# Ribet's Converse to Herbrand: Part I

Tomorrow I’m giving the STAGE talk on Ribet’s converse to Herbrand’s theorem, after I’ll try and post more notes, but for now here’s a little intro to get us thinking about the problem.

# Ribet's converse to Herbrand

We are interested in the class groups of cyclotomic fields \begin{equation*} h_p = h_{\mathbf{Q}(\mu_p)}\text{.} \end{equation*} Lets list the first few of these

##### DefinitionRegular primes

We'll call primes for which $p\nmid h_p$ regular primes. Otherwise irregular primes.

Why is this important from a number theory perspective?

It's hard to tell when a prime is a regular prime, you'd have to compute the class group.

##### DefinitionBernoulli numbers

The Bernoulli numbers are the sequence of integers given by the exponential generating function \begin{equation*} \frac{x}{e^x - 1} + \frac x2 - 1 = \sum_{n\ge 2}^\infty B_k\frac{x^k}{k!}\text{.} \end{equation*}

These have a number of cool properties, such as:

But most important for us is the relation to class numbers:

This is a great theorem relating class numbers to the Bernoulli numbers, but can we do better? What if I know a specific $k$ so that $p|B_k\text{,}$ can I say anything more specific about the class group? Yes; there is a strengthening of this theorem due in this form to Herbrand (in one direction) and Ribet (later, in the other direction).

First we need to recall the mod $p$ cyclotomic character $\chi\colon \operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \mathbf{F}_p^*$ defined by \begin{equation*} \zeta_p^{\chi(\sigma)} = \sigma (\zeta_p)\text{.} \end{equation*}

The $\Leftarrow$ direction was proved by Herbrand in 1932. And the $\Rightarrow$ direction by Ribet in 1974.

Now for completeness here is a table of factorisations of Bernoulli number numerators.