Showing papers on "Ring of integers published in 2003"

Les vari\'et\'es sur le corps \`a un \'el\'ement

Abstract: We propose a definition of varieties over the field with one element. These have extensions of scalars to the ring of integers which are varieties in the usual sense. We show that toric varieties can be defined over the field with one element. We also discuss zeta functions for such objects. We give a motivic interpretation of the image of the J-homomorphism defined by Adams. ~ ~ ~ ~

Geometry of Hilbert modular varieties over totally ramified primes

Abstract: Let L be a totally real field with ring of integers OL. Let N ≥ 4 be an integer and let M(μN) be the fine moduli scheme over Z of polarized abelian varieties with real multiplication (RM) and μN-level structure, satisfying the Deligne-Pappas condition. For every scheme S, we let M(S, μN) = M(μN) ×Z S be the moduli scheme over S; see Definition 2.1. Many aspects of the geometry of the modular varieties M(Fp, μN) are obtained via local deformation theory that factorizes according to the decomposition of p in OL. The unramified case was considered in [9] (see also [8]). Given that, one may restrict one’s attention to the case p = p in OL. We discuss here only the case e = g, that is, p is totally ramified in L. The ramified case was first treated by Deligne and Pappas in [6] (the case g = 2 was considered in [2]). We recall some of their results under the assumption that p is totally ramified. Let A/k be a polarized abelian variety with RM, defined over a field k of characteristic p. Fix an isomorphism OL⊗Zk ∼ = k[T ]/(T). One knows that HdR(A) is a free k[T ]/(T)-module of rank 2. The elementary divisors theorem furnishes us with k[T ]/(T)generators α and β for HdR(A) such that

Algebraic cycles on Hilbert modular fourfolds and poles of L-functions

Abstract: In this paper we give some evidence for the Tate (and Hodge) conjecture(s) for a class of Hilbert modular fourfolds X, whose connected components arise as arithmetic quotients of the fourfold product of the upper half plane by congruence subgroups \Gamma of SL(2, O_K), where O_K denotes the ring of integers of a quartic, Galois, totally real number field K. The expected relationship to the orders of poles of the associated L-functions is verified for abelian extensions of \Q. Also shown is the existence of homologically non-trivial cycles of codimension two which are not intersections of divisors.

Algebres de Hecke affines generiques

Abstract: Let $H$ be a generic affine Hecke algebra (Iwahori-Matsumoto definition) over a polynomial algebra with a finite number of indeterminates over the ring of integers. We prove the existence of an integral Bernstein-Lusztig basis related to the Iwahori-Matsumoto basis by a strictly upper triangular matrix, from which we deduce that the center $Z$ of $H$ is finitely generated and that $H$ is a finite type $Z$-module (this was proved after inversion of the parameters by Bernstein-Lusztig), and we give some applications to the theory of $H$-modules where the parameters act by 0. These results are related to the smooth $p$-adic or mod $p$ representations of reductive $p$-adic groups. We introduce the supersingular modules of the affine Hecke algebra of GL(n) with parameter 0, probably analogues of the Barthel-Livne supersingular mod $p$ representations of GL(2).

Semi-stable models for rigid-analytic spaces

Abstract: Let R be a complete discrete valuation ring with field of fractions K and let X K be a smooth, quasi-compact rigid-analytic space over Sp K. We show that there exists a finite separable field extension K' of K, a rigid-analytic space X' K' over Sp K' having a strictly semi-stable formal model over the ring of integers of K', and an etale, surjective morphism f : X' K' →X K of rigid-analytic spaces over Sp K. This is different from the alteration result of A.J. de Jong [dJ] who does not obtain that f is etale. To achieve this property we have to work locally on X K , i.e. our f is not proper and hence not an alteration.

The minimum index of a non-congruence subgroup of sl2 over an arithmetic domain

Abstract: LetA be an arithmetic Dedekind ring with only finitely many units. It is known that (i)A = ℤ, the ring of rational integers, (ii)A =Od, the ring of integers of the imaginary quadratic field\(\mathbb{Q}\left( {\sqrt { - d} } \right)\), whered is a square-free positive integer, or (iii)A=C=C(C,P,k), the coordinate ring of the affine curve obtained by removing a closed pointP from a smooth projective curveC over afinite fieldk. serre has shown that, in comparison with other low rank arithmetic groups, the groups SL2(A) have “many” non-congruence subgroups.

The syntomic regulator for the K-theory of fields

Abstract: We define complexes analogous to Goncharov's complexes for the K-theory of discrete valuation rings of characteristic zero. Under suitable assumptions in K-theory, there is a map from the cohomology of those complexes to the K-theory of the ring under consideration. In case the ring is a localization of the ring of integers in a number field, there are no assumptions necessary. We compute the composition of our map to the K-theory with the syntomic regulator. The result can be described in terms of a p-adic polylogarithm. Finally, we apply our theory in order to compute the regulator to syntomic cohomology on Beilinson's cyclotomic elements. The result is again given by the p-adic polylogarithm. This last result is related to one by Somekawa and generalizes work by Gros.

A bound for the torsion in the -theory of algebraic integers.

Abstract: Let m be an integer bigger than one, A a ring of algebraic integers, F its fraction field, and Km(A) the m-th Quillen K-group of A. We give a (huge) explicit bound for the order of the torsion subgroup of Km(A) (up to small primes), in terms of m, the degree of F over Q, and its absolute discriminant. Let F be a number field, A its ring of integers and Km(A) the m-th Quillen K-group of A. It was shown by Quillen that Km(A) is finitely generated. In this paper we shall give a (huge) explicit bound for the order of the torsion subgroup of Km(A) (up to small primes), in terms of m, the degree of F over Q, and its absolute discriminant. Our method is similar to the one developed in [13] for F = Q. Namely, we reduce the problem to a bound on the torsion in the homology of the general linear group GLN (A). Thanks to a result of Gabber, such a bound can be obtained by estimating the number of cells of given dimension in any complex of free abelian groups computing the homology of GLN (A). Such a complex is derived from a contractible CW -complex W on which GLN (A) with compact quotient. We shall use the construction of W given by Ash in [1] . It consists of those hermitian metrics h on A which have minimum equal to one and are such that their set M(h) of minimal vectors has rank equal to N in F . To count cells in W/GLN (A), one will exhibit an explicit compact subset of A ⊗ZR which, for every h ∈ W , contains a translate of M(h) by some matrix of GLN (A) (Proposition 2). The proof of this result relies on several arguments from the geometry of numbers using, among other things, the number field analog of Hermite’s constant [4]. Documenta Mathematica · Extra Volume Kato (2003) 761–788 762 Christophe Soule The bound on the K-theory of A implies a similar upper bound for the etale cohomology of Spec (A[1/p]) with coefficients in the positive Tate twists of Zp, for any (big enough) prime number p. However, this bound is quite large since it is doubly exponential both in m and, in general, the discriminant of F . We expect the correct answer to be polynomial in the discriminant and exponential in m (see 5.1). The paper is organized as follows. In Section 1 we prove a few facts on the geometry of numbers for A, including a result about the image of A by the regulator map (Lemma 3), which was shown to us by H. Lenstra. Using these, we study in Section 2 hermitian lattices over A, and we get a bound on M(h) when h lies in W . The cell structure of W is described in Section 3. The main Theorems are proved in Section 4. Finally, we discuss these results in Section 5, where we notice that, because of the Lichtenbaum conjectures, a lower bound for higher regulators of number fields would probably provide much better upper bounds for the etale cohomology of Spec (A[1/p]). We conclude with the example of K8(Z) and its relation to the Vandiver conjecture. 1 Geometry of algebraic numbers 1.1 Let F be a number field, and A its ring of integers. We denote by r = [F : Q] the degree of F over Q and by D = |disc (K/Q)| the absolute value of the discriminant of F over Q. Let r1 (resp. r2) be the number of real (resp. complex) places of F . We have r = r1 + 2 r2. We let Σ = Hom (F,C) be the set of complex embeddings of F . These notations will be used throughout. Given a finite set X we let # (X) denote its cardinal. 1.2 We first need a few facts from the geometry of numbers applied to A and A. The first one is the following classical result of Minkowski: Lemma 1. Let L be a rank one torsion-free A-module. There exists a non zero element x ∈ L such that the submodule spanned by x in L has index #(L/Ax) ≤ C1 , where C1 = r! rr · 42 π2 √ D in general, and C1 = 1 when A is principal. Proof. The A-module L is isomorphic to an ideal I in A. According to [7], V §4, p. 119, Minkowski’s first theorem implies that there exists x ∈ I the norm of which satisfies |N(x)| ≤ C1 N(I) . Documenta Mathematica · Extra Volume Kato (2003) 761–788 A Bound for the Torsion . . . 763 Here |N(x)| = #(A/Ax) and N(I) = # (A/I), therefore # (I/Ax) ≤ C1. The case where A is principal is clear. q.e.d. 1.3 The family of complex embeddings σ : F → C, σ ∈ Σ, gives rise to a canonical isomorphism of real vector spaces of dimension r F ⊗Q R = (C) , where (·)+ denotes the subspace invariant under complex conjugation. Given α ∈ F we shall write sometimes |α|σ instead of |σ(α)|. Lemma 2. Given any element x = (xσ) ∈ F ⊗Q R, there exists a ∈ A such that ∑ σ∈Σ |xσ − σ(a)| ≤ C2 , with C2 = 41 π2 rr−2 r! √ D in general, and C2 = 1/2 if F = Q . Proof. Define a norm on F ⊗Q R by the formula ‖x‖ = ∑

Ideal Membership in Polynomial Rings over the Integers

Abstract: We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials $f_0,f_1,...,f_n\in\Z[X]$, where $X=(X_1,...,X_N)$ is an $N$-tuple of indeterminates, are there $g_1,...,g_n\in\Z[X]$ such that $f_0=g_1f_1+...+g_nf_n$? We show that the degree of the polynomials $g_1,...,g_n$ can be bounded by $(2d)^{2^{O(N^2)}}(h+1)$ where $d$ is the maximum total degree and $h$ the maximum height of the coefficients of $f_0,...,f_n$. Some related questions, primarily concerning linear equations in $R[X]$, where $R$ is the ring of integers of a number field, are also treated.

Modular and p-adic cyclic codes

Abstract: This paper presents some basic theorems giving the structure of cyclic codes of length n over the ring of integers modulo p^a and over the p-adic numbers, where p is a prime not dividing n. An especially interesting example is the 2-adic cyclic code of length 7 with generator polynomial X^3 + lambda X^2 + (lambda - 1) X - 1, where lambda satisfies lambda^2 - lambda + 2 =0. This is the 2-adic generalization of both the binary Hamming code and the quaternary octacode (the latter being equivalent to the Nordstrom-Robinson code). Other examples include the 2-adic Golay code of length 24 and the 3-adic Golay code of length 12.

Division-ample sets and the Diophantine problem for rings of integers

Abstract: We prove that Hilbert's Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over K). We relate division-ample sets to arithmetic of abelian varieties.

Representation by spinor genera of ternary quadratic forms

Abstract: over the ring of integers Z, where A and B are non-degenerate and symmetric matrices of size m × m and n × n over Z respectively, and A is indefinite with m ≥ 3. It is a necessary condition for solubility of equation (1.1) that it is solvable over Zp for all primes p and the real numbers R. This necessary condition is already sufficient if m − n ≥ 3 [Kn1,Hs]. However the equation (1.1) is no longer a purely local problem when m − n ≤ 2. By the Hasse principle, the necessary condition implies there is a rational solution of (1.1). In the previous papers [CX] and [X1], one of us has given conditions that allow to decide for m−n ≤ 2 whether the equation (1.1) is solvable over Z by looking at a given rational solution whose denominator is prime to the determinant of A. Can one also determine the solubility of (1.1) if the denominator of the rational solution is not prime to the determinant of A? In this note, we try to give such a condition. Notation and terminology are standard if not explained, or adopted from [CX] and [X1]. Let V be a quadratic space over a number field F with a non-degenerate symmetric bilinear form 〈x, y〉, Q(x) = 〈x, x〉 be the quadratic map on V and SO(V ) be the special orthogonal group of V . A lattice in V means a finitely generated oF module in V such that it generates a non-degenerate quadratic subspace of V . A full lattice means a lattice which generates the whole space. For a full lattice L, L denotes the dual lattice of L. For two lattices K and L in V , 〈K, L〉 denotes the fractional ideal generated by 〈x, y〉 for x ∈ K and y ∈ L. We use τz for the reflection if Q(z) 6= 0. We also denote n(L) and s(L) as norm and scale of a lattice L in the sense of [O] respectively. For any prime p of F , Vp (resp. Fp, etc.) denotes the local completion of V (resp. F , etc.). Let oF be the ring of integers of F . If p is a finite prime, the group of units of oFp is denoted by up, and πp is a uniformizer of Fp. We use θp to denote the spinor norm map of SO(Vp). For a lattice Kp and a full lattice Lp in Vp, let

Euler characteristics of arithmetic groups

Abstract: We develop a general method for computing the homological Euler characteristic of finite index subgroups G of GL_m(O_K) where O_K is the ring of integers in a number field K. With this method we find, that for large, explicitly computed dimensions m, the homological Euler characteristic of finite index subgroups of GL_m(O_K) vanishes. For other cases, some of them very important for spaces of multiple polylogarithms, we compute non-zero homological Euler characteristic. With the same method we find all the torsion elements in GL_3(Z) up to conjugation. Finally, our method allows us to obtain a formula for the Dedekind zeta function at -1 in terms of the ideal class set and the multiplicative group of quadratic extensions of the base ring.

On cyclic top-associative Generalized Galois Rings

Abstract: A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic p n , for a prime number p, such that its top-factor \(\overline{S} = S/pS\) is a finite semifield. It is well known that if S is an associative Galois Ring (GR) then the set \(S^* = S \ pS\) is a finite multiplicative abelian group. This group is cyclic if and only if S is either a finite field, or a residual integer ring of odd characteristic or the ring ℤ4. A GGR is called top-associative if \(\overline{S}\) is a finite field. In this paper we study the conditions for a top-associative not associative GGR S to be cyclic.

Duality and Hermitian Galois Module Structure

Abstract: Suppose $\mathcal{O}$ is either the ring of integers of a number field, the ring of integers of a $p$-adic local field, or a field of characteristic $0$. Let $\mathcal{X}$ be a regular projective scheme which is flat and equidimensional over $\mathcal{O}$ of relative dimension $d$. Suppose $G$ is a finite group acting tamely on $\mathcal{X}$. Define ${\rm HCl}(\mathcal{O} G)$ to be the Hermitian class group of $\mathcal{O} G$. Using the duality pairings on the de Rham cohomology groups $H^*(X, \Omega^\bullet_{X / F})$ of the fiber $X$ of $\mathcal{X}$ over $F = {\rm Frac}(\mathcal{O})$, we define a canonical invariant $\chi_H(\mathcal{X}, G)$ in ${\rm HCl}(\mathcal{O} G)$ . When $d = 1$ and $\mathcal{O}$ is either $\mathbb{Z}$, $\mathbb{Z}_p$ or $\mathbb{R}$, we determine the image of $\chi_H(\mathcal{X}, G)$ in the adelic Hermitian classgroup ${\rm Ad\,HCl}(\mathbb{Z} G)$ by means of $\epsilon$-constants. We also show that in this case, the image in ${\rm Ad\,HCl}(\mathbb{Z} G)$ of a closely related Hermitian Euler characteristic $\chi_{H}(\mathcal{X}, G)(0)$ both determines and is determined by the $\epsilon_0$-constants of the symplectic representations of $G$.

On dedekind's criterion and monogenicity over dedekind rings

Abstract: We give a practical criterion characterizing the monogenicity of the integral closure of a Dedekind ring R, based on results on the resultant Res(P , Pi) of the minimal polynomial P of a primitive integral element and of its irreducible factors Pi modulo prime ideals of R. We obtain a generalization and an improvement of the Dedekind criterion (Cohen, 1996) and we give some applications in the case where R is a discrete valuation ring or the ring of integers of a number field, generalizing some well-known classical results.

Minimal norm Jordan splittings of quadratic lattices over complete dyadic discrete valuation rings

Abstract: In [5], the so called minimal norm Jordan splitting over a ring of integers of dyadic local fields is introduced for determining the generators of integral orthogonal groups for the purpose of computing integral spinor norms. Such a normalization of Jordan splittings turns out to be useful in dyadic theory (see also [6] and [7]). In this note, we give a more conceptual proof and extend this result to a complete dyadic discrete valuation ring, where the residue field is not necessarily perfect. As an application, we discuss the Witt cancellation theorem and also give a proof of Theorem 10 in [4, Chapter 10], where the rigorous proof is not available. It should be pointed out that [3] gives some variation of the classification theorem over $\mathbb{Z}_2$ but not the detailed proof.

Unicity of types for supercuspidals

Abstract: Let $F$ be a non-Archimedean local field, with the ring of integers $\mathfrak{o}_F. Let $G=GL_N(F)$, $K=GL_N(\mathfrak{o}_F)$ and $\pi$ a supercuspidal representation of $G$. We show that there exist a unique irreducible smooth representation $\tau$ of $K$, such that the restriction to $K$ of a smooth irreducible representation $\pi'$ of $G$ contains $\tau$ if and only if $pi'$ is isomorphic to $\pi\otimes\chi\circ\det$, where $\chi$ is an unramified quasicharacter of $F^{\times}$. Moreover, we show that $\pi$ contains $\tau$ with the multiplicity 1. As a corollary we obtain a kind of inertial local Langlands correspondence.

Non-isogenous superelliptic jacobians

Abstract: In his previous papers (J. reine angew. Math. 544 (2002), 91--110; math.AG/0103203) the author introduced a certain explicit construction of superelliptic jacobians, whose endomorphism ring is the ring of integers in the $p$th cyclotomic field. (Here $p$ is an odd prime.) In the present paper we discuss when these jacobians are mutually non-isogenous. (The case of hyperelliptic jacobians was treated in author's e-print math.NT/0301173 .)

Coates-wiles towers for cm-abelian varieties

Abstract: When A/Q is an elliptic curve of Mordell-Weil-rank zero with complex multiplication, Coates and Wiles showed that A satisfies the weak Birch and Swinnerton-Dyer conjecture. An alternate proof was given by Stark and Gupta, by computing conductors in what he called the Coates-Wiles tower. The main goal of this thesis is to approach the Birch and Swinnerton-Dyer problem for abelian varieties with complex multiplication by generalizing Gupta's work on Coates-Wiles towers. In particular, let K be a Galois CM-field, A an abelian variety defined over K with full complex multiplication by the ring of integers of K, and p a prime of K at which A has good reduction. Then we want to describe congruence relations on units in Kn, the extension of K generated by the pn -torsion points of A. We assume the existence of a point of infinite order Q ∈ A( K), and we let Ln be the extension of Kn generated by the pn -division values of Q (i.e., Ln = Kn( 1pn Q )). Then there exists a positive integer e such that Ln/Kn is an unramified extension, for n < e, and Le/Ke is ramified. We demonstrate that the conductor of Le/K e possesses an analogous form to the conductors calculated by Gupta. In the elliptic case, conductors of the form we calculate determine congruence relations on units of Ke. In the general case, we compute congruence relations on exterior products of units in Ke. These congruence relations relate to a recent conjecture of de Shalit.

A formula for the central value of certain Hecke L-functions

Abstract: Let N = 1 mod 4 be the negative of a prime, K=Q(sqrt{N}) and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters $\psi_{\D}$ of K with conductor $(\D)$ and infinite type $(1,0)$. Their L-series L(\psi_\D,s)$ are associated to a CM elliptic curve E(N,\D) defined over the Hilbert class field of $K$. We will prove a Waldspurger-type formula for L(\psi_\D,s) of the form L(\psi_\D,1) = \Omega \sum_{[\A],I} r(\D,[\A],I) m_{[\A],I}([\D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [\A] are class group representatives of $K$. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level $7|D|$ over $K$.

A note on Poincaré sums for finite groups

Abstract: A simple and beautiful idea of Poincare on Poincare series in automorphic functions can be applied to an arbitrary ring R acted by a group G When G is finite, the key is to look at the 0-dimensional Tate cohomology of (G, R) twisted by the 1-cohomology class of the group of units of R As a simplest case, we examine when R is the ring of integers of a quadratic field

Number of Representations of p-One-Dimensional Forms by a Genus

Abstract: We study branching of representations of a locally p-one-dimensional form by a genus of positive definite integral quadratic forms. We give a complete list of minimal representations by a genus for forms of square level. Gauss―Minkowski formulas are obtained for heights of representations over the ring of integers. As an application, we obtain formulas for heights of primitive representations by genera for specific forms constructed by the method of orthogonal complement. Bibliography: 6 titles.

A Class of Constraint Satisfaction Problems and Its Algorithms

Abstract: ?A new definition is put forward to a class of constraint satisfaction problems(CSP) in form of analytic constraints with multi-assignment to single variable. Based on the new definition, a series of concepts and 3 algorithms, i.e, the integer programming, inequility set and direct solution to indefinite equation, are set up through analyzing a particalar type of CSP, of which the 3rd algorithm is discussed in detail with its time complexity in worst case presented. Thus, the general analysis procedure of a class of CSPs can be described clearly to reveal the relation between CSP and classical integer programming,number theory and integer ring theory.