## BCN Spring 2016 Workshop: Number Theory & K-theory

### Advanced course on: Anderson's cyclotomic units" and special $L$-values

Bruno Anglès

#### Descripció

Let $p$ be an odd prime number and let $X$ be the $p$-Sylow subgroup of the ideal class group of $\mathbb Q(e^{\frac{2i\pi}{p}}).$

Let $\Delta={\rm Gal}(\mathbb Q(e^{\frac{2i\pi}{p}})/\mathbb Q)\simeq (\frac{\mathbb Z}{p\mathbb Z})^\times$ and let $\widehat{\Delta} ={\rm Hom}(\Delta, \mathbb Z_p^\times).$

Observe that $X$ is a $\mathbb Z_p[\Delta]$-module.  For $\chi \in \widehat {\Delta},$ let:

$$X(\chi)=e_\chi X,$$

where $e_\chi =\frac{1}{\mid \Delta\mid} \sum_{\delta \in \Delta} \chi( \delta) \delta^{-1}\in \mathbb Z_p[\Delta].$ Let $\omega_p \in \widehat{\Delta}$ be the $p$-adic Teichm\" uller character, and let $n\equiv 1\pmod {2},$ $n\in \{ 3, \ldots, p-2\}.$

Then K. Ribet proved the following result:

$$X(\omega_p^n)\not =\{0\} \Leftrightarrow B_{p-n}\equiv 0\pmod{p},$$

where $B_n$ denotes the $n$-th Bernoulli number.

Our aim in these lectures is to present a proof as self-contained as possible of Taelman's analogue (for the Carlitz module, see Goss reference, chapter 3) of the above Theorem. Our exposition will follow quite closely the approach of L. Taelman and the orator in Anglès and Taelman reference; more precisely, we will show how Anderson's cyclotomic units, an equivariant class  formula in the spirit of Taelman paper, and  some arithmetic properties of Thakur's Gauss sums can be used to prove Taelman's Herbrand-Ribet Theorem. If we will have enough time, we will also explain how these latter ideas can be generalized for  Drinfeld modules defined over Tate algebras.

1. On Anderson's cyclotomic units" and special $L$-values I (Bruno Anglès)
2. On Anderson's cyclotomic units" and special $L$-values II (Bruno Anglès)
3. On Anderson's cyclotomic units" and special $L$-values III (Bruno Anglès)