We study the relationship between the $p$-rank of a curve and the $p$-ranks of the Prym varieties of its cyclic covers in characteristic $p >0$. For arbitrary $p$, $g \geq 3$ and $0 \leq f \leq g$, we generalize a result of Nakajima by proving that the Prym varieties of all unramified cyclic degree $\ell \not = p$ covers of a generic curve $X$ of genus $g$ and $p$-rank $f$ are ordinary. Furthermore, when $p \geq 5$, we prove that there exists a curve of genus $g$ and $p$-rank $f$ having an unramified degree $\ell=2$ cover whose Prym is almost ordinary. Using work of Raynaud, we use these two theorems to prove results about the (non)-intersection of the $\ell$-torsion group scheme with the theta divisor of the Jacobian of a generic curve $X$ of genus $g$ and $p$-rank $f$. The proofs involve geometric results about the $p$-rank stratification of the moduli space of Prym varieties.

中文翻译：

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普通和几乎普通的 Prym 品种

我们研究了曲线的 $p$-rank 与其循环覆盖的 Prym 变体的 $p$-ranks 之间的关系，其中特征 $p>0$。对于任意的 $p$、$g \geq 3$ 和 $0 \leq f \leq g$，我们通过证明所有未分枝循环度 $\ell \not = p$ 的 Prym 变体来概括 Nakajima 的结果属 $g$ 和 $p$-rank $f$ 的一般曲线 $X$ 是普通曲线。进一步，当$p\geq 5$时，我们证明存在一个属$g$和$p$-rank$f$的曲线具有无分支度$\ell=2$覆盖，其Prym几乎是普通的。使用雷诺的工作，我们使用这两个定理来证明 $\ell$-扭转群方案与属 $g$ 的泛型曲线 $X$ 的雅可比的 theta 除数的（非）交集的结果和$p$-等级 $f$。