Showing papers on "Ring of integers published in 2015"

The big de Rham–Witt complex

Abstract: This paper gives a new and direct construction of the multi-prime big de Rham–Witt complex, which is defined for every commutative and unital ring; the original construction by Madsen and myself relied on the adjoint functor theorem and accordingly was very indirect. The construction given here also corrects the 2-torsion which was not quite correct in the original version. The new construction is based on the theory of modules and derivations over a λ-ring which is developed first. The main result in this first part of the paper is that the universal derivation of a λ-ring is given by the universal derivation of the underlying ring together with an additional structure depending directly on the λ-ring structure in question. In the case of the ring of big Witt vectors, this additional structure gives rise to divided Frobenius operators on the module of Kahler differentials. It is the existence of these divided Frobenius operators that makes the new construction of the big de Rham–Witt complex possible. It is further shown that the big de Rham–Witt complex behaves well with respect to etale maps, and finally, the big de Rham–Witt complex of the ring of integers is explicitly evaluated.

Hypersurfaces in projective schemes and a moving lemma

Abstract: Let X / S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H / S of X / S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in X / S containing a given closed subscheme C and intersecting properly a closed set F . Assume now that the base S is the spectrum of a ring R such that for any finite morphism Z → S , Pic ( Z ) is a torsion group. This condition is satisfied if R is the ring of integers of a number field or the ring of functions of a smooth affine curve over a finite field. We prove in this context a moving lemma pertaining to horizontal 1 -cycles on a regular scheme X quasi-projective and flat over S . We also show the existence of a finite surjective S -morphism to P S d for any scheme X projective over S when X / S has all its fibers of a fixed dimension d .

Normal zeta functions of the Heisenberg groups over number rings I: the unramified case

Abstract: Let K be a number field with ring of integers OK . We compute the local factors of the normal zeta functions of the Heisenberg groups H(OK) at rational primes which are unramified in K. These factors are expressed as sums, indexed by Dyck words, of functions defined in terms of combinatorial objects such as weak orderings. We show that these local zeta functions satisfy functional equations upon the inversion of the prime.

An arithmetic intersection formula for denominators of Igusa class polynomials

Abstract: In this paper we prove an explicit formula for the arithmetic intersection number $({\rm CM}(K).{\rm G}_1)_{\ell}$ on the Siegel moduli space of abelian surfaces, generalizing the work of Bruinier-Yang and Yang. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus $2$ curves for use in cryptography. Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for the intersection number $({\rm CM}(K).{\rm G}_1)_{\ell}$ under strong assumptions on the ramification of the primitive quartic CM field $K$. Yang later proved this conjecture assuming that $\cal{O}_K$ is freely generated by one element over the ring of integers of the real quadratic subfield. In this paper, we prove a formula for $({\rm CM}(K).{\rm G}_1)_{\ell}$ for more general primitive quartic CM fields, and we use a different method of proof than Yang. We prove a tight bound on this intersection number which holds for {\it all} primitive quartic CM fields. As a consequence, we obtain a formula for a multiple of the denominators of the Igusa class polynomials for an arbitrary primitive quartic CM field. Our proof entails studying the Embedding Problem posed by Goren and Lauter and counting solutions using our previous article that generalized work of Gross-Zagier and Dorman to arbitrary discriminants.

Adaptive Compute-and-Forward with Lattice Codes Over Algebraic Integers

Abstract: We consider the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all principal ideal domains (PID). A novel scheme called adaptive compute-and-forward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. This is enabled by generalizing the famous Construction A from PID to other rings of imaginary quadratic integers which may not form PID and by showing such the construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive compute-and-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely.

The catenary degree of krull monoids ii

Abstract: Let be a Krull monoid with finite class group such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree of is the smallest integer with the following property: for each and each pair of factorizations of , there exist factorizations of such that, for each , arises from by replacing at most atoms from by at most new atoms. To exclude trivial cases, suppose that . Then the catenary degree depends only on the class group and we have , where denotes the Davenport constant of . The cases when have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldinger et al. [‘The catenary degree of Krull monoids I’, J. Theor. Nombres Bordeaux 23 (2011), 137–169], we determine the class groups satisfying . Apart from the extremal cases mentioned, the precise value of is known for no further class groups.

Integrality in the Steinberg module and the top-dimensional cohomology of SL_n(O_K)

Abstract: We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this integrality by proving that the complex of partial bases of O^n is Cohen-Macaulay. We apply this to prove new vanishing and nonvanishing results for H^{vcd}(SL_n(O_K); Q), where O_K is the ring of integers in a number field and vcd is the virtual cohomological dimension of SL_n(O_K). The (non)vanishing depends on the (non)triviality of the class group of O_K. We also obtain a vanishing theorem for the cohomology H^{vcd}(SL_n(O_K); V) with twisted coefficients V.

Quantitative spectral gap for thin groups of hyperbolic isometries

Abstract: Let $\Lambda$ be a subgroup of an arithmetic lattice in $\mathrm{SO}(n+1 , 1)$. The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to ideals in a ring of integers. We establish a super-strong approximation result for Zariski-dense $\Lambda$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002)

On double cyclic codes over Z_4

Abstract: Let $R=\mathbb{Z}_4$ be the integer ring mod $4$. A double cyclic code of length $(r,s)$ over $R$ is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as $R[x]$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. In this paper, we determine the generator polynomials of this family of codes as $R[x]$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. Further, we also give the minimal generating sets of this family of codes as $R$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual.

Distribution of orders in number fields

Abstract: In this paper, we study the distribution of orders of bounded discriminants in number fields. We use the zeta functions introduced by Grunewald, Segal, and Smith. In order to carry out our study, we use p-adic and motivic integration techniques to analyze the zeta function. We give an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field. We also obtain non-trivial bounds for higher degree number fields.

Exterior Powers of Lubin-Tate Groups

Abstract: Let O be the ring of integers of a non-Archimedean local field of characteristic zero and π a fixed uniformizer of O. We prove that the exterior powers of a π-divisible module of dimension at most 1 over a locally Noetherian scheme exist and commute with arbitrary base change. We calculate the height and dimension of the exterior powers in terms of the height of the given π-divisible module. In the case of p-divisible groups, the existence of the exterior powers are proved without any condition on the basis.

Rankin–Selberg L-functions and the reduction of CM elliptic curves

Abstract: Let q be a prime and \(K={\mathbb Q}(\sqrt{-D})\) be an imaginary quadratic field such that q is inert in K. If \(\mathfrak {q}\) is a prime above q in the Hilbert class field of K, there is a reduction map $$\begin{aligned} r_{\mathfrak q}:\;{\mathcal {E\ell \ell }}({\mathcal {O}}_K) \longrightarrow {\mathcal {E\ell \ell }}^{ss}({\mathbb F}_{q^2}) \end{aligned}$$ from the set of elliptic curves over \(\overline{{\mathbb Q}}\) with complex multiplication by the ring of integers \({\mathcal {O}}_K\) to the set of supersingular elliptic curves over \({\mathbb {F}}_{q^2}.\) We prove a uniform asymptotic formula for the number of CM elliptic curves which reduce to a given supersingular elliptic curve and use this result to deduce that the reduction map is surjective for \(D \gg _{\varepsilon } q^{18+\varepsilon }.\) This can be viewed as an analog of Linnik’s theorem on the least prime in an arithmetic progression. We also use related ideas to prove a uniform asymptotic formula for the average $$\begin{aligned} \sum _{\chi }L(f \times \Theta _\chi ,1/2) \end{aligned}$$ of central values of the Rankin–Selberg L-functions \({L(f \times {\Theta _{\chi}},s)}\) where f is a fixed weight 2, level q arithmetically normalized Hecke cusp form and \(\Theta _\chi \) varies over the weight 1, level D theta series associated to an ideal class group character \(\chi \) of K. We apply this result to study the arithmetic of Abelian varieties, subconvexity, and \(L^4\) norms of autormorphic forms.

Ring-LWE Cryptography for the Number Theorist.

Abstract: In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems [EHL, ELOS] gives rise to interesting questions about number fields. We extend these attacks and survey related open problems in number theory, including spectral distortion of an algebraic number and its relationship to Mahler measure, the monogenic property for the ring of integers of a number field, and the size of elements of small order modulo q.

Integral $p$-adic Hodge theory — announcement

Abstract: Let C be an algebraically closed, nonarchimedean field of mixed characteristic which is complete under a rank one valuation; let O ⊆ C be its ring of integers, with maximal ideal m and residue field k. The reader may assume that C is the completion of an algebraic closure of Qp. The main aim of this note is to outline a proof of the following result, which was first announced by the third author during his series of Fall 2014 lectures at the MSRI. Details, generalisations, and further results will be presented in a forthcoming article. In particular, this will include a “comparison isomorphism”-style result.

From pro-$p$ Iwahori-Hecke modules to $(\varphi,\Gamma)$-modules I

Abstract: Let ${\mathfrak o}$ be the ring of integers in a finite extension $K$ of ${\mathbb Q}_p$, let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb Q}_p$, let $T$ be a maximal split torus in $G$. Let ${\mathcal H}(G,I_0)$ be the pro-$p$-Iwahori Hecke ${\mathfrak o}$-algebra. Given a semiinfinite reduced chamber gallery (alcove walk) $C^{({\bullet})}$ in the $T$-stable apartment, a period $\phi\in N(T)$ of $C^{({\bullet})}$ of length $r$ and a homomorphism $\tau:{\mathbb Z}_p^{\times}\to T$ compatible with $\phi$, we construct a functor from the category ${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$ of finite length ${\mathcal H}(G,I_0)$-modules to \'{e}tale $(\varphi^r,\Gamma)$-modules over Fontaine's ring ${\mathcal O}_{\mathcal E}$. If $G={\rm GL}_{d+1}({\mathbb Q}_p)$ there are essentially two choices of ($C^{({\bullet})}$, $\phi$, $\tau$) with $r=1$, both leading to a functor from ${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$ to \'{e}tale $(\varphi,\Gamma)$-modules and hence to ${\rm Gal}_{{\mathbb Q}_p}$-representations. Both induce a bijection between the set of absolutely simple supersingular ${\mathcal H}(G,I_0)\otimes_{\mathfrak o} k$-modules of dimension $d+1$ and the set of irreducible representations of ${\rm Gal}_{{\mathbb Q}_p}$ over $k$ of dimension $d+1$. We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of $G$ over $K$. For $d=1$ we recover Colmez' functor (when restricted to ${\mathfrak o}$-torsion ${\rm GL}_{2}({\mathbb Q}_p)$-representations generated by their pro-$p$-Iwahori invariants)

Generalized jacobians and Pellian polynomials

Abstract: Pell equations over the ring of integers are the forerunners of Thue equations. In fact, they too often have only finitely many solutions, when set over polynomial rings in characteristic zero. How often this happens has been the theme of recent work of D. Masser and U. Zannier. We pursue this study by considering Pell equations with non square-free discriminants over such rings. On the occasion of A. Thue’s 150th birthday

Transcendental Brauer groups of products of CM elliptic curves

Abstract: Let $L$ be a number field and let $E/L$ be an elliptic curve with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface $E\times E$. The results for the odd order torsion also apply to the Brauer group of the K3 surface $\textrm{Kum}(E\times E)$. We describe explicitly the elliptic curves $E/\mathbb{Q}$ with complex multiplication by $\mathcal{O}_K$ such that the Brauer group of $E\times E$ contains a transcendental element of odd order. We show that such an element gives rise to a Brauer-Manin obstruction to weak approximation on $\textrm{Kum}(E\times E)$, while there is no obstruction coming from the algebraic part of the Brauer group.

Finiteness theorems for vanishing cycles of formal schemes

Abstract: Let k be a non-Archimedean field with nontrivial valuation, and k po its ring of integers. In this paper we prove constructibility of vanishing cycles sheaves for arbitrary formal schemes locally finitely presented over k po as well as special formal schemes over k po (for discretely valued k). This allows us to extend continuity results, established earlier for locally algebraic formal schemes, to the whole classes of formal schemes.

An explicit Baker type lower bound of exponential values

Abstract: Let I denote an imaginary quadratic eld or the eld Q of rational numbers and ZI its ring of integers. We shall prove a new explicit Baker type lower bound for a ZIlinear form in the numbers 1; e 1 ;:::; e m ; m 2, where 0 = 0, 1;:::; m, are m + 1 dierent numbers from the eld I. Our work gives substantial improvements to the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies on m are improved.

$(1+2u)$-constacyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$.

Abstract: Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes over $R$ of odd lengths with the help of cyclic codes over $R$. It is proved that the Gray image of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic codes of length $2n$ over $\mathbb{Z}_4$. Further, a number of linear codes over $\mathbb{Z}_4$ as the images of $(1+2u)$-constacyclic codes over $R$ are obtained.

A Note on Finitely Generated Z-module and Algebraic Integers

Abstract: The theory of algebraic integer has its many applications, such as in algebraic coding, cryptology, information system and other fields. The research of algebraic integer can not leave finitely generated module, and the finitely generated module itself be also applied in group theory, ring theory, and some applied science. In this paper, we research the theory of algebraic integer using finitely generated module as tool, we obtained necessary and sufficient condition that an element is algebraic integer, and an intrinsic connects between algebraic number field and finitely generated Z-module.

Log Rigid Syntomic Cohomology for Strictly Semistable Schemes

Abstract: We construct log rigid syntomic cohomology for strictly semistable schemes over the ring of integers of a p-adic field, and prove that it is interpreted as the extension group of the complex of admissible filtered (φ,N)-modules.

A Note On Non-ordinary Primes

Abstract: Suppose that $O_L$ is the ring of integers of a number field $L$, and suppose that $f(z)=\sum_{n=1}^\infty a_f(n)q^n\in S_k\cap O_L[[q]]$ (note: $q := e^{2\pi iz}$) is a normalized Hecke eigenform for $\mathrm{SL}_2(\mathbb{Z})$. We say that $f$ is non-ordinary at a prime $p$ if there is a prime ideal $\mathfrak{p}\subset O_L$ above $p$ for which $a_f(p)\equiv 0 \ (mod\ {\mathfrak{p}})$. For any finite set of primes $S$, we prove that there are normalized Hecke eigenforms which are non-ordinary for each $p\in S$. The proof is elementary and follows from a generalization of work of Choie, Kohnen and the third author.

Diophantine exponents for standard linear actions of $\mathrm{SL}_2$ over discrete rings in $\mathbb{C}$

Abstract: We give upper and lower bounds for various Diophantine exponents associated with the standard linear actions of $\mathrm{SL}_2 ( \mathcal{O}_K )$ on the punctured complex plane $\mathbb{C}^2 \setminus \{ \mathbf{0} \}$, where $K$ is a number field whose ring of integers $\mathcal{O}_K$ is discrete and within a unit distance of any complex number. The results are similar to those of Laurent and Nogueira for $\mathrm{SL}_2 ( \mathbb{Z} )$ action on $\mathbb{R}^2 \setminus \{ \mathbf{0} \}$ albeit for us, uniformly nice bounds are obtained only outside of a set of null measure.

The Apollonian structure of Bianchi groups

Abstract: We study the orbit of $\widehat{\mathbb{R}}$ under the Mobius action of the Bianchi group $\operatorname{PSL}_2(\mathcal{O}_K)$ on $\widehat{\mathbb{C}}$, where $\mathcal{O}_K$ is the ring of integers of an imaginary quadratic field $K$. The orbit $\mathcal{S}_K$, called a Schmidt arrangement, is a geometric realisation, as an intricate circle packing, of the arithmetic of $K$. We give a simple geometric characterisation of certain subsets of $\mathcal{S}_K$ generalizing Apollonian circle packings, and show that $\mathcal{S}_K$, considered with orientations, is a disjoint union of all primitive integral such $K$-Apollonian packings. These packings are described by a new class of thin groups of arithmetic interest called $K$-Apollonian groups. We make a conjecture on the curvatures of these packings, generalizing the local-to-global conjecture for Apollonian circle packings.

Minimal dimension of faithful representations for $p$-groups

Abstract: For a group $G$, we denote by $m_{faithful}(G)$, the smallest dimension of a faithful complex representation of $G$. Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the maximal ideal $\mathfrak{p}$. In this paper, we compute the precise value of $m_{faithful}(G)$ when $G$ is the Heisenberg group over $\mathcal{O}/\mathfrak{p}^n$. We then use the Weil representation to compute the minimal dimension of faithful representations of the group of unitriangular matrices over $\mathcal{O}/\mathfrak{p}^n$ and many of its subgroups. By a theorem of Karpenko and Merkurjev, our result yields the precise value of the essential dimension of the latter finite groups.

Conjugacy classes of torsion in GL_n(Z)

Abstract: The problem of integral similarity of block-triangular matrices over the ring of integers is connected to that of finding representatives of the classes of an equivalence relation on general integer matrices. A complete list of representatives of conjugacy classes of torsion in the 4 A— 4 general linear group over ring of integers is given. There are 45 distinct such classes and each torsion element has order of 1, 2, 3, 4, 5, 6, 8, 10 or 12.

On the realizable classes of the square root of the inverse different in the unitary class group

Abstract: Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and let $G$ be a finite abelian group of odd order. Given a $G$-Galois $K$-algebra $K_h$, let $A_h$ denote its square root of the inverse different, which exists by Hilbert's formula. If $K_h/K$ is weakly ramified, then the pair $(A_h,Tr_h)$ is locally $G$-isometric to $(\mathcal{O}_KG,t_K)$ and hence defines a class in the unitary class group $\mbox{UCl}(\mathcal{O}_KG)$ of $\mathcal{O}_KG$. Here $Tr_h$ denotes the trace of $K_h/K$ and $t_K$ the symmetric bilinear form on $\mathcal{O}_KG$ for which $t_K(s,t)=\delta_{st}$ for all $s,t\in G$. We study the collection of all such classes and show that a subset of them is in fact a subgroup of $\mbox{UCl}(\mathcal{O}_KG)$.

Rational discrete first degree cohomology for totally disconnected locally compact groups

Abstract: It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $\mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $\mathrm{dH}^1(G,\mathrm{Bi}(G))$, where $\mathrm{Bi}(G)$ is the rational discrete standard bimodule of $G$, and $\mathrm{dH}^\bullet(G,\_)$ denotes rational discrete cohomology as introduced in [6]. As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B).