On the Eisenstein ideal over function fields

TL;DR

This paper investigates the Eisenstein ideal in Drinfeld modular curves over function fields, revealing its relation to cuspidal divisors and Jacobian component groups, and characterizing Eisenstein primes in certain levels.

Contribution

It provides new insights into the structure of the Eisenstein ideal for small levels and establishes the characteristic of the function field as an Eisenstein prime under specific conditions.

Findings

Eisenstein ideal relates to cuspidal divisor groups and Jacobian component groups.

Characteristic of the function field is an Eisenstein prime for non square-free levels.

Proves the Eisenstein ideal properties for levels not equal to a prime square.

Abstract

We study the Eisenstein ideal of Drinfeld modular curves of small levels, and the relation of the Eisenstein ideal to the cuspidal divisor group and the component groups of Jacobians of Drinfeld modular curves. We prove that the characteristic of the function field is an Eisenstein prime number when the level is an arbitrary non square-free ideal of $\mathbb{F}_q[T]$ not equal to a square of a prime.