Showing papers on "Ring of integers published in 2002"

Revêtements étales et groupe fondamental (SGA 1)

Abstract: Le texte pr\'esente les fondements d'une th\'eorie du groupe fondamental en G\'eom\'etrie Alg\'ebrique, dans le point de vue ``kroneckerien'' permettant de traiter sur le m\^eme pied le cas d'une vari\'et\'e alg\'ebrique au sens habituel, et celui d'un anneau des entiers d'un corps de nombres, par exemple. The text presents the foundations of a theory of the fundamental group in Algebraic Geometry from the Kronecker point of view, allowing one to treat on an equal footing the case of an algebraic variety in the usual sense, and that of the ring of integers in a number field, for instance.

Multiplicités modulaires et représentations de ${\rm GL}\sb 2(\mathbf {Z}\sb p)$ et de ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ en $\ell=p$. Appendice par Guy Henniart. Sur l'unicité des types pour ${\rm GL}\sb 2$

Abstract: We formulate a conjecture giving a link between the various rings parametrizing the $2$-dimensional potentially semistable $p$-adic representations of ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ with Hodge-Tate weights $(0,k-1)(k\in \mathbf {Z},1

Multiplicity and Hilbert-Kunz Multiplicity of Monoid Rings

Abstract: In this paper, we will give a method to compute the multiplicity and the Hilbert-Kunz multiplicity of monoid rings. The multiplicity and the Hilbert-Kunz multiplicity are fundamental invariants of rings. For example, the multiplicity (resp. the Hilbert-Kunz multiplicity) of a regular local ring equals to one. Monoid rings are defined by lattice ideals, which are binomial ideals I in a polynomial ring R over a field such that any monomial is a non zero divisor on R/I. Affine semigrouprings are monoidrings. Hencewe want to extendthe thoery of affine semigroup rings to that of monoid rings. 1. Main Result. Let N> 0b e an integer andZ the ring of integers. For α ∈ Z N , we denote the i-th entry of α by αi .W e say α> 0i fα � 0a ndαi ≥ 0 for each i .A nd α>α � if α − α � > 0. Let R = k[X1, ··· ,X N ] be a polynomial ring over a field k .F or α> 0, we simply write X α in place of N=1 X αi i . For a positive submodule V of Z N of rank r, we define an ideal I( V )of R ,w hich is generatedby all binomials X α −X β with α−β ∈ V (we say that V is positiveif it is contained in the kernel of a map Z N → Z which is defined by positive integers). Put d = N − r .T hen R/I (V ) is naturally a Z d -graded ring, which is called a monoid ring. Further, there is a

Asymptotic variation of L functions of one-variable exponential sums

Abstract: Let d>2 and let p be a prime coprime to d. Let Z_pbar be the ring of integers of Q_pbar. Suppose f(x) is a degree-d polynomial over Qbar and Z_pbar. Let P be a prime ideal over p in the ring of integers of Q(f), where Q(f) is the number field generated by coefficients of f in Qbar. Let A^d be the dimension-d affine space over Qbar, identified with the space of coefficients of degree-d monic polynomials. Let NP(f mod P) denote the p-adic Newton polygon of L(f mod P;T). Let HP(A^d) denote the p-adic Hodge polygon of A^d. We prove that there is a Zariski dense open subset U defined over Q in A^d such that for every geometric point f(x) in U(Qbar) we have lim_{p-->oo} NP(f mod P) = HP(A^d), where P is any prime ideal in the ring of integers of Q(f) lying over p. This proves a conjecture of Daqing Wan.

Systèmes dynamiques non archimédiens et corps des norms (Non-Archimedean Dynamic Systems and Fields of Norms)

Abstract: Let \(\mathcal{O}\)k be the ring of integers of a finite extension k of the field \(\mathbb{Q}\)p of p-adic numbers. The endomorphisms of a formal group law defined over \(\mathcal{O}\)k provide nontrivial examples of commuting formal series with coefficients in \(\mathcal{O}\)k. This article deals with the inverse problem formulated by Jonathan Lubin within the context of non-Archimedean dynamical systems. We present a large family of series, with coefficients in \(\mathbb{Z}\)p, which satisfy Lubin's conjecture. These series are constructed with the help of Lubin–Tate formal group laws over \(\mathbb{Q}\)p. We introduce the notion of minimally ramified series which turn out to be modulo p reductions of some series of this family. The commutant monoids of these minimally ramified series are determined by using the Fontaine–Wintenberger theory of the field of norms which allows an interpretation of them as automorphisms of \(\mathbb{Z}\)p-extensions of local fields of characteristic zero. A particularly effective example illustrating the paper is given by a family of series generalizing Cebysev polynomials

Cyclic covers of the projective line, their jacobians and endomorphisms

Abstract: We study the endomorphism ring $End(J(C))$ of the complex jacobian $J(C)$ of a curve $y^p=f(x)$ where $p$ is an odd prime and $f(x)$ is a polynomial with complex coefficiens of degree $n>4$ and without multiple roots. Assume that all the coefficients of $f$ lie in a (sub)field $K$ and the Galois group of $f$ over $K$ is either the full symmetric group $S_n$ or the alternating group $A_n$. Then we prove that $End(J(C))$ is the ring of integers in the in the $p$th cyclotomic field, if $p$ is a Fermat prime (e.g., $p=3,5,17,257$). Similar results for $p=2$ (the case of hyperelliptic curves) were obtained by the author in Math. Res. Lett. 7(2000), 123--132.

CTRU, a polynomial analogue of NTRU

Abstract: CTRU, a new public-key cryptosystem is introduced. In this analogue of NTRU, the ring of integers is replaced by the ring of polynomials in one variable over a finite field. Attacks based on either the LLL algorithm or the Chinese Remainder Theorem are avoided. An important tool of cryptanalys- is is the Popov normal form of matrices with polynomial entries. The speed of encryption/decryption of CTRU is the same as NTRU for the same value of N. An implementation in Aldor is described.

Arithmetic knots in closed 3-manifolds

Abstract: Let Od denote the ring of integers in . An orientable finite volume cusped hyperbolic 3-manifold M is called arithmetic if the faithful discrete representation of π1(M) into PSL(2,C) is conjugate to a group commensurable with some Bianchi group PSL(2,Od). If M is a closed orientable 3-manifold, we say a link L ⊂ M is arithmetic if M\L is arithmetic. In the paper, we show that there exist closed orientable 3-manifolds which do not contain anarithmetic knot. Our methods give much more precise informations for non-hyperbolic 3-manifolds. For certain Lens Spaces, we can give fairly complete statements.

Quelques classes caractéristiques en théorie des nombres

Abstract: Let A be an arbitrary ring. We introduce a Dennis trace map mod n, from K_1(A;Z/n) to the Hochschild homology group with coefficients HH_1(A;Z/n). If A is the ring of integers in a number field, explicit elements of K_1(A,Z/n) are constructed and the values of their Dennis trace mod n are computed. If F is a quadratic field, we obtain this way non trivial elements of the ideal class group of A. If F is a cyclotomic field, this trace is closely related to Kummer logarithmic derivatives; this trace leads to an unexpected relationship between the first case of Fermat last theorem, K-theory and the number of roots of Mirimanoff polynomials.

Algebraic Jf-theory and Trace Invariants

Abstract: The cyclotomic trace of Bokstedt-Hsiang-Madsen, the subject of Bokstedt's lecture at the congress in Kyoto, is a map of pro-abelian groups K-fiA) ^.TR;(A;p) from Quillen's algebraic A"-theory to a topological refinement of Connes' cyclic homology. Over the last decade, our understanding of the target and its relation to A"-theory has been significantly advanced. This and possible future development is the topic of my lecture. The cyclotomic trace takes values in the subset fixed by an operator F called the Frobenius. It is known that the induced map K*(A,Z/pv) -^ TR;(A;P,Z/PV)F=1 is an isomorphism, for instance, if A is a regular local Fp-algebra, or if A is a henselian discrete valuation ring of mixed characteristic (0,p) with a separably closed residue field. It is possible to evaluate A"-theory by means of the cyclotomic trace for a wider class of rings, but the precise connection becomes slightly more complicated to spell out. The pro-abelian groups TR*(A;p) are typically very large. But they come equipped with a number of operators, and the combined algebraic structure is quite rigid. There is a universal example of this structure — the de Rham-Witt complex — which was first considered by Bloch-Deligne -Illusie in connection with Grothendieck's crystalline cohomology. In general, the canonical map W.QqA^TR-q(A-p) is an isomorphism, if q < 1, and the higher groups, too, can often be expressed in terms of the de Rham-Witt groups. This is true, for example, if A is a regular Fp-algebra, or if A is a smooth algebra over the ring of integers in a local number field. The calculation in the latter case verifies the LichtenbaumQuillen conjecture for focal number fields, or more generally, for henselian discrete valuation fields of geometric type.

Monogenesis of the rings of integers in certain imaginary abelian fields

Abstract: In this paper we consider a subfield $K$ in a cyclotomic field $k_m$ of conductor $m$ such that $\left[k_m : K\right] = 2$ in the cases of $m = \ell p^n$ with a prime $p,$ where $\ell = 4$ or $p > \ell = 3$ Then the theme is to know whether the ring of integers in $K$ has a power basis or does not

On reduction of Hilbert-Blumenthal varieties

Abstract: Let $O_F$ be the ring of integers of a totally real field $F$ of degree $g$. We study the reduction of the moduli space of separably polarized abelian $O_F$-varieties of dimension $g$ modulo $p$ for a fixed prime $p$. The invariants and related conditions for the objects in the moduli space are discussed. We construct a scheme-theoretic stratification by $a$-numbers on the Rapoport locus and study the relation with the slope stratification. In particular, we recover the main results of Goren and Oort [GO, J. Alg. Geom. 2000] on the stratifications when $p$ is unramified in $O_F$. We also prove the strong Grothendieck conjecture for the moduli space in some restricted cases, particularly when $p$ is totally ramified in $O_F$.

Integral representations for the alternating groups

Abstract: We show that every complex representation of an alternating group can be realized over the ring of integers of a small abelian number field.

Euler measure as generalized cardinality

Abstract: Schanuel has pointed out that there are mathematically interesting categories whose relationship to the ring of integers is analogous to the relationship between the category of finite sets and the semi-ring of non-negative integers. Such categories are inherently geometrical or topological, in that the mapping to the ring of integers is a variant of Euler characteristic. In these notes, I sketch some ideas that might be used in further development of a theory along lines suggested by Schanuel.

Comparing EQP and MOD_{p^k}P using Polynomial Degree Lower Bounds

Abstract: We show that an oracle A that contains either 1/4 or 3/4 of all strings of length n can be used to separate EQP from the counting classes MOD_{p^k}P. Our proof makes use of the degree of a representing polynomial over the finite field of size p^k. We show a linear lower bound on the degree of this polynomial. We also show an upper bound of O(n^{1/log_p m}) on the degree over the ring of integers modulo m, whenever m is a squarefree composite with largest prime factor p.

p-adic variation of L functions of one variable exponential sums, II

Abstract: Let d>2 and let p be a prime coprime to d. Let Z_pbar be the ring of integers of Q_pbar. Suppose f(x) is a degree-d polynomial over Qbar and Z_pbar. Let P be a prime ideal over p in the ring of integers of Q(f), where Q(f) is the number field generated by coefficients of f in Qbar. Let A^d be the dimension-d affine space over Qbar, identified with the space of coefficients of degree-d monic polynomials. Let NP(f mod P) denote the p-adic Newton polygon of L(f mod P;T). Let HP(A^d) denote the p-adic Hodge polygon of A^d. We prove that there is a Zariski dense open subset U defined over Q in A^d such that for every geometric point f(x) in U(Qbar) we have lim_{p-->oo} NP(f mod P) = HP(A^d), where P is any prime ideal in the ring of integers of Q(f) lying over p. This proves a conjecture of Daqing Wan.

The Hooley-Huxley contour method for problems in number fields II: factorization and divisibility

Abstract: Let L be a Galois extension of the number field K. Set n = nK = deg K/ℚ, nL = deg L/ℚ and nL/K = deg L/K. Let I = IL/K denote the group of fractional ideals of K whose prime decomposition contains no prime ideals that ramify in L, and let P = {(α)ΣI: αΣK*, α>0}. Following Hecke [9}, let (λ1, λ2, …, λn − 1) be a basis for the torsion-free characters on P that satisfy λi(α) = 1 (1≤i≤n − 1) for all units α>0 in , the ring of integers of K. Fixing an extension of each λi to a character on I, then λi,(α) (1 ≤i≤n − 1) are defined for all ideals α of K that do not ramify in L. So, for such ideals, we can define . Then the small region of K referred to above isfor 0

On cohomology rings of a cyclic group and a ring of integers

Abstract: We determine the ring homomorphism HH ∗ (Γ) → H ∗ (G, Γ) ex- plicitly, where G denotes the cyclic group of order p ν and Γ denotes the ring of integers of the cyclotomic field (ζ) for a primitive p ν -th root of unity ζ.

On cycles of polynomials with integral rational coefficients

Abstract: The paper deals with polynomial cycles in the rings of integers of cyclic algebraic number fields for polynomials with integral rational coefficients. In the first par t , a connection between the existence of cycles and the existence of power basis is shown. In the second part , properties of cycles for quadratic polynomials with rational integral coefficients are described. Let K be a ring. Recall that a finite subset {xx,x2,... , x n } of K is called a cycle for a polynomial f(x) if for i = 1, 2 , . . . , n — 1 one has f(xi) = xi+l, f(xn) = xx and xi ^ xfor i ^ j . The number n will be called the length of the cycle, and x^s, cyclic elements of order n. Denote by fi the zth iterate of / for i = 1, 2 , . . . , i.e. /-_ = / and / . + 1 = f(f{) for i = 1, 2 , . . . . Let K be an algebraic number field; denote by 7LK the ring of integers of K. The possible cycle-lengths in the rings of integers in quadratic number fields were determined independently by J. B o d u c h and by G. Baron. The result can be found in [6]. For fields K of larger degrees, the problem of determining all cycle-lengths in their rings of integers Z ^ is still open. Cycles of quadratic polynomials were recently studied by P. M o r t o n [5] and P. R u s s o , R. W a l d e [11]. In this paper, only cycle-lengths for polynomials with rational integral coefficients in the rings of integers 7LK of an algebraic number field K will be studied. First, a connection between the existence of power basis for 7LK over rational integers 7L and polynomial cycles for a polynomial f(x) G 7L\\x\\ will be investigated. Recall the definition of an order. 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : Primary 11R04, 11C08. K e y w o r d s : polynomial cycle, ring of integers of algebraic number field. This research was supported by GA of the Czech Academy of Sciences, Grant A1187101/01.

The support problem for abelian varieties

Abstract: Let $A$ be an abelian variety over a number field $K$. If $P$ and $Q$ are $K$-rational points of $A$ such that the order of the reduction of $Q$ divides that of $P$ for all but finitely many primes of the ring of integers of $K$, then there exists a $K$-endomorphism $\phi$ of $A$ and a positive integer $k$ such that $kQ = \phi(P)$.

Knapsack type public key cryptographic system, public key generation method and device therefor, and program and recording medium therefor

Abstract: PROBLEM TO BE SOLVED: To accelerate processing speed. SOLUTION: Elements Pij (i=1 to n) are generated from an integer ring OK of an algebraic number field K, from which prime ideals p1 to ph, generators gj (j=1 to h) of OK/pj (except zero), and pj are removed, in accordance with the integer ring OK of the field K and integers n, k, and h (101), and aij satisfying pij=gjaijmodpj is calculated (103), and p=p1.p2.....ph is obtained (102), and g=(g1 to gh) and ai=(ai1 to aih) are synthesized by a Chinese remainder theorem to generate an integer d, and hi=(ai+)mod(N(P)-2) (N(P) is the norm of P) is calculated to obtain a public key bi.

Kernel Groups and nontrivial Galois module structure of imaginary quadratic fields

Abstract: Let $K$ be an algebraic number field with ring of integers $\Cal{O}_{K}$, $p>2$ be a rational prime and $G$ be the cyclic group of order $p $. Let $\Lambda$ denote the order $\Cal{O}_{K}[G].$ Let $Cl(\Lambda)$ denote the locally free class group of $\Lambda$ and $D(\Lambda)$ the kernel group, the subgroup of $Cl(\Lambda)$ consisting of classes that become trivial upon extension of scalars to the maximal order. If $p$ is unramified in $K$, then $D(\Lambda) = T(\Lambda)$, where $T(\Lambda)$ is the Swan subgroup of $Cl(\Lambda).$ This yields upper and lower bounds for $D(\Lambda)$. Let $R(\Lambda)$ denote the subgroup of $Cl(\Lambda)$ consisting of those classes realizable as rings of integers, $\Cal{O}_{L},$ where $L/K$ is a tame Galois extension with Galois group $Gal(L/K) \cong G.$ We show under the hypotheses above that $T(\Lambda)^{(p-1)/2} \subseteq R(\Lambda) \cap D(\Lambda) \subseteq T(\Lambda)$, which yields conditions for when $T(\Lambda)=R(\Lambda) \cap D(\Lambda)$ and bounds on $R(\Lambda) \cap D(\Lambda)$. We carry out the computation for $K=\Bbb{Q}(\sqrt{-d}), d>0, d eq 1$ or $3.$ In this way we exhibit primes $p$ for which these fields have tame Galois field extensions of degree $p$ with nontrivial Galois module structure.