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.

Arithmetic Part of Faltings’s Proof

Abstract: In this chapter we will follow §5 of [22] quite closely. We start in the following situation: k is a number field, A k is an abelian variety over k, X k ⊂ A k is a subvariety such that \( X_{\bar k} \) does not contain any translate of a positive dimensional abelian subvariety of \( A_{\bar k} \), m is a sufficiently large integer as in Chapter IX, Lemma 1 Let R be the ring of integers in k. Since we want to apply Faltings’s version of Siegel’s Lemma (see Ch. X, Lemma 4) we need lattices in things like г(X k m , line bundle). We obtain such lattices as: г(proper model of X k m over R, extension of line bundle).

Graded cancellation properties of graded rings and graded unit-regular Leavitt path algebras

Abstract: We raise the following general question regarding a ring graded by a group: "If $P$ is a ring-theoretic property, how does one define the graded version $P_{\operatorname{gr}}$ of the property $P$ in a meaningful way?". Some properties of rings have straightforward and unambiguous generalizations to their graded versions and these generalizations satisfy all the matching properties of the nongraded case. If $P$ is either being unit-regular, having stable range 1 or being directly finite, that is not the case. The first part of the paper addresses this issue. Searching for appropriate generalizations, we consider graded versions of cancellation, internal cancellation, substitution, and module-theoretic direct finiteness. In the second part of the paper, we turn to Leavitt path algebras. If $K$ is a trivially graded field and $E$ is an oriented graph, the Leavitt path algebra $L_K(E)$ is naturally graded by the ring of integers. If $E$ is a finite graph, we present a property of $E$ which is equivalent with $L_K(E)$ being graded unit-regular. This property critically depends on the lengths of paths to cycles making it stand out from other known graph conditions which characterize algebraic properties of $L_K(E).$ It also further illustrates that graded unit-regularity is quite restrictive in comparison to the alternative generalization of unit-regularity which we consider in the first part of the paper.

Endomorphisms of superelliptic jacobians

Abstract: Let K be a field of characteristic zero, n>4 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group is doubly transitive simple non-abelian group. Let p be an odd prime, Z[\zeta_p] the ring of integers in the p-th cyclotomic field, C_{f,p}:y^p=f(x) the corresponding superelliptic curve and J(C_{f,p}) its jacobian. Assuming that either n=p+1 or p does not divide n(n-1), we prove that the ring of all endomorphisms of J(C_{f,p}) coincides with Z[\zeta_p].

On the Probability of Relative Primality in the Gaussian Integers

Abstract: This paper studies the interplay between probability, number theory, and geometry in the context of relatively prime integers in the ring of integers of a number field. In particular, probabilistic ideas are coupled together with integer lattices and the theory of zeta functions over number fields in order to show that $$P(\gcd(z_{1},z_{2})=1) = \frac{1}{\zeta_{\Q(i)}(2)}$$ where $z_{1},z_{2} \in \mathbb{Z}[i]$ are randomly chosen and $\zeta_{\Q(i)}(s)$ is the Dedekind zeta function over the Gaussian integers. Our proof outlines a lattice-theoretic approach to proving the generalization of this theorem to arbitrary number fields that are principal ideal domains.