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摘要翻译：
设$x$是定义在完全D.V.R的分数域$k$上的光滑曲线。$R$，让$k'/k$是一个驯服的扩展名。我们研究了$X_{k'}$上的$g=\gal(k'/k)$-action扩展到$X_{k'}$上的某些规则模型，即$r$在$k'$中的积分闭包。特别地，我们考虑了这种正则模型的特殊纤维结构束上同调群的诱导作用，得到了群元在上同调群的交替和上诱导的自同态的Brauer迹的公式。利用这些结果，我们用封闭的、幂次的子群格式，研究了特殊纤维的N\'eron Jacobian$x$N\'eron模型的自然过滤。我们证明了这种过滤中的跳跃只依赖于严格法向交叉的最小正则模型的特殊纤维的纤维类型，特别是与剩余特性无关。此外，我们获得关于这些跳跃发生的位置的信息。对于属1和属2的曲线，我们还计算了有限多个可能的纤维类型中的每一种的跳跃。
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英文标题：
《Galois actions on Neron models of Jacobians》
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作者：
Lars Halvard Halle
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Let $X$ be a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, and let $K'/K$ be a tame extension. We study extensions of the $G = \Gal(K'/K)$-action on $ X_{K'} $ to certain regular models of $X_{K'}$ over $R'$, the integral closure of $R$ in $K'$. In particular, we consider the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model, and obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. We apply these results to study a natural filtration of the special fiber of the N\'eron model of the Jacobian of $X$ by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for $X$ over $R$, and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber types for curves of genus 1 and 2.
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PDF链接：
https://arxiv.org/pdf/0805.3080
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