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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, the twisted Kloosterman sum ∑ α∈(O/qO)∗ χ(N(α))ζ T (α)+z/N(β) q, where T and N denote trace and norm, and where χ is a Dirichlet character (mod q).
Abstract: Let O denote the ring of integers of an algebraic number field of degree m which is totally and tamely ramified at the prime p. Write ζq = exp(2πi/q), where q = pr. We evaluate the twisted Kloosterman sum ∑ α∈(O/qO)∗ χ(N(α))ζ T (α)+z/N(α) q , where T and N denote trace and norm, and where χ is a Dirichlet character (mod q). This extends results of Salie for m = 1 and of Yangbo Ye for prime m dividing p− 1. Our method is based upon our evaluation of the Gauss sum ∑ α∈(O/qO)∗ χ(N(α))ζ T (α) q , which extends results of Mauclaire for m = 1.
4 citations
TL;DR: In this article, the authors study K3 surfaces with complex multiplication and provide an algorithm that given a number field K and a CM number field E, returns a finite list of groups which contain Br ( X ‾ ) G K for any K3 surface X / K that has CM by the ring of integers of E. They run their algorithm when E is a quadratic imaginary field (a condition that translates into X having maximal Picard rank).
4 citations
TL;DR: In this paper, it was shown that there is a real quadratic field (with the assumption of the generalized Riemann hypothesis) such that the fundamental group of the ring of integers is isomorphic to the A_5 group.
Abstract: Under the assumption of the generalized Riemann hypothesis (GRH), we show that there is a real quadratic field $$K$$
such that the $${\acute{\mathrm{e}}}$$
tale fundamental group $$\pi ^{\acute{\mathrm{et}}}_1(\mathrm {Spec}\;\mathcal {O}_K)$$
of the spectrum of the ring of integers $$\mathcal {O}_K$$
of $$K$$
is isomorphic to $$A_5$$
. The proof uses standard methods involving Odlyzko bounds, as well as the proof of Serre’s modularity conjecture. To the best of the author’s knowledge, this is the first example of a number field $$K$$
for which $$\pi ^{\acute{\mathrm{et}}}_1(\mathrm {Spec}\;\mathcal {O}_K)$$
is finite, nonabelian and simple under the assumption of the GRH.
4 citations
TL;DR: In this paper, the authors dealt with group actions of one-dimensional formal groups defined over the ring of integers in a finite extension of the p-adic field, where the space acted upon is the maximal ideal in the ring.
Abstract: This paper deals with group actions of one-dimensional formal groups defined over the ring of integers in a finite extension of the p-adic field, where the space acted upon is the maximal ideal in the ring of integers of an algebraic closure of the p-adic field Given a formal group F as above, a formal flow is a series Φ(t,x) satisfying the conditions Φ(0,x)=x and Φ(F(s,t),x)=Φ(s,Φ(t,x)) With this definition, any formal group will act on the disk by left translation, but this paper constructs flows Φ with any specified divisor of fixed points, where a point ξ of the open unit disk is a fixed point of order ≤n if (x−ξ)
n
|(Φ(t,x)−x) Furthermore, if γ is an analytic automorphism of the open unit disk with only finitely many periodic points, then there is a flow Φ, an element α of the maximal ideal of the ring of constants, and an integer m such that the m-fold iteration of γ(x) is equal to Φ(α,x) All the formal flows constructed here are actions of the additive formal group on the unit disk Indeed, if the divisor of fixed points of a formal flow is of degree at least two, then the formal group involved must become isomorphic to the additive group when the base is extended to the residue field of the constant ring
4 citations
TL;DR: In this paper, it was shown that the invariants of an algebraic number field can be computed in terms of an effective prime ideal theorem due to Lagarias and Odlyzko.
Abstract: Consider an algebraic number field, $K$, and its ring of integers, $\mathcal{O}_K$. There exists a smallest $B_K>1$ such that for any $x>1$ we can find a prime ideal, $\mathfrak{p}$, in $\mathcal{O}_K$ with norm $N(\mathfrak{p})$ in the interval $[x,B_Kx]$. This is a generalization of Bertrand's postulate to number fields, and in this paper we produce bounds on $B_K$ in terms of the invariants of $K$ from an effective prime ideal theorem due to Lagarias and Odlyzko. We also show that a bound on $B_K$ can be obtained from an asymptotic estimate for the number of ideals in $\mathcal{O}_K$ less than $x$.
4 citations