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p-ranks of Prym Varieties Ozman, Ekin


We study the relationship between the p-rank of a curve and the $p$-ranks of the Prym varieties of its unramified cyclic covers in characteristic $p > 0$. For arbitrary primes $p$ and $\ell$ with $\ell \ne p$ and integers $g \ge 3$ and $0 \le f \le g$, we generalize a result of Nakajima by proving that the Prym varieties of all the unramified $\mathbb{Z}/\ell$-covers of a generic curve $X$ of genus $g$ and $p$-rank $f$ are ordinary. Furthermore, when $p \ge 5$ and $\ell \ne 2$, we prove that there exists a curve of genus $g$ and $p$-rank $f$ having an unramified double cover whose Prym has p-rank $f^\prime$ for each $g/2 − 1 \le f^\prime \le g − 2$; (these Pryms are not 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 unramified cyclic covers of curves. This is joint work with Rachel Pries.

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