MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $\zeta_m\subset \mathbb{F}_q$.

share|cite|improve this question
1  
You might want to have a look at the work of Greg Anderson. – Felipe Voloch Jul 2 '12 at 0:59
    
Thanks, Felipe. It looks promising, specially his paper "A two-dimensional analogue of Stickelberger’s theorem" and Robert Coleman's paper ams.org/journals/proc/1988-102-03/S0002-9939-1988-0928961-7/… – RPC Jul 4 '12 at 16:05

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.