General Uniformity of Zeta Functions
General Uniformity of Zeta Functions
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发表时间：
2012-09
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通讯作者：
L. Weng
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作者：
L. Weng
利用与稳定丛相关的解析挠率，我们引入了紧Riemann曲面的zeta函数。为了证明良好的定义，我们分析了退化的分析挠的边界上的模空间，奇异的分析挠Brill-Noether轨迹，和渐近行为的分析挠的程度。这些新的内在的zeta，无论是阿贝尔的还是非阿贝尔的，都有望在理解黎曼曲面的整体分析和几何学中发挥关键作用，例如由Atiyah-Bott搜索的黎曼曲面的Tamagawa数猜想，以及维滕的模空间的体积公式。与此相关的是，在我们关于齐塔人特殊一致性的理论中，我们将首先基于阿贝尔齐塔人和群对称性构建一个对称齐塔人，然后推测我们的非阿贝尔齐塔人与这些后来的具有对称性的齐塔人重合。所有这些，再加上数域和函数域的齐塔，就构成了我们的齐塔一般一致性理论。
Using analytic torsion associated to stable bundles, we introduce zeta functions for compact Riemann surfaces. To justify the well-definedness, we analyze the degenerations of analytic torsions at the boundaries of the moduli spaces, the singularities of analytic torsions at Brill-Noether loci, and the asymptotic behaviors of analytic torsions with respect to the degree. These new yet intrinsic zetas, both abelian and non-abelian, are expected to play key roles to understand global analysis and geometry of Riemann surfaces, such as the Tamagawa number conjecture for Riemann surfaces, searched by Atiyah-Bott, and the volumes formula of moduli spaces of Witten. Relating to this, in our theory on special uniformity of zetas, we will first construct a symmetric zetas based on abelian zetas and group symmetries, then conjecture that our non-abelian zetas coincide with these later zetas with symmetries. All this, together with that for zetas of number fields and function fields, then consists of our theory of general uniformity of zetas.