Abstract Lenny Taelman

The Herbrand-Ribet theorem relates the p-part of the class group of $\mathbb{Q} (\zeta_p)$ to the Bernoulli numbers $B_n$ It is a refinement of the theorem of Kummer which states that p divides the class number of $\mathbb{Q}(\zeta_p)$ if and only if p divides one of the Bernoulli numbers $B_n$ with $n \lt p-1$. In this talk I will discuss an analogue of this theorem for the ring $\mathbb{F}_q[t] $ . We will see how the Herbrand-Ribet theorem can be interpreted as a statement about the group scheme $\mu_p$, and what to replace $\mu_p$ by to obtain a meaningful analogue for $\mathbb{F}_q[t] $. Reference: http://arxiv.org/abs/1104.5363