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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: It is proved that for a finite abelian p-extension L/K, it has a relative normal integral basis (NIB) if and only if it is unramified outside S.
3 citations
TL;DR: In this paper, the authors give an explicit formula for the size of the isogeny class of a Hilbert-Blumenthal abelian variety over a finite field, where OL is the ring of integers in a totally real field dimension g over Q, N 0 and N N 0 are relatively prime square-free integers, and k is a relatively prime field of characteristic relatively prime to both N 0N and disc(L, Q).
3 citations
03 Jan 1990
TL;DR: It is shown that Newton's method provides polylog-time solutions for a broader class of linear systems than was previously supposed and can be implemented on a special-purpose architecture based on the three-dimensional mesh of trees.
Abstract: This thesis considers the mesh of trees architecture as both a special-purpose and a general-purpose parallel computer. A family of special-purpose VLSI architectures for computing an ($n\sb1 \times n\sb2 \times \cdots \times n\sb{d}$)-point multidimensional DFT over $\doubz\sb{M}$, the ring of integers modulo $M$, is proposed. Using the two-dimensional mesh of trees as a component, these architectures achieve optimal VLSI area $A$ = $\Theta((N\sp2\log\sp{2}M)/T\sp2)$ for any given computation time $T\ \epsilon$ ($\Omega(\log N),O(\sqrt{N\log M})\rbrack.$
The convergence properties of Newton's method are studied. By introducing and formalizing the notion of attunement of a linear system of equations, it is shown that Newton's method provides polylog-time solutions for a broader class of linear systems than was previously supposed. In particular, the system matrix need not be well-conditioned; all that is required is that the known vector be well-attuned to the system matrix. It is then shown that Newton's method can be implemented on a special-purpose architecture based on the three-dimensional mesh of trees. This same architecture can be used to construct the stiffness equations arising from a finite element approximation. Furthermore, it can be hybridized with a systolic array to achieve a processor-time or area-time tradeoff.
Then, in a different vein, the two-dimensional mesh of trees is studied as a general-purpose parallel computer. It is shown that this architecture can afford finer memory granularity and, thereby, reduce the memory redundancy required for deterministic P-RAM simulation. A distributed-memory, bounded-degree network model of parallel computation is proposed that allows one to take greater advantage of the potential for fine-grain memories without sacrificing other aspects of realism. The simulation scheme presented is admitted by this new model and achieves constant memory redundancy.
3 citations
TL;DR: In this paper, the authors showed that the set of all stable curves X over O_K with (K_{X/S})^2 / [K : Q] > heightFal(J(X_K)), where X is the canonically metrized dualizing sheaf of X over S = Spec(O_K) and heightFal is the Faltings modular height of the Jacobian of X_K, is finite under the following equivalence.
Abstract: Let K be a number field, O_K the ring of integers of K and X a stable curve over O_K of genus g >= 2. In this note, we will prove a strict inequality
( (K_{X/S})^2 / [K : Q] ) > Height_{Fal}(J(X_K)),
where $K_{X/S}$ is the canonically metrized dualizing sheaf of X over S = Spec(O_K) and Height_{Fal}(J(X_K)) is the Faltings modular height of the Jacobian of X_K. As corollary, for any constant A, the set of all stable curves X over O_K with
( (K_{X/S})^2 / [K : Q] ) <= A
is finite under the following equivalence. For stable curves X and Y, X is equivalent to Y if X is isomorphic to Y over O_{K'} for some finite extension field K' of K.
3 citations
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