论文标题
在有限场上作用在矢量空间上的群体的共同体学
Cohomology of groups acting on vector spaces over finite fields
论文作者
论文摘要
令$ {\ mathbf {f}} _ q $为有限字段，$ q = p^m $元素，$ g $为$ {\ rm {gl}} _ n（{\ mathbf {f}} _ q）的子组。 1987年出版的Nori著名定理指出，存在一个（无效的）常数$ c（n）$，仅取决于$ n $，因此，如果$ p> c> c（n）$和$ g $在$ {\ mathbf {f}}} _ p^n $，然后$ h^1（$ h^1（p^n $ p^n $ n $）上的$ {\ mathbf {f}} _ p^n $ {我们解决了一个长期存在的问题，也考虑了有效证明Nori定理的问题。我们的方法产生了最佳常数$ c（n）= n+2 $。我们还证明了Nori定理的更一般版本，即，对于所有功率$ q $ $ p $的$ q $，如果$ g $在$ {\ mathbf {f}} _ q^n $和$ p> n $ and $ p> n+2 $上的semisimply，则$ h^1（g，mathbf {\ mathbf {f} iS_ q^n）我们将这些结果应用于完善标准，该标准由Çiperiani和Stix证明，这为凯瑟尔（Cassels）在阿贝尔（Abelian）品种在数字字段上提出的经典问题提供了足够的条件。
Let ${\mathbf{F}}_q$ be the finite field with $q=p^m$ elements and $G$ be a subgroup of ${\rm{GL}}_n({\mathbf{F}}_q)$. A famous theorem of Nori published in 1987 states that there exists a (non-effective) constant $c(n)$, depending only on $n$, such that if $p>c(n)$ and $G$ acts semisimply on ${\mathbf{F}}_p^n$, then $H^1(G,{\mathbf{F}}_p^n)=0$. We solve the long-standing problem, also considered by Serre of giving an effective proof of Nori's Theorem. Our approach yields the optimal constant $c(n)=n+2$. We also prove a more general version of Nori's theorem, namely, that for all powers $q$ of $p$, if $G$ acts semisimply on ${\mathbf{F}}_q^n$ and $p>n+2$, then $H^1(G,{\mathbf{F}}_q^n)$ is trivial. We apply these results to refine a criterion, proved by Çiperiani and Stix, which gives sufficient conditions for an affirmative answer to a classical question posed by Cassels in the case of abelian varieties over number fields.
