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

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