Let $[X,\lambda ]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor ${\nu }_v([X,\lambda ])$ for each place $v$ of $\mathbb{Q}$, and show that the product of these factors essentially computes the size of the isogeny class of $[X,\lambda ]$.

The derivation of this mass formula depends on a formula of Kottwitz and on analysis of measures on the group of symplectic similitudes and, in particular, does not rely on a calculation of class numbers.

中文翻译：

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通过 Frobenius 密度计算有限域上的阿贝尔变体

让$[X,\lambda ]$是具有交换自同态环的有限域上的主极化阿贝尔簇；进一步假设$X$是普通的或领域是质数。受等分布启发式的启发，我们引入了一个因素${\nu }_v([X,\lambda ])$每个地方$v$的$\mathbb{Q}$, 并表明这些因素的乘积本质上计算了等基因类的大小$[X,\lambda ]$.

这个质量公式的推导依赖于 Kottwitz 的公式和对辛相似群的测度分析，特别是不依赖于类数的计算。