On the Eisenstein ideal of Drinfeld modular curves

TL;DR

This paper investigates the structure of the Eisenstein ideal in the Hecke algebra associated with Drinfeld modular curves, establishing its local principality and properties related to the characteristic of the base field.

Contribution

It proves that the Eisenstein ideal is locally principal and its quotient's order is not divisible by the characteristic, advancing understanding of the algebraic structure of Drinfeld modular curves.

Findings

The Eisenstein ideal is locally principal.

The characteristic p does not divide the order of the quotient.

Structural properties of the Hecke algebra are clarified.

Abstract

Let $\goth E(\goth p)$ denote the Eisenstein ideal in the Hecke algebra $\Bbb T(\goth p)$ of the Drinfeld modular curve $X_0(\goth p)$ parameterizing Drinfeld modules of rank two over $\Bbb F_q[T]$ of general characteristic with Hecke level $\goth p$-structure, where $\goth p\triangleleft\Bbb F_q[T]$ is a non-zero prime ideal. We prove that the characteristic $p$ of the field $\Bbb F_q$ does not divide the order of the quotient $\Bbb T(\goth p)/\goth E(\goth p)$ and the Eisenstein ideal $\goth E(\goth p)$ is locally principal.