Here is the setup to Stickelberger's theorem over number fields (following Washington's book *Intro. to cyclotomic fields*).

Let $M/\mathbb{Q}$ be a finite abelian extension with galois group $G$. Suppose $M\subset\mathbb{Q}(\zeta_m)$, where $\zeta_m$ is a primitive $m$-th root of unity and $m$ is minimal with this property. For an integer $a\mod m$ with $(a,m)=1$, the map $\sigma_a(\zeta_m)=\zeta_m^a$ defines an element of $G$.

Define the *Stickelberger element* of $M$ to be
$$
\theta=\theta(M):=\sum_{a\mod m\atop (a,m)=1}\left<\dfrac{a}{m}\right>\sigma_a^{-1}\in \mathbb{Q}[G],
$$
where $<>$ is the fractional part function, $<x>=x-[x]$.
Define the *Stickelberger ideal* to be $I(M)=\mathbb{Z}[G]\cap\theta\mathbb{Z}[G]$. Then Stickelberger proved

(Stickelberger's Theorem) The Stickelberger ideal $I$ annihilates the ideal class group of $M$.

I know that if we replace $M$ by an abelian extension of $\mathbb{F}_q(t)$ then David Hayes in *Stickelberger elements in function fields* proved an analogue (which I do not understand completely) of Stickelberger's theorem. I am wondering how tight this analogy is.

Here is a concrete example that I have in mind:

Let $K=\mathbb{F}_q(t)$ and consider $M=K(u)$ with $u^m=f(t)$, $(m,q)=1$ and $f(t)$ a square-free polynomial over $\mathbb{F}_q$. Let $\mathcal{O}_M$ be the integral closure of $\mathbb{F}_q[t]$ in $M$ and let ${Cl}_M$ be the ideal class group of $\mathcal{O}_M$. It is well-known that $Cl_M=J(\mathbb{F}_q)$, where $J=\operatorname{Jac}(\mathcal{C})$ is the jacobian of the smooth projective curve defined by $\mathcal{C}:u^m=f(t)$. Now if $Fr$ is the $q$-frobenius endomorphism of $J$ then we have that $1-Fr$ annihilates $J(\mathbb{F}_q)$.

My question is: can we define an explicit ideal $I\subset \mathbb{Z}[Gal(M/K)]$ such that $1-Fr\in I$?

We may want to assume that $m|(q-1)$ so that $\zeta_m\subset \mathbb{F}_q$.