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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Arithmetic surface ： ウィキペディア英語版 Arithmetic surface In mathematics, an arithmetic surface over a Dedekind domain ''R'' with fraction field $K$ is a geometric object having one conventional dimension, and one other dimension provided by the infinitude of the primes. When ''R'' is the ring of integers ''Z'', this intuition depends on the prime ideal spectrum Spec(''Z'') being seen as analogous to a line. Arithmetic surfaces arise naturally in diophantine geometry, when an algebraic curve defined over ''K'' is thought of as having reductions over the fields ''R''/''P'', where ''P'' is a prime ideal of ''R'', for almost all ''P''; and are helpful in specifying what should happen about the process of reducing to ''R''/''P'' when the most naive way fails to make sense. Such an object can be less informally defined as an R-scheme with a non-singular, connected projective curve $C/K$ for a generic fiber and unions of curves (possibly reducible, singular, non-reduced ) over the appropriate residue field for special fibers. ==Formal definition== In more detail, an arithmetic surface $S$ (over the Dedekind domain $R$) is a scheme with a morphism $p:S\backslash rightarrow\; \backslash mathrm(R)$ with the following properties: $S$ is integral, normal, excellent, flat and of finite type over $R$ and the generic fiber is a non-singular, connected projective curve over $\backslash mathrm(R)$ and for other $t$ in $\backslash mathrm(R)$, :$S\backslash underset\backslash mathrm(k\_t)$ is a union of curves over $R/t$.〔Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, p. 311.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Arithmetic surface」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.