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.

Embeddings into the integral octonions

Abstract: Coxeter discovered a maximal order R inO, which is unique up to the action of Aut(O), with the property that R=pR is an octonion algebra over Z=pZ for all primes p. We review the construction of the order R, and some of its properties, inx1. In x2, we let K be an imaginary quadratic eld, with ring of integers A and discriminant D. We count the number of ring embeddings of A into R, using the L-function L(";s) of the quadratic Dirichlet character " :( Z=DZ)!f 1g associated to K. Theorem 1. The number of embeddings of A into R is 252L("; 2). We give two dierent proofs of this result. The rst uses theta series and Eisenstein series of half-integral weight. The second uses the theory of Tamagawa measures, as developed by Siegel and Weil. From the formula in Theorem 1, it follows that the number of embeddings of A into R lies between 3jDj 5= 2 and 5jDj 5= 2 . Inx3 we let K be a denite quaternion algebra over Q, and let A be a maximal order in K. Let S be the nite set of primes which ramify in K; thus p2 S if and only if KQp is a division algebra over Qp. Using the theory of Tamagawa measures, we will prove the following. Theorem 2. The number of embeddings of A into R is 504 Q p2S (p 2 1). Our interest in octonions dates from a lecture that Serre gave at Harvard on the subject, in the fall of 1990. The embedding problems which we study are generalizations of the results of Hasse and Eichler (cf. [14, p. 92.]) on the embeddings of rings of integers in imaginary quadratic elds into certain orders in rational quaternion algebras. Since Olga loved the arithmetic of quaternion algebras, we felt it was appropriate to dedicate this paper to her memory.

Statistics of $K$-groups modulo $p$ for the ring of integers of a varying quadratic number field

Abstract: For each odd prime $p$, we conjecture the distribution of the $p$-torsion subgroup of $K_{2n}(\mathcal{O}_F)$ as $F$ ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the $3$-torsion subgroup of $K_{2n}(\mathcal{O}_F)$ is as predicted by this conjecture.

Forms of differing degrees over number fields

Abstract: Consider a system of polynomials in many variables over the ring of integers of a number field $K$. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_K^m$ satisfies the Hasse principle, weak approximation and the Manin-Peyre conjecture, if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where $K=\mathbb{Q}$. Our main tool is Skinner's number field version of the Hardy-Littlewood circle method. As a by-product, we point out and correct an error in Skinner's treatment of the singular integral.

A family of monogenic $S_4$ quartic fields arising from elliptic curves

Abstract: We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial $3$-torsion fields for a certain one-parameter family of non-CM elliptic curves, we describe a power basis. As a result, we show that the one-parameter family of quartic $S_4$ fields given by $T^4 - 6T^2 - \alpha T - 3$ for $\alpha \in \mathbb{Z}$ such that $\alpha \pm 8$ are squarefree, are monogenic.