A Herbrand-Ribet theorem for function fields
type: Item
instance of: scholarly article
title: A Herbrand-Ribet theorem for function fields
zbMATH Open document ID: 1278.11102
author: Lenny Taelman
published in: Inventiones Mathematicae
publication date: 2012-05-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1104.5363
review text: Let \(F={\mathbb Q}(\zeta_p)\) be the \(p\)-th cyclotomic number field and \(\mathrm{Pic}\;{\mathcal O}_F\) its class group. Let \(w: \mathrm{Gal} (F/{\mathbb Q})\to {\mathbb F}_p^{\ast}\) be the cyclotomic character. Then \[ {\mathbb F}_p\otimes_{\mathbb Z} \mathrm{Pic}\;{\mathcal O}_F = \bigoplus_{n=1}^{p-1} ({\mathbb F}_p\otimes_{\mathbb Z} \mathrm{Pic}\;{\mathcal O}_F)(w^n). \] The \textit{Bernoulli numbers} are defined by the identity \[ \frac{z}{\exp(z)-1}=\sum_{n=0}^{\infty}B_n\frac{z^n}{n!}. \] The \textit{Herbrand-Ribet theorem} states that if \(n\) is even and \(1<n<p\), then \[ ({\mathbb F}_p\otimes_{\mathbb Z} \mathrm{Pic}\;{\mathcal O}_F)(w^{1-n})\neq 0\;\text{if and only if}\;p\mid B_n. \] The \textit{Kummer-Vandiver conjecture} asserts that for all odd \(n\) we have \[ ({\mathbb F}_p\otimes_{\mathbb Z} \mathrm{Pic}\;{\mathcal O}_F)(w^{1-n})=0. \] The author studies in this paper the function field analogue of both, the Herbrand-Ribet theorem and the Kummer-Vandiver conjecture. Let \(C\) be the Carlitz module over \(\mathrm{Spec}\;A\) where \(A={\mathbb F}_q[t]\). Let \(K={\mathbb F}_q(t)\). For a nonzero prime \({\mathfrak p}\) of \(A\), we have the isomorphism \(\chi: G=\mathrm{Gal}(K(\lambda_{\mathfrak p})/K)\to (A/{\mathfrak p})^{\ast}\) where \(\lambda_{\mathfrak p}\) is a generator of the \({\mathfrak p}\)-torsion of \(C\). \(\chi\) is induced by the Galois representation \[ \rho:\mathrm{Gal}(K^{\text{sep}}/K)\to (A/{\mathfrak p})^{\ast}. \] Let \(R\) be the normalization (integral closure) of \(A\) in \(L=K(\lambda_{\mathfrak p})\) and \(Y=\mathrm{Spec}\;R\). Let \(Y_{\text{fl}}\) be the flat site on \(Y\). The \({\mathfrak p}\)-torsion \(C[{\mathfrak p}]\) of \(C\) is a finite flat group scheme of rank \(q^d\) over \(\mathrm{Spec}\;A\) where \(d\) is the degree of \({\mathfrak p}\). Let \(C[{\mathfrak p}]^D\) be the Cartier dual of \(C[{\mathfrak p}]\) and consider the decomposition \[ H^1(Y_{\text{fl}}, C[{\mathfrak p}]^D)=\bigoplus_{n=1}^{q^d…
Mathematics Subject Classification ID: 11R60
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 6037981
zbMATH Keywords: Herbrand-Ribet theorem
zbMATH Keywords: Carlitz module
zbMATH Keywords: cyclotomic function fields
zbMATH Keywords: Bernoulli-Carlitz numbers
zbMATH Keywords: Bernoulli numbers
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
OpenAlex ID: W1967150711
arXiv ID: 1104.5363
cites work: Log-algebraicity of twisted \(A\)-harmonic series and special values of \(L\)-series in characteristic \(p\)
cites work: On Gekeler's conjecture for function fields
cites work: Duality in the flat cohomology of curves
cites work: The class number of cyclotomic function fields
cites work: On regularity of small primes in function fields
cites work: Q4714153
cites work: Class-groups of function fields
cites work: Notes on étale cohomology of number fields
cites work: Q3754095
cites work: Kummer's theory for function fields
cites work: Schémas en groupes de type $(p,\ldots,p)$
cites work: A modular construction of unramified p-extensions of \(\mathbb{Q}(\mu_p)\)
cites work: The Carlitz shtuka
cites work: Special \(L\)-values of Drinfeld modules
cites work: Group schemes of prime order
cites work: Q5690489
DOI: 10.1007/S00222-011-0346-3
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description: scientific article; zbMATH DE number 6037981