AI 中文总结
该研究针对有限域上$\boldsymbol{\text{G}}_m^d$的反常层，证明一致分层一般消失定理并推广Katz的等分布定理，得到Frobenius共轭类在极大紧子群共轭类空间中等分布的结论。
AI 中文摘要
我们针对有限域上$\boldsymbol{\text{G}}_m^d$上的反常层（perverse sheaves）证明了一个一致分层的一般消失定理，其常数仅依赖于维度和Sawin意义下的复杂度。作为推论，我们证明了$\boldsymbol{\text{G}}_m^d$上的一个等分布定理，推广了Katz针对$\boldsymbol{\text{G}}_m$的定理。该推论适用于具有有界复杂度、共同Tannakian单值群的反常层序列，且在特征各异的有限域上成立。当域的基数趋于无穷时，与乘法特征相关的Frobenius共轭类在极大紧子群的共轭类空间中等分布。
英文摘要
We prove a uniform stratified generic vanishing theorem for perverse sheaves on $\mathbb{G}_m^d$ over finite fields, with constants depending only on the dimension and on the complexity à la Sawin. As a corollary, we prove an equidistribution theorem for $\mathbb{G}_m^d$ extending Katz's theorem for $\mathbb{G}_m$. The corollary applies to sequences of perverse sheaves of bounded complexity with common tannakian monodromy group, over finite fields of varying characteristic. As the cardinalities of the fields tend to infinity, the Frobenius conjugacy classes attached to the multiplicative characters equidistribute in the space of conjugacy classes of a maximal compact subgroup.
Commentsv2: 39 pp, minor corrections
