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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 article, a toroidal embedding associated to the Newton polygon of a smooth projective curve over a discretely valued field K is constructed for the case of affine equations f(x,y) = 0.
Abstract: Let C be a smooth projective curve over a discretely valued field K, defined by an affine equation f(x,y)=0. We construct a model of C over the ring of integers of K using a toroidal embedding associated to the Newton polygon of f. We show that under 'generic' conditions it is regular with normal crossings, and determine when it is minimal, the global sections of its relative dualising sheaf, and the tame part of the first etale cohomology of C.
12 citations
TL;DR: In this paper, it was shown that the set of complex numbers which are badly approximable by ratios of elements of the ring of integers in, where D ∈ {1, 2, 3, 7, 11, 19, 43, 67, 163} has maximal Hausdorff dimension.
Abstract: We show that the set of complex numbers which are badly approximable by ratios of elements of the ring of integers in , where D ∈ {1, 2, 3, 7, 11, 19, 43, 67, 163} has maximal Hausdorff dimension. In addition, the intersection of these sets is shown to have maximal dimension. The results remain true when the sets in question are intersected with a suitably regular fractal set.
12 citations
TL;DR: In this article, the authors give a simple proof of (i) without using representation theory, which will give a more explict form of n.i.b. The fact (ii) is important on the normal integral basis problem of number fields.
12 citations
TL;DR: In this article, it was shown that the set of elements of Cl (M ) which are realizable by some tamely ramified extensions with Galois groups isomorphic to Θ using some Stickelberger elements, and proved that it is a subgroup of Cl(M ).
12 citations
TL;DR: In this paper, it was shown that the Borel-serre compactification of the quotient of hyperbolic 3-space H by a finite index subgroup Γ in a Bianchi group can be used to decompose the kernel of α into its parts associated to each cusp.
12 citations