由有理阿贝尔簇定义的Frobenius迹的算术性质（J-P的两个附录。塞尔）
Arithmetic properties of the Frobenius traces defined by a rational abelian variety (with two appendices by J-P. Serre)
摘要：
令$A$是$\mathbb{Q}$上维数为$g$的阿贝尔簇，使得其相关联的绝对伽罗瓦表示$ρ_A$的像在$\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$中是开的。本文研究了在$ρ_A$下，在$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$中的Frobenius在$p$上的迹$a_{1, p}$的算法，模变素数$p$ .特别是，我们得到的计数函数$\#\{p \leq x: a_{1, p} = t\}$的上界，我们证明了一个Erdös-Kac型定理的数量的素因子的$a_{1, p}$。我们还提出了一个关于$\#\{p \leq x: a_{1, p} = t\}$的渐近行为的猜想，它推广了S. Lang和H. Trotter从1976年开始研究椭圆曲线。
英文摘要：
Let $A$ be an abelian variety over $\mathbb{Q}$ of dimension $g$ such that the image of its associated absolute Galois representation $ρ_A$ is open in $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. We investigate the arithmetic of the traces $a_{1, p}$ of the Frobenius at $p$ in $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ under $ρ_A$, modulo varying primes $p$. In particular, we obtain upper bounds for the counting function $\#\{p \leq x: a_{1, p} = t\}$ and we prove an Erdös-Kac type theorem for the number of prime factors of $a_{1, p}$. We also formulate a conjecture about the asymptotic behaviour of $\#\{p \leq x: a_{1, p} = t\}$, which generalizes a well-known conjecture of S. Lang and H. Trotter from 1976 about elliptic curves.
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