On the Iwasawa invariants of certain real abelian fields. II (Q2785291)

Let \(k\) be a totally real number field and \(p\) a prime number. R. Greenberg conjectured that the Iwasawa invariant \(\lambda_p(k)=0\) for the cyclotomic \(\mathbb{Z}_p\)-extension of \(k\). The authors give a computational method for verifying this conjecture when \(k/\mathbb{Q}\) is abelian. In particular, they prove that \(\lambda_3(k)=0\) for all real quadratic fields \(\mathbb{Q}(\sqrt{m})\) with \(1<m<10000\). This result has also been obtained by \textit{J. Kraft} and \textit{R. Schoof} [Compos. Math. 97, 135-155 (1995; Zbl 0840.11043); erratum 103, 241 (1996; Zbl 0857.11056)] for the fields in which 3 is not split. The present method is computationally faster than that of Kraft and Schoof, though this latter method obtains the \(\mathbb{Z}_3[[T]]\)-structure of the class group. NEWLINENEWLINENEWLINEThe authors assume for simplicity that \(p\) is an odd prime such that the exponent of \(\Delta=\text{Gal}(k/\mathbb{Q})\) divides \(p-1\). Let \(\chi\) be a nontrivial character of \(\Delta\) and let \(L_p(s,\chi)\) be the corresponding \(p\)-adic \(L\)-function. Let \(f\) be the conductor of \(\chi\) and let \(q=\text{lcm}(p,f)\). Let \(g_{\chi}(T)\in \mathbb{Z}_p[[T]]\) be such that \(g_{\chi}((1+q)^{1-s}-1)=L_p(s,\chi)\) and write \(g_{\chi}(T)=P_{\chi}(T)U(T)\), where \(U(T)\) is a unit in \(\mathbb{Z}_p[[T]]\) and \(P_{\chi}(T)\) is a distinguished polynomial. Factor \(P_{\chi}(T)=P_1(T)^{e_1}\cdots P_r(T)^{e_r}\) into irreducible polynomials in \(\mathbb{Z}_p[T]\). Let \(\omega_n(T)=(1+T)^{p^n}-1\) and let \(p^{a_{i,n}}\) be the exponent of \(\mathbb{Z}_p[[T]]/(P_i, \omega_n)\). Define \(X_{i,n}(T)\in \mathbb{Z}_p[T]\) by \(X_{i,n}P_i\equiv p^{a_{i,n}} \pmod {\omega_n}\) and choose \(Y_{i,n}(T)\in \mathbb{Z}[T]\) congruent to \(X_{i,n}\) mod \(p^{a_{i,n}}\). The authors define a certain cyclotomic unit \(c_{i,n}\) of \(k_n\), the \(n\)th level of the \(\mathbb{Z}_p\)-extension of \(k\), and prove that (when \(\chi(p)\neq 1\) and \(\chi\omega^{-1}(p)\neq 1\), where \(\omega\) is the Teichmüller character; the other cases require slight modifications) \(\lambda_{\chi}=0\) if and only if for each \(i\) with \(1\leq i\leq r\), there exists \(n\geq 0\) such that \((c_{i,n})^{Y_{i,n}(T)}\not\in (k_n^{\times})^{p^{a_{i,n}}}\). This condition can be checked using rational arithmetic by working modulo a prime ideal of degree 1.