Néron models, Lie algebras, and reduction of curves of genus one
Néron models, Lie algebras, and reduction of curves of genus one
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Néron 模型、李代数和亏格一曲线的简化
DOI：
10.1007/s00222-004-0342-y
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发表时间：
2004
影响因子：
3.1
通讯作者：
M. Raynaud
中科院分区：
文献类型：
--
作者：
Qing Liu;Dino J. Lorenzini;M. Raynaud
设K是一个离散赋值域。设OK表示K的整数环，K是OK的特征p ≥ 0的剩余域.令S：= SpecOK。设X K是K上亏格为1的光滑几何连通射影曲线。用EK表示X K的雅可比矩阵。设X/S和E/S分别是X K和EK的极小正则模型。在这篇文章中，我们研究了特殊纤维Xk和Ek之间的可能关系。在这样做的时候，我们被引导去研究Picard函子Pic X/S的几何，当X/S不一定是上同调平坦的。作为本研究的一个应用，我们能够在充分的一般性证明戈登的定理之间的等价性的Artin-Tate和Birch-SwinnertonDyer图。回想一下，当k是代数闭的时，椭圆曲线的特殊纤维根据它们的科代拉型来分类，用符号T ∈ {In，I <$n，n ∈ Z≥0，II，II <$，III，III <$，IV，IV <$}表示。给定一个类型T和一个正整数m，我们用mT表示通过将T的所有重数乘以m而从T获得的新类型。当k是代数闭的时，亏格为1的曲线的类型与其雅可比矩阵的类型之间的关系可以总结如下。
Let K be a discrete valuation field. Let OK denote the ring of integers of K , and let k be the residue field of OK , of characteristic p ≥ 0. Let S := SpecOK . Let X K be a smooth geometrically connected projective curve of genus 1 over K . Denote by EK the Jacobian of X K . Let X/S and E/S be the minimal regular models of X K and EK , respectively. In this article, we investigate the possible relationships between the special fibers Xk and Ek . In doing so, we are led to study the geometry of the Picard functor Pic X/S when X/S is not necessarily cohomologically flat. As an application of this study, we are able to prove in full generality a theorem of Gordon on the equivalence between the Artin-Tate and Birch-SwinnertonDyer conjectures. Recall that when k is algebraically closed, the special fibers of elliptic curves are classified according to their Kodaira type, which is denoted by a symbol T ∈ {In, I∗n, n ∈ Z≥0, II, II∗, III, III∗, IV, IV∗}. Given a type T and a positive integer m, we denote by mT the new type obtained from T by multiplying all the multiplicities of T by m. When k is algebraically closed, the relationships between the type of a curve of genus 1 and the type of its Jacobian can be summarized as follows.