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Let O be the ring of integers of a number field K. For an O-algebra R which is torsion free as an O-module we define what we mean by a Lambda_O-ring structure on R. We can determine whether a finite etale K-algebra E with Lambda_O-ring structure has an integral model in terms of a Deligne-Ribet monoid of K. This a commutative monoid whose invertible elements form a ray class group.
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We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and their behavior under certain quotients. We illustrate the power of our ideas in a new modular normal form algorithm for modules over rings of integers, vastly outperforming classical algorithms.
Let $K$ be a number field with ring of integers $R$. Given a modulus $mathfrak{m}$ for $K$ and a group $Gamma$ of residues modulo $mathfrak{m}$, we consider the semi-direct product $Rrtimes R_{mathfrak{m},Gamma}$ obtained by restricting the multiplicative part of the full $ax+b$-semigroup over $R$ to those algebraic integers whose residue modulo $mathfrak{m}$ lies in $Gamma$, and we study the left regular C*-algebra of this semigroup. We give two presentations of this C*-algebra and realize it as a full corner in a crossed product C*-algebra. We also establish a faithfulness criterion for representations in terms of projections associated with ideal classes in a quotient of the ray class group modulo $mathfrak{m}$, and we explicitly describe the primitive ideals using relations only involving the range projections of the generating isometries; this leads to an explicit description of the boundary quotient. Our results generalize and strengthen those of Cuntz, Deninger, and Laca and of Echterhoff and Laca for the C*-algebra of the full $ax+b$-semigroup. We conclude by showing that our construction is functorial in the appropriate sense; in particular, we prove that the left regular C*-algebra of $Rrtimes R_{mathfrak{m},Gamma}$ embeds canonically into the left regular C*-algebra of the full $ax+b$-semigroup. Our methods rely heavily on Lis theory of semigroup C*-algebras.
We consider generalized $Lambda$-structures on algebras and schemes over the ring of integers $mathit{O}_K$ of a number field $K$. When $K=mathbb{Q}$, these agree with the $lambda$-ring structures of algebraic K-theory. We then study reduced finite flat $Lambda$-rings over $mathit{O}_K$ and show that the maximal ones are classified in a Galois theoretic manner by the ray class monoid of Deligne and Ribet. Second, we show that the periodic loci on any $Lambda$-scheme of finite type over $mathit{O}_K$ generate a canonical family of abelian extensions of $K$. This raises the possibility that $Lambda$-schemes could provide a framework for explicit class field theory, and we show that the classical explicit class field theories for the rational numbers and imaginary quadratic fields can be set naturally in this framework. This approach has the further merit of allowing for some precise questions in the spirit of Hilberts 12th Problem. In an interlude which might be of independent interest, we define rings of periodic big Witt vectors and relate them to the global class field theoretical mathematics of the rest of the paper.
A set $mathcal{A}$ is said to be an additive $h$-basis if each element in ${0,1,ldots,hn}$ can be written as an $h$-sum of elements of $mathcal{A}$ in {it at least} one way. We seek multiple representations as $h$-sums, and, in this paper we make a start by restricting ourselves to $h=2$. We say that $mathcal{A}$ is said to be a truncated $(alpha,2,g)$ additive basis if each $jin[alpha n, (2-alpha)n]$ can be represented as a $2$-sum of elements of $mathcal{A}$ in at least $g$ ways. In this paper, we provide sharp asymptotics for the event that a randomly selected set is a truncated $(alpha,2,g)$ additive basis with high or low probability.
We determine higher topological Hochschild homology of rings of integers in number fields with coefficients in suitable residue fields. We use the iterative description of higher THH for this and Postnikov arguments that allow us to reduce the necessary computations to calculations in homological algebra, starting from the results of Bokstedt and Lindenstrauss-Madsen on (ordinary) topological Hochschild homology.
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