Ring of integers
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Integrality in the Steinberg module and the top-dimensional cohomology of SLn OK Open
We prove a new structural result for the spherical Tits building attached to\nSL_n(K) for many number fields K, and more generally for the fraction fields of\nmany Dedekind domains O: the Steinberg module St_n(K) is generated by integral\n…
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The regular representations of GLN over finite local principal ideal rings Open
Let
\no
\no be the ring of integers in a non-Archimedean local field with finite residue field,
\np
\np its maximal ideal, and
\nr
\n⩾
\n2
\nr⩾2 an integer. An irreducible representation of the finite group
\nG
\nr
\n=
\nGL
\nN
\n(
\no…
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On error distributions in ring-based LWE Open
Since its introduction in 2010 by Lyubashevsky, Peikert and Regev, the ring learning with errors problem (ring-LWE) has become a popular building block for cryptographic primitives, due to its great versatility and its hardness proof consi…
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Properly integral polynomials over the ring of integer-valued polynomials on a matrix ring Open
Let $D$ be a domain with fraction field $K$, and let $M_n(D)$ be the ring of\n$n \\times n$ matrices with entries in $D$. The ring of integer-valued\npolynomials on the matrix ring $M_n(D)$, denoted ${\\rm Int}_K(M_n(D))$,\nconsists of tho…
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Calculation of Fourier-Galois transforms in reduced binary number systems Open
The paper proposes a new method for calculating Fourier-Galois transforms (number-theoretical transforms), which are a modular analog of the discrete Fourier transform. A number of specific problems related to the calculation of transforms…
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Universal quadratic forms and indecomposables in number fields: A survey Open
We give an overview of universal quadratic forms and lattices, focusing on the recent developments over the rings of integers in totally real number fields. In particular, we discuss indecomposable algebraic integers as one of the main too…
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A formal proof of Hensel's lemma over the p-adic integers Open
The field of $p$-adic numbers $\\mathbb{Q}_p$ and the ring of $p$-adic\nintegers $\\mathbb{Z}_p$ are essential constructions of modern number theory.\nHensel's lemma, described by Gouv\\^ea as the "most important algebraic property\nof the…
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p-adic estimates of exponential sums oncurves Open
The purpose of this article is to prove a ``Newton over Hodge'' result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve…
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Moduli of Langlands Parameters Open
Let $F$ be a nonarchimedean local field of residue characteristic $p$, let $\hat{G}$ be a split reductive group over $\mathbb{Z}[1/p]$ with an action of $W_F$, and let $^LG$ denote the semidirect product $\hat{G}\rtimes W_F$. We construct …
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Generalization of low rank parity-check (LRPC) codes over the ring of integers modulo a positive integer Open
Following the work of Gaborit et al. (in: The international workshop on coding and cryptography (WCC 13), 2013) defining LRPC codes over finite fields, Renner et al. (in: IEEE international symposium on information theory, ISIT 2020, 2020)…
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Computation of lattice isomorphisms and the integral matrix similarity problem Open
Let K be a number field, let A be a finite-dimensional K -algebra, let $\operatorname {\mathrm {J}}(A)$ denote the Jacobson radical of A and let $\Lambda $ be an $\mathcal {O}_{K}$ -order in A . Suppose that each simple component of the se…
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Imaginary biquadratic Polya fields of the form Q(sqrt(d), sqrt(-2)) Open
A number field k, with ring of integers Ok, is called a Polya field if the module of integer-valued polynomials over Ok has a regular basis. In this paper, we characterize all imaginary biquadratic P\olya fields of the form ℚ(√d, √-2) wher…
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Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields Open
Let K be a real quadratic field and OK its ring of integers. Let Γ be a congruence subgroup of SL2(OK) and M(k1,k2)(Γ) be the finite dimensional space of Hilbert modular forms of weight (k1, k2) for Γ. Given a form f(z) ∈ M(k1,k2)(Γ), how …
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Distribution of Modular Symbols in ℍ3 Open
We introduce a new technique for the study of the distribution of modular symbols, which we apply to the congruence subgroups of Bianchi groups. We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O…
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Polynomial Dedekind domains with finite residue fields of prime characteristic Open
We show that every Dedekind domain $R$ lying between the polynomial rings $\\Z[X]$ and $\\Q[X]$ with the property that its residue fields of prime characteristic are finite fields is equal to a generalized ring of integer-valued polynomial…
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Splines over integer quotient rings Open
Given a graph with edges labeled by elements in $\mathbb{Z}/m\mathbb{Z}$, a generalized spline is a labeling of each vertex by an integer $\mod m$ such that the labels of adjacent vertices agree modulo the label associated to the edge conn…
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Universally defining finitely generated subrings of global fields Open
It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in…
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Datatype defining rewrite systems for the ring of integers, and for natural and integer arithmetic in unary view. Open
A datatype defining rewrite system (DDRS) is a ground-complete term rewriting system, intended to be used for the specification of datatypes. As a follow-up of an earlier paper we define two concise DDRSes for the ring of integers, each co…
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Infinitesimal Comparisons: Homomorphisms between Giordano’s Ring and the Hyperreal Field Open
The primary purpose of this paper is to analyze the relationship between the familiar non-Archimedean field of hyperreals from Abraham Robinson’s nonstandard analysis and Paolo Giordano’s ring extension of the real numbers containing nilpo…
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On the relative Galois module structure of rings of integers in tame extensions Open
Let $F$ be a number field with ring of integers $O_F$ and let $G$ be a finite\ngroup. We describe an approach to the study of the set of realisable classes in\nthe locally free class group $Cl(O_FG)$ of $O_FG$ that involves applying the\nw…
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On the Distribution of <i>αp</i> Modulo One in Quadratic Number Fields Open
We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and…
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Non-Wieferich primes in number fields and $abc$-conjecture Open
Let $K/\mathbb{Q}$ be an algebraic number field of class number one and $\mathcal{O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal{O}_K$ under the assumpt…
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Diophantine problems in rings and algebras: undecidability and reductions to rings of algebraic integers Open
We study systems of equations in different families of rings and algebras. In each such structure $R$ we interpret by systems of equations (e-interpret) a ring of integers $O$ of a global field. The long standing conjecture that $\mathbb{Z…
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Abelian $n$-division fields of elliptic curves and Brauer groups of product Kummer & abelian surfaces Open
Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$ . In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\,Y/\te…
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Representations of SL over finite local rings of length two Open
Let Fqbe a finite field of characteristic pand let W2(Fq)be the ring of Witt vectors of length two over Fq. We prove that for any integer nsuch that pdivides n, the groups SLn(Fq[t]/t2)and SLn(W2(Fq)) have the same number of irreducible re…
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An Alternative Approach to Polynomial Modular Number System Internal Reduction Open
International audience
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Chinese remainder theorem secret sharing in multivariate polynomials Open
This paper deals with a generalization of the secret sharing using Chinese remainder theorem over the integers to multivariate polynomials over a finite field. We work with the ideals and their Gröbner bases instead of integer moduli. Ther…
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Period Relations for Standard $L$-functions of Symplectic Type Open
This article is to understand the critical values of $L$-functions $L(s,Π\otimes χ)$ and to establish the relation of the relevant global periods at the critical places. Here $Π$ is an irreducible regular algebraic cuspidal automorphic rep…
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Theoretical and Experimental Approach to p-Class Field Towers of Cyclic Cubic Number Fields Open
Cyclic number fields of odd prime degree are constructed as ray class fields over the rational number field. They are collected in multiplets sharing a common conductor and discriminant. The algorithms are implemented in Magma and applied …
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Definability and decidability for rings of integers in totally imaginary fields Open
We show that the ring of integers of is existentially definable in the ring of integers of , where denotes the field of all totally real numbers. This implies that the ring of integers of is undecidable and first‐order nondefinable in . Mo…