Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

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.

Quaternion-based, efficient fully-homomorphic cryptosystem

Abstract: The present invention relates to an efficient fully-homomorphic cryptosystem based on a new homomorphic and probabilistic transform between ring ℤ⁄2ℤ (ring of the residual integers modulo 2) and the modular Lipschitz integer ring. The cryptosystem of the invention can process clear messages in the form of input bits and supply ciphertexts in the form of Lipschitz quaternion matrices modulo a large even natural number η in a non-deterministic manner. The security of the present cryptosystem is based on the difficulty of solving a system of multivariate polynomial equations in a non-commutative ring. The cryptosystem can be used to further reduce multiplication computation time in fully-homomorphic encryption algorithms, since it allows the size of a secret key to be minimised and the expansion of ciphertexts to be reduced.

Zeta Functions for the Adjoint Action of GL(n) and Density of Residues of Dedekind Zeta Functions

Abstract: We define zeta functions for the adjoint action of \(\mathop{\mathrm{GL}} olimits _{n}\) on its Lie algebra and study their analytic properties. For n ≤ 3 we are able to fully analyse these functions. If n = 2, we recover the Shintani zeta function for the prehomogeneous vector space of binary quadratic forms. Our construction naturally yields a regularisation, which is necessary to improve the analytic properties of these zeta function, in particular for the analytic continuation if n ≥ 3.We further obtain upper and lower bounds on the mean value \(X^{-\frac{5} {2} }\sum _{E}\mathop{ \mathrm{res}}_{s=1}\zeta _{E}(s)\) as X → ∞, where E runs over totally real cubic number fields whose second successive minimum of the trace form on its ring of integers is bounded by X. To prove the upper bound we use our new zeta function for \(\mathop{\mathrm{GL}} olimits _{3}\). These asymptotic bounds are a first step towards a generalisation of density results obtained by Datskovsky in case of quadratic field extensions.

On a reduction map for Drinfeld modules

Abstract: In this paper we investigate a local to global principle for Mordell-Weil group defined over a ring of integers ${\cal O}_K$ of $t$-modules that are products of the Drinfeld modules ${\widehat\varphi}={\phi}_{1}^{e_1}\times \dots \times {\phi}_{t}^{e_{t}}.$ Here $K$ is a finite extension of the field of fractions of $A={\mathbb F}_{q}[t].$ We assume that the ${\mathrm{rank}}(\phi)_{i})=d_{i}$ and endomorphism rings of the involved Drinfeld modules of generic characteristic are the simplest possible, i.e. ${\mathrm{End}}({\phi}_{i})=A$ for $ i=1,\dots , t.$ Our main result is the following numeric criterion. Let ${N}={N}_{1}^{e_1}\times\dots\times {N}_{t}^{e_t}$ be a finitely generated $A$ submodule of the Mordell-Weil group ${\widehat\varphi}({\cal O}_{K})={\phi}_{1}({\cal O}_{K})^{e_{1}}\times\dots\times {\phi}_{t}({\cal O}_{K})^{{e}_{t}},$ and let ${\Lambda}\subset N$ be an $A$ - submodule. If we assume $d_{i}\geq e_{i}$ and $P\in N$ such that $r_{\cal W}(P)\in r_{\cal W}({\Lambda}) $ for almost all primes ${\cal W}$ of ${\cal O}_{K},$ then $P\in {\Lambda}+N_{tor}.$ We also build on the recent results of S.Bara{n}czuk \cite{b17} concerning the dynamical local to global principle in Mordell-Weil type groups and the solvability of certain dynamical equations to the aforementioned $t$-modules.

An automorphic generalization of the Hermite-Minkowski theorem.

Abstract: We show that for any integer $N$, there are only finitely many cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,1,,23\}$ More generally, we define a simple sequence $(r(w))_{w \geq 0}$ such that for any integer $w$, any number field $E$ whose root-discriminant is less than $r(w)$, and any ideal $N$ in the ring of integers of $E$, there are only finitely many cuspidal algebraic automorphic representations of general linear groups over $E$ whose conductor is $N$ and whose weights are in the interval $\{0,1,,w\}$ Assuming a version of GRH, we also show that we may replace $r(w)$ with $8 \pi e^{\gamma-H_w}$ in this statement, where $\gamma$ is Euler's constant and $H_w$ the $w$-th harmonic number The proofs are based on some new positivity properties of certain real quadratic forms which occur in the study of the Weil explicit formula for Rankin-Selberg $L$-functions Both the effectiveness and the optimality of the methods are discussed