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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 paper, the complement of the ring of integers in a number field K is Diophantine and the set of ring integers in K can be written as {t in K | for all x_1,..., x_N in K, f(t,x_1,..., X, N) is not 0}.
Abstract: We show that the complement of the ring of integers in a number field K is Diophantine. This means the set of ring of integers in K can be written as {t in K | for all x_1, ..., x_N in K, f(t,x_1, ..., x_N) is not 0}. We will use global class field theory and generalize the ideas originating from Koenigsmann's recent result giving a universal first order formula for Z in Q.
12 citations
TL;DR: In this paper, it was shown that a hyperelliptic curve has a Weierstrass model over the ring of integers of a number field with height effectively bounded only in terms of the genus of the curve.
Abstract: Let $C$ be a hyperelliptic curve of genus $g\geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\geq 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside $S$.
12 citations
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TL;DR: In this article, a finite extension with ring of integers is associated to a finitely generated module over the Fontaine ring over $o $ endowed with a semilinear ''etale action of the monoid $T+}.
Abstract: Let $K/\mathbb Q_{p}$ be a finite extension with ring of integers $o$, let $G$ be a connected reductive split $\mathbb Q_{p}$-group of Borel subgroup $P=TN$ and let $\alpha$ be a simple root of $T$ in $N$. We associate to a finitely generated module $D$ over the Fontaine ring over $o $ endowed with a semilinear \'etale action of the monoid $T_{+} $ (acting on the Fontaine ring via $\alpha$), a $G(\mathbb Q_{p})$-equivariant sheaf of $o$-modules on the compact space $G(\mathbb Q_{p})/P(\mathbb Q_{p})$. Our construction generalizes the representation $D\boxtimes \mathbb P^{1} $ of $ GL(2,\mathbb Q_{p})$ associated by Colmez to a $(\varphi,\Gamma)$-module $D$ endowed with a character of $\mathbb Q_{p}^{*}$.
12 citations
TL;DR: For all odd primes p, it was shown in this paper that p does not divide h+ if and only if each cyclic extension of K, lying in the minus part H− of the Hilbert p-class field H of k, is contained in a Z p-extension of K.
11 citations
01 Jan 1980
TL;DR: In this paper, the Artin-Hasse exponentials are replaced by a functor WF(- which is essentially the functor of q-typical curves in a (twisted) Lubin-Tate formal group law over A, where A is a discrete valuation ring that admits a Frobenius-like endomorphism a (we require o(a) =_ a q mod m for all a E A where m is the maximal idea of A).
Abstract: For any ring R let A(R) denote the multiplicative group of power series of the form I + a It + * * * with coefficients in R. The Artin-Hasse exponential mappings are homomorphisms Wp, ,(k) -+ A( Wp,, (k)), which satisfy certain additional properties. Somewhat reformulated, the Artin-Hasse exponentials turn out to be special cases of a functorial ring homomorphism E: Wp .(-) -> W ,,,0( Wp(-)), where Wp is the functor of infinite-length Witt vectors associated to the prime p. In this paper we present ramified versions of both Wp,4(-) and E, with Wp .(-) replaced by a functor WF(-), which is essentially the functor of q-typical curves in a (twisted) Lubin-Tate formal group law over A, where A is a discrete valuation ring that admits a Frobenius-like endomorphism a (we require o(a) =_ a q mod m for all a E A, where m is the maximal idea of A). These ramified-Witt-vector functors W!F,(-) do indeed have the property that, if k = A/rm is perfect, A is complete, and I/k is a finite extension of k, then Wqo.(l) is the ring of integers of the unique unramified extension LIK covering I/k.
11 citations