BCN Spring 2016 Workshop: Number Theory & K-theory

On Anderson's ``cyclotomic units" and special $L$-values I


Bruno Anglès


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 (\cite{RIB}):
$$X(\omega_p^n)\not =\{0\} \Leftrightarrow B_{p-n}\equiv 0\pmod{p},$$
where $B_n$ denotes the $n$-th Bernoulli number.\par

Our aim in these lectures is to present a proof as self-contained as possible of Taelman's analogue (for the Carlitz module, see \cite{GOS} chapter 3) of the above Theorem (\cite{TAE3}). Our exposition will follow quite closely the approach of L. Taelman and the orator in \cite{ANG&TAE1}; more precisely, we will show how  Anderson's cyclotomic units (\cite{AND}, \cite{ANDE}), an equivariant class  formula in the spirit of \cite{TAE2}, and  some arithmetic properties of Thakur's Gauss sums (\cite{THA}) 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 (\cite{APTR}).\par
 \bibitem{AND} G. Anderson, Log-algebraicity of twisted $A$-harmonic series and special values of $L$-series in characteristic $p$, {\it  Journal of  Number Theory} {\bf 60} (1996), 165-209.
 \bibitem{ANDE} G. Anderson, Rank one elliptic $A$-modules and $A$-harmonic series, {\it Duke Mathematical Journal}
 {\bf 73} (1994), 491-542.
 \bibitem{APTR} B. Angl\`es, F. Pellarin,  F. Tavares Ribeiro,  Arithmetic of positive characteristic $L$-series values in Tate algebras, {\it Compositio Mathematica},  {\bf 152} (2016), 1-61.
 \bibitem{ANG&TAE1} B. Angl\`es, L. Taelman, Arithmetic of characteristic $p$ special $L$-values,  {\it Proceedings of the  London Mathematical  Society} {\bf 110} (2015), 1000-1032.
  \bibitem{GOS} D. Goss, {\it Basic Structures of Function Field Arithmetic}, Springer, Berlin, 1996.
  \bibitem{RIB} K. A. Ribet, A modular construction of unramified $p$-extensions of $\mathbb Q(\mu_p),$ {\it Inventiones mathematicae} {\bf 34} (1976), 151-162.
  \bibitem{TAE2} L. Taelman, Special $L$-values of Drinfeld modules, {\it Annals of Mathematics} {\bf 75} (2012), 369-391.
  \bibitem{TAE3} L. Taelman, A Herbrand-Ribet theorem for function fields, {\it Inventiones  mathematicae} {\bf 188} (2012), 253-275.
  \bibitem{THA} D. Thakur, Gauss sums for $\mathbb F_q[t],$   {\it Inventiones  mathematicae}   {\bf 94} (1988), 105-112 .



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