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

TL;DR

This paper investigates the p-rank of the reduction mod p of Jacobians of ramified cyclic covers of curves over cyclotomic fields, extending Deuring polynomial concepts to higher genus and linking to Hecke characters.

Contribution

It introduces a method to analyze the p-rank of Jacobians for higher genus curves using Jacobi sums and L-functions, generalizing Deuring polynomial concepts beyond elliptic curves.

Findings

Derived analogues of Deuring polynomial for higher genus curves.

Connected the p-rank of Jacobians to Hecke characters in cyclotomic fields.

Utilized Jacobi sums and L-functions for the analysis.

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 $\rm{mod}\, p$ 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. 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.