We study towers of varieties over a finite field such as $y^2=f(x^{{\ell }^n})$ and prove that the characteristic polynomials of the Frobenius on the étale cohomology show a surprising $\ell$-adic convergence. We prove this by proving a more general statement about the convergence of certain invariants related to a skew-abelian cohomology group. The key ingredient is a generalization of Fermat’s little theorem to matrices. Along the way, we will prove that many natural sequences of polynomials ${(p_n(x))}_{n\ge 1}\in {\mathbb{Z}}_{\ell }{[x]}^{\mathbb{N}}$ converge $\ell$-adically and give explicit rates of convergence.

中文翻译：

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斜阿贝尔岩泽塔中 Frobenius 特征值的变化

我们研究有限域上的品种塔，例如$y^2=F（X^{{\ell }^n}）$并证明 étale 上同调上的 Frobenius 特征多项式表现出令人惊讶的结果$\ell$-adic收敛。我们通过证明关于与斜阿贝尔上同调群相关的某些不变量的收敛性的更一般的陈述来证明这一点。关键要素是将费马小定理推广到矩阵。一路上，我们将证明许多多项式的自然序列${（p_n（X））}_{n\ge 1}\epsilon {\mathbb{Z}}_{\ell }{[X]}^{\mathbb{N}}$收敛$\ell$-adically 并给出明确的收敛速度。