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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, a genus 2 curve is defined over a finite field and a quartic CM field and an algorithm for computing the field of definition of, and the action of Frobenius on, the subgroups for prime powers is presented.
Abstract: We present algorithms which, given a genus 2 curve $C$ defined over a finite field and a quartic CM field $K$, determine whether the endomorphism ring of the Jacobian $J$ of $C$ is the full ring of integers in $K$. In particular, we present probabilistic algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups $J[\ell^d]$ for prime powers $\ell^d$. We use these algorithms to create the first implementation of Eisentrager and Lauter's algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem \cite{el}, and we demonstrate the algorithm for a few small examples. We observe that in practice the running time of the CRT algorithm is dominated not by the endomorphism ring computation but rather by the need to compute $p^3$ curves for many small primes $p$.

27 citations

Journal ArticleDOI
TL;DR: In this paper, the authors define for each positive n and each prime ideal p of an algebraic number field a nonnegative integer rn( p ) as follows: if n = Σi = 0tkiqi is the q-adic expansion of n where q = N p, set rn (p) = (n − σ σ)/(q − 1) where the product is taken over all prime ideals of θ.

27 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the curvature of the Schmidt arrangement in the Bianchi group and showed that it is a disjoint union of all primitive integral groups of Apollonian packings.
Abstract: We study the orbit of $\widehat{\mathbb{R}}$ under the Mobius action of the Bianchi group $\operatorname{PSL}_2(\mathcal{O}_K)$ on $\widehat{\mathbb{C}}$, where $\mathcal{O}_K$ is the ring of integers of an imaginary quadratic field $K$. The orbit $\mathcal{S}_K$, called a Schmidt arrangement, is a geometric realisation, as an intricate circle packing, of the arithmetic of $K$. We give a simple geometric characterisation of certain subsets of $\mathcal{S}_K$ generalizing Apollonian circle packings, and show that $\mathcal{S}_K$, considered with orientations, is a disjoint union of all primitive integral such $K$-Apollonian packings. These packings are described by a new class of thin groups of arithmetic interest called $K$-Apollonian groups. We make a conjecture on the curvatures of these packings, generalizing the local-to-global conjecture for Apollonian circle packings.

27 citations

Journal ArticleDOI
01 Mar 2004
TL;DR: In this article, it was shown that the ring of integers in the cyclotomic field of the affine curve is canonically isomorphic to the ring in the full symmetric group.
Abstract: Suppose that $K$ is a field of characteristic zero, $K_a$ is its algebraic closure, and that $f(x) \in K[x]$ is an irreducible polynomial of degree $n \ge 5$ , whose Galois group coincides either with the full symmetric group $\Sn$ or with the alternating group $\An$ . Let $p$ be an odd prime, $\Z[\zeta_p]$ the ring of integers in the $p$ th cyclotomic field $\Q(\zeta_p)$ . Suppose that $C$ is the smooth projective model of the affine curve $y^p\,{=}\,f(x)$ and $J(C)$ is the jacobian of $C$ . We prove that the ring $\End(J(C))$ of $K_a$ -endomorphisms of $J(C)$ is canonically isomorphic to $\Z[\zeta_p]$

27 citations

Journal ArticleDOI
TL;DR: In this paper, the relative algebraic K-group K0(OK[G], K) is computed as an abstract abelian group and the discrete logarithm problem is solved in the case K = Q.
Abstract: Let G be finite group and K a number field or a p-adic field with ring of integers OK. In the first part of the manuscript we present an algorithm that computes the relative algebraic K-group K0(OK[G], K) as an abstract abelian group. We also give algorithms to solve the discrete logarithm problems in K0(OK[G], K) and in the locally free class group cl(OK[G]). All algorithms have been implemented in Magma for the case K = Q.In the second part of the manuscript we prove formulae for the torsion subgroup of K0(Z[G], Q) for large classes of dihedral and quaternion groups.

26 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202250
2021117
2020121
2019111
201896