论文标题
Riemann的假设，用于有限场上曲线的非亚洲Zeta函数
Riemann Hypothesis for Non-Abelian Zeta Functions of Curves over Finite Fields
论文作者
论文摘要
在本文中，我们针对Riemann假设开发了一些基本技术，用于在有限字段$ \ mathbb f_q $上$ g $的积分常规投影曲线的高级非亚伯Zeta函数。作为这些真正的Zeta功能的Riemann假设的应用，我们在$ x/\ Mathbb f_q $ $ x $ $ x $ and $ x $ and $ n，\，Q $ g $ g $ g $ g $ n，\ n，Q $ n，Q $ n，Q $ n，Q $ n，b的基本非亚伯利亚$α$ - 和$β$ -Invariants上获得了一些明确的界限。 \ sum_ {v} \ frac {q^{h^0（x，x，v）} -1} {\＃\ m马理\ sum_ {v} \ frac {1} {\＃{\ mathrm aut}（v）} \ qquad（m \ in \ mathbb z）$$，其中$ v $通过所有排名$ n $ n $ n $ n $ semi-stable $ n $ n $ n $ semi-stable $ \ mathbb f_q $ f_q $ - f_q $ - Quq $ - Quq $ - 政治矢量bundles $ x $ x $ x $ $ x $ a $ mn。特别是，$$ \ prod_ {k = 1}^{n} \ frac {\ \ big（\ sqrt q^k-1 \ big）^{2G-1} \} {（\ sqrt q^k+1）} β_{x，\ mathbb f_q; n}（0）\ leq \ prod_ {k = 1}^{n}^{n} \ frac {\ big（1+ \ sqrt q^k \ big）^{2G-1} {2G-1} {2G-1} \} {（在建立较高等级Zetas的Riemann假设中的核心作用。
In this paper, we develop some basic techniques towards the Riemann hypothesis for higher rank non-abelian zeta functions of an integral regular projective curve of genus $g$ over a finite field $\mathbb F_q$. As an application of the Riemann hypothesis for these genuine zeta functions, we obtain some explicit bounds on the fundamental non-abelian $α$- and $β$-invariants of $X/\mathbb F_q$ in terms of $X$ and $n,\, q$ and $g$: $$α_{X,\mathbb F_q;n}(mn) = \sum_{V}\frac{q^{h^0(X,V)}-1}{\#\mathrm{Aut}(V)} \qquad{\rm and}\qquad β_{X,\mathbb F_q;n}(mn ):= \sum_{V}\frac{1}{\#{\mathrm Aut}(V)}\qquad(m\in \mathbb Z)$$ where $V$ runs through all rank $n$ semi-stable $\mathbb F_q$-rational vector bundles on $X$ of degree $mn$. In particular, $$ \prod_{k=1}^{n}\frac{\ \big( \sqrt q^k-1\big)^{2g-1}\ }{(\sqrt q^k+1)}\leq q^{-\binom{n}{2}(g-1)} β_{X,\mathbb F_q;n}(0) \leq \prod_{k=1}^{n}\frac{\ \big( 1+\sqrt q^k\big)^{2g-1}\ }{(\sqrt q^k-1)}, $$ Finally, we demonstrate that the related bounds in lower ranks in turn play a central role in establishing the Riemann hypothesis for higher rank zetas.
