solved in an expected time of o q g 2 g 0 1 1 g 0 and in an expected time of o cl 0 c 2 g 0 1 1 g0 as a complementary result we prove that for any fixed n with n 2 the discrete logarithm problem in the groups of rational points of elliptic curves over finite fields f q n q a prime power can be solved in an expected time of o q 2 2 n furthermore we give an algorithm for the efficient construction of a uniformly randomly distributed effective divisor of a specific degree given the curve and its l polynomial

p adic heights of heegner points and λ adic regulators let e be an elliptic curve defined over q the aim of this paper is to make it possible to compute heegner l functions and anticyclotomic adic regulators of e which were studied by mazur rubin and howard we generalize results of cohen and watkins and thereby compute heegner points of nonfundamental discriminant we then prove a relationship between the denominator of a point of e defined over a number field and the leading coefficient of the minimal polynomial of its xcoordinate using this relationship we recast earlier work of mazur stein and tate to produce effective algorithms to compute p adic heights of points of e defined over number fields these methods enable us to give the first explicit examples of heegner l functions and anticyclotomic adic regulators

on taking square roots without quadratic nonresidues over finite fields we present a novel idea to compute square roots over finite fields without being given any quadratic nonresidue and without assuming any unproven hypothesis the algorithm is deterministic and the proof is elementary in some cases the square root algorithm runs in o log 2 q bit operations over finite fields with q elements as an application we construct a deterministic primality proving algorithm which runs in o log 3 n for some integers n

on the change of the jordan form under the transition from the adjacency matrix of a vertex transitive digraph to its principal submatrix of co order one abstract let j n 1 n k be the set of matrices a such that is an eigenvalue of a and n 1 n k are the sizes of the jordan blocks associated with for a given index v of a denote by a v the principal submatrix of co order one obtained from a by deleting the v th row and column in the present paper all possible changes of the part of the jordan form corresponding to under the transition from a to a v are determined for matrices a j n 1 n k such that for the eigenvalue of both a and a there exists a jordan chain of the largest length n k whose eigenvector has nonzero v th entry in particular it is shown that for almost every matrix a j n 1 n k n 1 n k 1 are the sizes of jordan blocks for considered as an eigenvalue of a v moreover it is also proved that if a is the adjacency matrix of a vertex transitive digraph and k 2 then the change n 1 n k n 1 n k 2 2 n k 1 1 holds for the eigenvalue under the transition from a to a v in the case of k 1 is a simple eigenvalue of a and does not belong to the spectrum of a v

singular graph and extension of jordan chains of an m matrix abstract let s be a singular graph we study conditions on the vertices of s to extend jordan chains for the generalized nullspace of each minus m matrix a with singular graph s the conditions are the interlevel the strong interlevel and the strong predecessor properties

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

miniversal deformations of marked matrices abstract given the set of square matrices m m n m c that keep the subspace w c n x 0 c n m invariant we obtain the implicit form of a miniversal deformation of a matrix a m and we compute it explicitely when this matrix is marked this is if there is a permutation matrix p m n m c such that p 1 ap is a jordan matrix we derive some applications to tackle the classical carlson problem

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

reducing elliptic curve logarithms to logarithms in a finite field elliptic curve cryptosystems have the potential to provide relatively small block size high security public key schemes that can be efficiently implemented as with other known public key schemes such as rsa and discrete exponentiation in a finite field some care must be exercised when selecting the parameters involved in this case the elliptic curve and the underlying field specific classes of curves that give little or no advantage over previously known schemes are discussed the main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logarithm problem in the multiplicative group of an extension of the underlying finite field for the class of supersingular elliptic curves the reduction takes probabilistic polynomial time thus providing a probabilistic subexponential time algorithm for the former problem

sous espaces invariants d un operateur nilpotent sur un espace de banach resume soit a un operateur lineaire continu nilpotent sur un espace de banach reel ou complexe nous montrons que pour tout sous espace invariant separable m de a il existe un commutant de a dont l image est dense dans m

description of characteristic non hyperinvariant subspaces over the field gf 2 given a square matrix a an a invariant subspace is called hyperinvariant respectively characteristic if and only if it is also invariant for all matrices t respectively nonsingular matrices t that commute with a shoda s theorem gives a necessary and sufficient condition for the existence of characteristic non hyperinvariant subspaces for a nilpotent matrix in gf 2 here we present an explicit construction for all subspaces of this type

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

modular equations for hyperelliptic curves we dene modular equations describing the torsion subgroups of the jacobian of a hyperelliptic curve over a nite base eld we prove factorization properties that extend the well known results used in atkin s improvement of schoof s genus 1 point counting algorithm

on the cyclotomic main conjecture for the prime 2 we complete the proof of the equivariant tamagawa number conjecture for tate motives over absolutely abelian fields by proving a refined cyclotomic main conjecture at the prime 2

on the number of elliptic pseudoprimes for an elliptic curve e with complex multiplication by an order in k q v d a point p of infinite order on e and any prime p with d i p 1 we have that p 1 p o mod p where 0 is the point at infinity and calculations are done using the addition law for e any composite number which satisfies these conditions is called an elliptic pseudoprime in this paper it is shown that assuming the generalized riemann hypothesis elliptic pseudoprimes are less numerous than primes in particular on the grh the number of elliptic pseudoprimes less than x is o x log log x log2 x for certain curves it is shown that infinitely many elliptic pseudoprimes exist

note on taking square roots modulo n in this article it is shown how gauss 1981 famous cyclotomic sum formula can be used for extracting square roots modulo n

algorithms for the arithmetic of elliptic curves using iwasawa theory we explain how to use results from iwasawa theory to obtain information about p parts of tate shafarevich groups of specific elliptic curves over q our method provides a practical way to compute x e q p in many cases when traditional p descent methods are completely impractical and also in situations where results of kolyvagin do not apply e g when the rank of the mordell weil group is greater than 1 we apply our results along with a computer calculation to show that x e q p 0 for the 1 534 422 pairs e p consisting of a non cm elliptic curve e over q with conductor 30 000 rank 2 and good ordinary primes p with 5 p 1000 and surjective mod p representation

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

extensions of jordan bases for invariant subspaces of a matrix a characterization is obtained for the matrices a with the property that every some jordan basis of every a invariant subspace can be extended to a jordan basis of a these results are based on a criterion for a jordan basis of an invariant subspace to be extendable to a jordan basis of the whole space the criterion involves two concepts the constancy property and the depth property

constructing invariant subspaces as kernels of commuting matrices abstract given an n n matrix a over c and an invariant subspace n a straightforward formula constructs an n n matrix n that commutes with a and has n ker n for q a matrix putting a into jordan canonical form j q 1 a q we get n q 1 m where m ker m is an invariant subspace for j with m commuting with j in the formula m p z t 1 p t the matrices z and t are m m and p is an n m row selection matrix if n is a marked subspace m n and z is an n n block diagonal matrix and if n is not a marked subspace then m n and z is an m m near diagonal block matrix strikingly each block of z is a monomial of a finite dimensional backward shift each possible form of z is easily arranged in a lattice structure isomorphic to and thereby displaying the complete invariant subspace lattice l a for a

sur l orbite d un espace de banach sous l action du commutant d un operateur nilpotent resume soit a un operateur nilpotent sur un espace de banach e nous demontrons que l orbite de e par l action du commutant de a contient les f lat a qui sont de dimension finie

the radius of convergence of the p adic sigma function the purpose of this article is to investigate the radius of convergence of the p adic sigma function of elliptic curves especially when p is a prime of supersingular reduction as an application we prove certain p divisibility of critical values of hecke l functions of imaginary quadratic fields at inert primes

using partial smoothness of p 1 for factoring polynomials modulo p let an arbitrarily small positive constant less than 1 and a poly nomial f with integer coefficients be fixed we prove unconditionally that j modulo p can be completely factored in deterministic polynomial time if p 1 has a lnp o 1 smooth divisor exceeding p we also address the issue of factoring f modulo p over finite extensions of the prime field f p and show that p 1 can be replaced by p k 1 k n for explicit classes of primes p

continuous dependence results for non linear neumann type boundary value problems abstract we obtain estimates on the continuous dependence on the coefficient for second order non linear degenerate neumann type boundary value problems our results extend previous work of cockburn et al jakobsen and karlsen and gripenberg to problems with more general boundary conditions and domains a new feature here is that we account for the dependence on the boundary conditions as one application of our continuous dependence results we derive for the first time the rate of convergence for the vanishing viscosity method for such problems we also derive new explicit continuous dependence on the coefficients results for problems involving bellman isaacs equations and certain quasilinear equation

computing l l isogenies in polynomial time on jacobians of genus 2 curves in this paper we compute l isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l when l is an odd prime which is coprime to the characteristic we use level n symmetric theta structure where n 2 or n 4 in a second part of this paper we explain how to convert between mumford coordinates of jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4 combined with the preceding algorithm this gives a method to compute l l isogenies in polynomial time on jacobians of genus 2 curves

on the orbit of invariant subspaces of linear operators in finite dimensional spaces new proof of a halmos s result abstract p r halmos proved that for a linear operator a over a finite dimensional complex vector space e every a invariant subspace of e is the range of a commutant of a his proof was based on a generalization of the concept of eigenvector in this note we give an invariant proof of this halmos s theorem

on the existence of a common eigenvector for all matrices in the commutant of a single matrix the main purpose of this paper is to study common invariant subspaces of any matrix in the centralizer of a given matrix a mn f where f denotes an algebraically closed field in particular we obtain a necessary and sufficient condition for the existence of a common eigenvector for all the matrices in this set

peak characteristic and nonnegative signature abstract we introduce the concepts of peak characteristic of an m matrix a and of peak signature and nonnegative signature of subsets of the generalized nullspace e a of a and we discuss the relations between them we also study completions of sets to height bases and to level bases our results are then applied to the study of the nonnegativity of height bases for e a in particular we show that a sequence can serve as the nonnegative signature of some height basis for e a if and only if it is less than or equal to the peak characteristic of a

the distribution of lucas and elliptic pseudoprimes let l x denote the counting function for lucas pseudoprimes and e x denote the elliptic pseudoprime counting function we prove that for large x l x x l x 1 2 and e x x l x 1 3 where l x exp log xlogloglog x loglog x

characteristic and hyperinvariant subspaces over the field gf 2 let f be an endomorphism of a vector space v over a field k an f invariant subspace x v is called hyperinvariant respectively characteristic if x is invariant under all endomorphisms respectively automorphisms that commute with f if k 2 then all characteristic subspaces are hyperinvariant if k 2 then there are endomorphisms f with invariant subspaces that are characteristic but not hyperinvariant in this paper we give a new proof of a theorem of shoda which provides a necessary and sufficient condition for the existence of characteristic non hyperinvariant subspaces

explicit units and the equivariant tamagawa number conjecture let l k be a finite abelian extension of number fields of group g we study the tamagawa number t l k of h 0 spec l considered as a motive defined over k and with coefficients inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i g for a large class of extensions l k we interpret the conjectural vanishing of t l k in terms of the existence of s units in l satisfying a variety of explicit conditions these explicit conditions are in the same spirit as but are in general much finer than the conditions studied by rubin and popescu by using this approach we are able to complete the proof that t l k vanishes for all of the genus field extensions considered by frohlich in the course of proving this result we find that for certain extensions the vanishing of t l k is a refinement of the main result of solomon concerning wild euler systems

counting points on hyperelliptic curves in average polynomial time let g 1 and let q2 z x be a monic squarefree polynomial of degree 2g 1 for an odd prime p not dividing the discriminant of q let zp t denote the zeta function of the hyperelliptic curve of genus g over the nite eld fp obtained by reducing the coecients of the equation y 2 q x modulo p we present an explicit deterministic algorithm that given as input q and a positive integer n computes zp t simultaneously for all such primes p n whose average complexity per prime is polynomial in g logn and the number of bits required to represent q