The $p$-rank of the reduction $\mathrm{mod}\mathrm{p}$ of jacobians and Jacobi sums

Abstract

Let $Y_K\to X_K$ be a ramified cyclic covering of curves, where $K$ is a cyclotomic field. In this work we study the $p$-rank of the reduction $\mathrm{mod}\mathrm{p}$ of a model of the jacobian of $Y_K$. In this way, we obtain counterparts of the Deuring polynomial, defined for elliptic curves, for genus greater than one. Moreover, we show that curves $Y_K$ give Hecke characters for cyclotomic fields. To carry out this study we use Jacobi sums and certain $L$-functions.

24 pages

Cited by in corpus (1)

• A calculation of $L$-series in terms of Jacobi sums