Topic
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.
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TL;DR: In this paper, a complete proof of a duality theorem for the fppf cohomology of either a curve over a finite field or a ring of integers of a number field is given.
Abstract: We provide a complete proof of a duality theorem for the fppf cohomology of either a curve over a finite field or a ring of integers of a number field, which extends the classical Artin–Verdier Theorem in etale cohomology. We also prove some finiteness and vanishing statements.
8 citations
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TL;DR: In this article, the restriction of irreducible representations of a non-Archimedean local field to a Borel subgroup of the local field was studied. But the restriction was not restricted to the representation theory of the original local field.
Abstract: Let $F$ be a non-Archimedean local field and let $p$ be the residual characteristic of $F$. Let $G=GL_2(F)$ and let $P$ be a Borel subgroup of $G$. In this paper we study the restriction of irreducible representations of $G$ on $E$-vector spaces to $P$, where $E$ is an algebraically closed field of characteristic $p$. We show that in a certain sense $P$ controls the representation theory of $G$. We then extend our results to smooth $\oK[G]$- modules of finite length and unitary $K$-Banach space representations of $G$, where $\oK$ is the ring of integers of a complete discretely valued field $K$, with residue field $E$.
8 citations
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TL;DR: In this article, all indecomposable unimodular hermitian lattices in dimension 13 over the ring of integers in Q ( −3 ) were determined, and all of them were shown to be convex.
8 citations
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TL;DR: In this paper, it was shown that any formal subgroup of the generic fiber of A whose closure in A is smooth over an open subset of Spec O� k>>\s is a subvariety of A. The proof is of a transcendental nature and uses the Arakelovian formalism introduced by Bost [3].
Abstract: In this paper, we generalize the result of [12] in the following sense. Let A be an abelian variety over a number field k, let ? be the Neron model of A over the ring of integers O
k
of k. Completing ? along its zero section defines a formal group
$\widehat{\mathcal{A}}$
over O
k
. We prove that any formal subgroup of the generic fiber of
$\widehat{\mathcal{A}}$
whose closure in
$\widehat{\mathcal{A}}$
is smooth over an open subset of Spec O
k
arises in fact from an abelian subvariety of A. The proof is of a transcendental nature and uses the Arakelovian formalism introduced by Bost [3].
8 citations
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TL;DR: Fraction-free forms for this classical test are presented that enhance it with the property that the testing of a polynomial with Gaussian or real integer coefficients can be completed over the respective ring of integers.
Abstract: The Routh test is the simplest and most efficient algorithm to determine whether all the zeros of a polynomial have negative real parts. However, the test involves divisions that may decrease its numerical accuracy and are a drawback in its use for various generalized applications. The paper presents fraction-free forms for this classical test that enhance it with the property that the testing of a polynomial with Gaussian or real integer coefficients can be completed over the respective ring of integers. Two types of algorithms are considered one, named the G-sequence, which is most efficient (as an integer algorithm) for Gaussian integers, and another, named the R-sequence, which is most efficient for real integers. The G-sequence can be used also for the real case, but the R-sequence is by far more efficient for real integer polynomials. The count of zeros with positive real parts for normal polynomials is also presented for each algorithm.
8 citations