The p-rank of the reduction mod p of Jacobians and Jacobi sums

摘要

Let Y_K → 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 mod 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. We provide a new point of view of this subject in terms of L-functions. To carry out this study we use the relationship between Jacobi sums and L-functions. This is established in [A. Weil, Jacobi sums as "Groessencharaktere", Trans. Amer. Math. Soc. 73 (1952) 487-495] for the case of Fermat curves. We also give a new proof of a result of Deligne concerning the constant terms of these L-functions and Jacobi sums.