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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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33 citations
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TL;DR: In this article, Poonen's weak vertical method was used to produce new examples of rings of algebraic numbers (including rings of integers) where Z and/or the ring of integers of a subfield are existentially definable and where Mazur's conjecture on the topology of rational points does not hold.
Abstract: Using Poonen's version of the "weak vertical method" we produce new examples of "large" and "small" rings of algebraic numbers (including rings of integers) where Z and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur's conjecture on the topology of rational points does not hold.
33 citations
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TL;DR: In this paper, the Deligne-Pappas condition was considered for the case e = g, that is, p is totally ramified in a totally real field with ring of integers OL.
Abstract: Let L be a totally real field with ring of integers OL. Let N ≥ 4 be an integer and let M(μN) be the fine moduli scheme over Z of polarized abelian varieties with real multiplication (RM) and μN-level structure, satisfying the Deligne-Pappas condition. For every scheme S, we let M(S, μN) = M(μN) ×Z S be the moduli scheme over S; see Definition 2.1. Many aspects of the geometry of the modular varieties M(Fp, μN) are obtained via local deformation theory that factorizes according to the decomposition of p in OL. The unramified case was considered in [9] (see also [8]). Given that, one may restrict one’s attention to the case p = p in OL. We discuss here only the case e = g, that is, p is totally ramified in L. The ramified case was first treated by Deligne and Pappas in [6] (the case g = 2 was considered in [2]). We recall some of their results under the assumption that p is totally ramified. Let A/k be a polarized abelian variety with RM, defined over a field k of characteristic p. Fix an isomorphism OL⊗Zk ∼ = k[T ]/(T). One knows that HdR(A) is a free k[T ]/(T)-module of rank 2. The elementary divisors theorem furnishes us with k[T ]/(T)generators α and β for HdR(A) such that
33 citations
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TL;DR: In this paper, the canonical pairing associated with a one-dimensional formal group law over the ring of integers of a finite extension of and an isogeny was investigated, where the Hilbert symbol was associated with the multiplicative law and the amogeny raising to the th power.
Abstract: This paper investigates the canonical pairing associated with a one-dimensional formal group law over the ring of integers of a finite extension of and an isogeny , just as the Hilbert symbol is associated with the multiplicative law and the isogeny raising to the th power. Formulas are obtained which generalize the formulas of Artin-Hasse, Iwasawa, and Wiles. The formulas describe the values of the symbol in terms of -adic differentiation, the logarithm of the formal group law, the norm, and the trace. Bibliography: 8 titles.
33 citations
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TL;DR: In particular, it is not weakly mixing and has zero metric entropy as discussed by the authors, which is the property of the Z^d-action on a compact abelian group.
Abstract: Let $K/\mathbf{Q}$ be a degree-$d$ extension. Inside the ring of integers
$\mathscr O_K$ we define the set of $k$-free integers $\mathscr F_k$ and a
natural $\mathscr O_K$-action on the space of binary $\mathscr O_K$-indexed sequences,
equipped with an $\mathscr O_K$-invariant probability measure associated to
$\mathscr F_k$. We prove that this action is ergodic, has pure point
spectrum, and is isomorphic to a $\mathbf Z^d$-action on a compact abelian
group. In particular, it is not weakly mixing and has zero
measure-theoretical entropy. This work generalizes the work of
Cellarosi and Sinai [ J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1343--1374] that considered the case $K=\mathbf{Q}$ and $k=2$.
32 citations