Showing papers on "Field (mathematics) published in 2005"

Algebraic Number Fields

Abstract: Subrings of fields Complete fields Decomposition groups and the Artin map Analytic methods and ray classes Class field theory Quadratic fields Appendix References Index.

The GL 2 Main Conjecture for Elliptic Curves without Complex Multiplication

Abstract: Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is isomorphic to Z p. We prove the existence of a canonical Ore set S* of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S*, we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its p-primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an elliptic curve E over Q, without complex multiplication, over the field F generated by the coordinates of all its p-power division points; here p is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over Q.

Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center

Abstract: The SL(2,Z) representation $\pi$ on the center of the restricted quantum group U_{q}sl(2) at the primitive 2p-th root of unity is shown to be equivalent to the SL(2,Z) representation on the extended characters of the logarithmic (1,p) conformal field theory model. The multiplicative Jordan decomposition of the U_{q}sl(2) ribbon element determines the decomposition of $\pi$ into a ``pointwise'' product of two commuting SL(2,Z) representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2,Z) representation on the (1,p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of U_{q}sl(2) at the primitive 2p-th root of unity is shown to coincide with the fusion algebra of the (1,p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of~U_{q}sl(2).

Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2

Abstract: First we describe the Skjelbred-Sund method for classifying nilpotent Lie algebras. Then we use it to classify 6-dimensional nilpotent Lie algebras over any field of characteristic not 2. The proof of this classification is essentially constructive: for a given 6-dimensional nilpotent Lie algebra L, following the steps of the proof, it is possible to find a Lie algebra M that occurs in the list, and an isomorphism L --> M.

Finite hochschild cohomology without finite global dimension

Abstract: Dieter Happel asked the following question: If the $n$-th Hochschild cohomology group of a finite dimensional algebra $\Gamma$ over a field vanishes for all sufficiently large $n$, is the global dimension of $\Gamma$ finite? We give a negative answer to this question.

About mutually unbiased bases in even and odd prime power dimensions

Abstract: Mutually unbiased bases generalize the X, Y and Z qubit bases. They possess numerous applications in quantum information science. It is well known that in prime power dimensions N = pm (with p prime and m a positive integer), there exists a maximal set of N + 1 mutually unbiased bases. In the present paper, we derive an explicit expression for those bases, in terms of the (operations of the) associated finite field (Galois division ring) of N elements. This expression is shown to be equivalent to the expressions previously obtained by Ivanovic (1981 J. Phys. A: Math. Gen. 14 3241) in odd prime dimensions, and Wootters and Fields (1989 Ann. Phys. 191 363) in odd prime power dimensions. In even prime power dimensions, we derive a new explicit expression for the mutually unbiased bases. The new ingredients of our approach are, basically, the following: we provide a simple expression of the generalized Pauli group in terms of the additive characters of the field, and we derive an exact groupal composition law between the elements of the commuting subsets of the generalized Pauli group, renormalized by a well-chosen phase-factor.

Cyclic homology, cdh-cohomology and negative K-theory

Abstract: We prove a blow-up formula for cyclic homology which we use to show that infinitesimal $K$-theory satisfies $cdh$-descent. Combining that result with some computations of the $cdh$-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic $K$-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.

Rectifying curves as centrodes and extremal curves

Abstract: Rectifying curves are introduced in [2] as space curves whose position vector always lies in its rectifying plane. In this article, we establish a surprising simple relationship between rectifying curves and the notion of centrodes in mechanics. Furthermore, we show that rectifying curves are indeed the extremal curves which satisfy the equality case of a general inequality. Further geometric properties of rectifying curves are also presented. 1. Rectifying curves. Let E denote Euclidean three-space, with its inner product 〈 , 〉 and let S be the unit sphere in E centered at the origin. Consider a unit-speed space curve x : I → E, where I is a real interval, that has at least four continuous derivatives. Let t denote x. It is possible, in general, that t(s) = 0 for some s; however, we assume that this never happens. Then we can introduce a unique vector field n and positive function κ so that t = κn. We call t the curvature vector field, n the principal normal vector field, and κ the curvature of the given curve. Since t is a constant length vector field, n is orthogonal to t. The binormal vector field is defined by b = t × n. It is a unit vector field orthogonal to both t Received by the editors April 22, 2004. AMS 2000 Subject Classification: Primary: 53A15; Secondary 53C40, 53C42.

The rational numbers as an abstract data type

Abstract: We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0−1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting systemq

On the distribution of the maximum of a Gaussian field with d parameters

Abstract: Let I be a compact d-dimensional manifold, let X:I\\to R be a Gaussian process with regular paths and let F_I(u), u\\in R, be the probability distribution function of sup_{t\\in I}X(t). We prove that under certain regularity and nondegeneracy conditions, F_I is a C^1-function and satisfies a certain implicit equation that permits to give bounds for its values and to compute its asymptotic behavior as u\\to +\\infty. This is a partial extension of previous results by the authors in the case d=1. Our methods use strongly the so-called Rice formulae for the moments of the number of roots of an equation of the form Z(t)=x, where Z:I\\to R^d is a random field and x is a fixed point in R^d. We also give proofs for this kind of formulae, which have their own interest beyond the present application.

Some results and questions on castelnuovo-mumford regularity

Abstract: §1. The two most classical definitions of Castelnuovo-Mumford regularity Let R := k[X1, . . . , Xn] be a polynomial ring over a field k and M a finitely generated graded R-module. The two most popular definitions of Castelnuovo-Mumford regularity are the one in terms of graded Betti numbers and the one using local cohomology. — Local cohomology modules. Set m := (X1, . . . , Xn) = R>0, then H m(M) := {x ∈M | ∃N, mx = 0} and the functors H m(—) can be defined as right derived functors of H 0 m(—) in the category of R-modules. A more concrete way of considering these modules, is to see them as the cohomology modules of the Cech complex C• m :

Pseudorandom generators for low degree polynomials

Abstract: We investigate constructions of pseudorandom generators that fool polynomial tests of degree d in m variables over finite fields F. Our main construction gives a generator with seed length O(d4 log m (1 + log(d ⁄ e) ⁄ log log m) + log |F|) bits that achieves arbitrarily small bias e and works whenever |F| is at least polynomial in d, log m, and 1⁄e. We also present an alternate construction that uses a seed that can be described by O(c2d8m6⁄(c-2) log(d⁄e) + log |F|) bits (more precisely, O(c2d8m6⁄(c-2)) field elements, each chosen from a set of size poly(cd⁄e), plus two field elements ranging over all of F), works whenever |F| is at least polynomial in c, d, and 1⁄e, and has the property that every element of the output is a function of at most c field elements in the input. Both generators are computable by small arithmetic circuits. The main tool used in the construction is a reduction that allows us to transform any "dense" hitting set generator for polynomials into a pseudorandom generator.

On the Equivariant Tamagawa Number Conjecture for Abelian Extensions of a Quadratic Imaginary Field

Abstract: Let k be a quadratic imaginary field, p a prime which splits in k/Q and does not divide the class number hk of k. Let L denote a finite abelian extension of k and let K be a subextension of L/k. In this article we prove the p-part of the Equivariant Tamagawa Number Conjecture for the pair (h 0 (Spec(L)), Z(Gal(L/K))).

Mutually unbiased phase states, phase uncertainties, and Gauss sums

Abstract: Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is a constant equal to \(1/\sqrt{d}\), with d the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality d+1 have been derived for prime power dimensions d=pm using the tools of abstract algebra. Presumably, for non prime dimensions the cardinality is much less. Here we reinterpret MUBs as quantum phase states, i.e. as eigenvectors of Hermitian phase operators generalizing those introduced by Pegg and Barnett in 1989. We relate MUB states to additive characters of Galois fields (in odd characteristic p) and to Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for general pure quantum electromagnetic states and find them to be related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally, we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in the study of entanglement and its information aspects.

Heights and preperiodic points of polynomials over function fields

Abstract: Let K be a function field in one variable over an arbitrary field F. Given a rational function f(z) in K(z) of degree at least two, the associated canonical height on the projective line was defined by Call and Silverman. The preperiodic points of f all have canonical height zero; conversely, if F is a finite field, then every point of canonical height zero is preperiodic. However, if F is an infinite field, then there may be non-preperiodic points of canonical height zero. In this paper, we show that for polynomial f, such points exist only if f is isotrivial. In fact, such K-rational points exist only if f is defined over the constant field of K after a K-rational change of coordinates.

Some properties of solutions to polynomial systems of differential equations

Abstract: In (7) and (8), Parker and Sochacki considered iterative methods for computing the power series solution to y 0 = G y where G is a polynomial from R n to R n , including truncations of Picard iteration. The authors demon- strated that many ODE's may be transformed into computationally feasible polynomial problems of this type, and the methods generalize to a broad class of initial value PDE's. In this paper we show that the subset of the real an- alytic functions A consisting of functions that are components of the solution to polynomial dierential equations is a proper subset of A and that it shares the field and near-field structure of A, thus making it a proper sub-algebra. Consequences of the algebraic structure are investigated. Using these results we show that the Maclaurin or Taylor series can be generated algebraically for a large class of functions. This finding can be used to generate ecient numer- ical methods of arbitrary order (accuracy) for initial value ordinary dierential equations. Examples to indicate these techniques are presented. Future ad- vances in numerical solutions to initial value ordinary dierential equations are indicated.

Représentations lisses de GL(m, D) II : β-extensions

Abstract: This work is concerned with type theory for reductive groups over a non Archimedean field. Given such a field F, and a division algebra D of finite dimension over its center F, we obtain results concerning the construction of simple types for the group GL(m, D), . More precisely, for each simple stratum of the matrix algebra M(m, D), we produce a set of β-extensions in the sense of Bushnell and Kutzko.

Finitely ramified iterated extensions

Abstract: Let K be a number field, t a parameter, F = K(t), and '(x) ∈ K(x) a polyno- mial of degree d ≥ 2. The polynomialn(x, t) = ' ◦n (x)−t ∈ F(x), where ' ◦n = '◦'◦� � �◦ ' is the n-fold iterate of ', is absolutely irreducible over F; we compute a recursion for its discriminant. Let F' be the field obtained by adjoining to F all roots (in a fixed F) of �n(x, t) for all n ≥ 1; its Galois group Gal(F'/F) is the iterated monodromy group of '. The iterated extension F' is finitely ramified over F if and only if ' is post-critically finite. We show that, moreover, for post-critically finite ', every specialization of F'/F at t = t0 ∈ K is finitely ramified over K, pointing to the possibility of studying Galois groups of number fields with restricted ramification via tree representations associated to iterated monodromy groups of post-critically finite polynomials. We discuss the wildness of rami- fication in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogene number fields that arise from the construction.

Generators of $D$–modules in positive characteristic

Abstract: Let R = k[x1, . . . , xd] or R = k[[x1, . . . , xd]] be either a polynomial or a formal power series ring in a finite number of variables over a field k of characteristic p > 0 and let DR|k be the ring of klinear differential operators of R. In this paper we prove that if f is a non-zero element of R then Rf , obtained from R by inverting f , is generated as a DR|k–module by 1 f . This is an amazing fact considering that the corresponding characteristic zero statement is very false. In fact we prove an analog of this result for a considerably wider class of rings R and a considerably wider class of DR|k-modules.

Rational curves and points on k3 surfaces

Abstract: We study the distribution of algebraic points on K3 surfaces. 1. Introduction. Let k be a field and k a fixed algebraic closure of k. We are interested in connections between geometric properties of algebraic varieties and their arithmetic properties over k, over its finite extensions k ' /k or over k. Here we study certain varieties of intermediate type, namely K3 surfaces and their higher dimensional generalizations, Calabi-Yau varieties. To motivate the following discussion, let 5 be a K3 surface over k. In positive characteristic, 5 may be unirational and covered by rational curves. Examples are supersingular K3 surfaces over fields of characteristic two or the surface

Motivation for Hodge cycles

Abstract: Given two smooth projective varieties X and Y over a field, we say that X motivates Y if the (suitably defined) motive of Y is contained in the category generated from X by taking sums, summands and products. This notion has appeared implicitly in many places, but it seems useful to make it explicit. Some techniques are given for checking this condition, but in a nutshell it involves building a correspondence which "dominates" Y. Among the more interesting examples dealt with are moduli spaces. We show that in number of cases moduli spaces of sheaves over curves or surfaces are motivated by underly curve or surface. This allows us to check (or recheck) the (generalized) Hodge and Lefschetz standard conjectures for some of these examples. The dominating correspondence in these examples is built from the universal sheaf, and can, in some instances, be realized as a kind of Fourier-Mukai transform.

Asymptotic Behavior of the Length of Local Cohomology

Abstract: Let be a field of characteristic be a polynomial ring, and its maximal homogeneous ideal Let be a homogeneous ideal in Let denote the length of an -module In this paper, we show that always exists This limit has been shown to be for -primary ideals in a local Cohen–Macaulay ring, where denotes the multiplicity of But we find that this limit may not be rational in general We give an example for which the limit is an irrational number thereby showing that the lengths of these extension modules may not have polynomial growth

Galois theory of q-difference equations

Abstract: Choose $q\in {\mathbb C}$ with 0<|q|<1. The main theme of this paper is the study of linear q-difference equations over the field K of germs of meromorphic functions at 0. It turns out that a difference module M over K induces in a functorial way a vector bundle v(M) on the Tate curve $E_q:={\mathbb C}^*/q^{\mathbb Z}$. As a corollary one rediscovers Atiyah's classification of the indecomposable vector bundles on the complex Tate curve. Linear q-difference equations are also studied in positive characteristic in order to derive Atiyah's results for elliptic curves for which the j-invariant is not algebraic over ${\mathbb F}_p$. A universal difference ring and a universal formal difference Galois group are introduced. Part of the difference Galois group has an interpretation as `Stokes matrices', the above moduli space is the algebraic tool to compute it. It is possible to provide the vector bundle v(M) on E_q, corresponding to a difference module M over K, with a connection $ abla_M$. If M is regular singular, then $ abla_M$ is essentially determined by the absense of singularities and `unit circle monodromy'. More precisely, the monodromy of the connection $(v(M), abla_M)$ coincides with the action of two topological generators of the universal regular singular difference Galois group. For irregular difference modules, $ abla_M$ will have singularities and there are various Tannakian choices for $M\mapsto (v(M), abla_M)$. Explicit computations are difficult, especially for the case of non integer slopes.

Zeta functions over F1

Abstract: We show basic properties of zeta functions over the one element field starting from an algebraic set over the integer ring. We calculate several examples and we investigate special values via the associated K-group identified as the stable homotopy group of spheres.

Modular forms and elliptic curves over imaginary quadratic fields

Abstract: The aim of this thesis is to contribute to an ongoing project to understand the correspondence between cusp forms, for imaginary quadratic fields, and elliptic curves. This contribution mainly takes the form of developing explicit constructions and computing particular examples. It is hoped that as well as being of interest in themselves, they will be helpful in guiding future theoretical developments. Cremona [7] began the programme of extending the classical techniques using modular symbols to the case of imaginary quadratic fields. He was followed by two of his students Whitley [25] and Bygott [5]. Together they have covered the cases where the class number of the field is equal to 1 or 2. This thesis extends their work to treat all fields of odd class number. It describes an algorithm, which holds for any such field, for determining the space of cusp forms, and for computing the eigenforms and eigenvalues for the action of the Hecke algebra on this space. The approach, using modular symbols, closely follows the work of the previous authors, but new techniques and theoretical simplifcations are obtained which hold in the case considered. All of the algorithms presented in this thesis have been implemented in a computer algebra package, Magma [3], and the results obtained for the fields Q(sqrt(-23)) and Q(sqrt(-31)) are included.