摘要翻译:
设$k$是一个完全实数域。对于每一个奇数$n\geq3$,我们构造了一个Dedekind zeta动机在$k$之上的混合Tate动机的类别$\mt(k)$中。通过直接计算它的Hodge实现,我们证明了它的周期是$\pi^{n[k:\q]}\zeta^*_k(1-n)$的有理倍数,其中$\zeta^*_k(1-n)$表示Dedekind zeta函数的特殊值$k$。我们推导出群$\ext^1_{\mt(k)}(\q(0),\q(n))$是由二次曲面相对于超平面的上同调生成的。这证明了某些基动络合物对于$k$的一个全面性结果,这些基动络合物可以计算群$\ext^1_{\mt(k)}(\q(0),\q(n))$。特别地,Dedekind zeta函数的特殊值是定义在$k$上的测地双曲单形体积的行列式。
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英文标题:
《Dedekind Zeta motives for totally real fields》
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作者:
Francis Brown
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最新提交年份:
2013
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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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一级分类:Mathematics 数学
二级分类:Number Theory 数论
分类描述:Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数,丢番图方程,解析数论,代数数论,算术几何,伽罗瓦理论
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英文摘要:
Let $k$ be a totally real number field. For every odd $n\geq 3$, we construct a Dedekind zeta motive in the category $\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $\pi^{n[k:\Q]}\zeta^*_k(1-n)$, where $\zeta^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculate the groups $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$. In particular, the special value of the Dedekind zeta function is a determinant of volumes of geodesic hyperbolic simplices defined over $k$.
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PDF链接:
https://arxiv.org/pdf/0804.1654