A Herbrand-Ribet theorem for function fields
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 \[ H^1(Y_{\text{fl}}, C[{\mathfrak p}]^D)=\bigoplus_{n=1}^{q^d-1} H^1(Y_{\text{fl}}, C[{\mathfrak p}]^D)(\chi^n) \] of the \(A/{\mathfrak p}\)-vector space \(H^1(Y_{\text{fl}}, C[{\mathfrak p}]^D)\) under the action of \(G\). Let \(e(z)\in Kz\) be the Carlitz exponential and let \(BC_n\in K\) be the \textit{Bernoulli-Carlitz numbers} defined by the series identity \[ \frac{z}{e(z)}=\sum_{n=0}^{\infty} BC_n\frac{z^n}{n!_C} \] where \(n!_C\) is the Carlitz exponential. The analogue of the Herbrand-Ribet theorem is Theorem 1: Let \(0<n<q^d-1\) be divisible by \(q-1\). Then \({\mathfrak p}\) divides \(BC_n\) if and only if \(H^1(Y_{\text{fl}}, C[{\mathfrak p}]^D)(\chi^{n-1})\) is nonzero. Related to the Kummer-Vandiver conjecture is the following Question 1: Does \(H^1(Y_{\text{fl}}, C[{\mathfrak p}]^D)(\chi^{n-1})\) vanish if \(n\) is not divisible by \(q-1\)? A later paper of the author and \textit{B. Anglès} [J. Number Theory 133, No. 3, 830--841 (2013; Zbl 1285.11138)] provides examples where Question 1 has a negative answer. If \(p\) is an odd prime number, \(F={\mathbb Q}(\zeta_p)\) and \(D=\mathrm{Spec}\;{\mathcal O}_F\), then \({\mathbb F}_p\otimes_{\mathbb Z} \mathrm{Pic}\;D= \mathrm{Ext}^2_{D_{\text{et}}}( {\mathbb Z}/p{\mathbb Z}, {\mathbb G}_{m,D})\) and \(H^1(D_{\text{et}}, {\mathbb Z}/ p{\mathbb Z})=H^1(D_{\text{fl}}, {\mathbb Z}/p{\mathbb Z})\). The Herbrand-Ribet theorem says that for even \(n\), \(1<n<p-1\), \(p\mid B_n\) if and only if \(H^1(D_{\text{fl}}, \mu_p^D)(\chi^{n-1})\neq 0\) in perfect analogy with the statement of Theorem 1.
scientific article; zbMATH DE number 6037981
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