Arithmetic invariant theory
Arithmetic invariant theory
复制标题
算术不变性理论
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
B. Gross
中科院分区：
文献类型：
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作者：
M. Bhargava;B. Gross
设k是域，G是k上的约化代数群，V是G的线性表示。几何不变理论涉及研究V上G-不变多项式的k-代数，以及这些不变量与V上G-轨道之间的关系，通常假设基域k是代数闭的。在有利的情况下，可以确定几何商（V /！/ G = mathrm{Spec}（mathrm{Sym}^{{ast}}（V ^{vee }）^{G}）），并且可以识别态射（V
ightarrow V/！/本文研究了当k不是代数闭时的类似问题。在这种情况下，轨道图中出现的额外复杂性就是我们所说的算术不变理论。我们举例说明了一些问题，通过考虑正规半单轨道，即，分裂奇特殊正交群（G = mathrm{SO}_{2n+1}）的三种富算术表示中稳定子维数最小的闭轨道.
Let k be a field, let G be a reductive algebraic group over k, and let V be a linear representation of G. Geometric invariant theory involves the study of the k-algebra of G-invariant polynomials on V, and the relation between these invariants and the G-orbits on V, usually under the hypothesis that the base field k is algebraically closed. In favorable cases, one can determine the geometric quotient (V /!/G = mathrm{Spec}(mathrm{Sym}^{{ast}}(V ^{vee })^{G})) and can identify certain fibers of the morphism (V
ightarrow V/!/G) with certain G-orbits on V. In this paper we study the analogous problem when k is not algebraically closed. The additional complexity that arises in the orbit picture in this scenario is what we refer to as arithmetic invariant theory. We illustrate some of the issues that arise by considering the regular semisimple orbits—i.e., the closed orbits whose stabilizers have minimal dimension—in three arithmetically rich representations of the split odd special orthogonal group (G = mathrm{SO}_{2n+1}).