A note on the Iwasawa \(\lambda\)-invariants of real quadratic fields
Let \(k\) be a real quadratic field, \(p\) an odd prime number and \(\chi\) the primitive Dirichlet character associated to \(k\). It is known from the Iwasawa main conjecture proved by \textit{B. Mazur} and \textit{A. Wiles} [Invent. Math. 76, 179-330 (1984; Zbl 0545.12005)] that the \(\lambda\)-invariant \(\lambda_p(k)\) of \(k\) is not greater than the \(\lambda\)-invariant \(\lambda^*_p(k)\) of the power series associated to the \(p\)-adic \(L\)-function \(L_p (s,\chi)\). In this paper, the author studies the conjecture \[ (*):\lambda_p(k) \leq\max \bigl\{0, \lambda^*_p (k)-1 \bigr\}, \] which is weaker than Greenberg's one that asserts \(\lambda_p (k)=0\). Let \(\chi^*\) be the primitive Dirichlet character associated to the product of the Teichmüller character and \(\chi^{-1}\). In an earlier work of \textit{H. Ichimura} and \textit{H. Sumida} [Int. Math. J. 7, No. 6, 721-744 (1996)], it was shown that \((*)\) is valid if \(\chi^*(p)=1\). In this paper, the author constructs two families of \(k\) which involve infinitely many \(k\) such that \(\chi^*(p)\neq 1\) and \((*)\) is valid. For a fixed squarefree positive integer \(r\) and each positive integer \(m\), let \(d_m^{(1)}= p^4r^2 m^2+r\), \(d_m^{(2)} =p^4 m^2+p\) and \(k_m^{(i)} =\mathbb{Q} (\sqrt {d_m^{(i)}} )\). Then \(p\) remains prime in \(k_m^{(1)}\) and ramifies in \(k_m^{(2)}\). Furthermore, \(\chi^*(p) \neq 1\) for \(k=k_m^{(i)}\). Using explicit formulas for fundamental units \(\varepsilon\) of \(k_m^{(i)}\), the author shows some congruent relations concerning \(\varepsilon\). And he concludes that if \(d_m^{(i)}\) is squarefree, then \(\lambda_p^* (k)\geq 1\) and \((*)\) is valid for \(k =k_m^{(i)}\). The irreducibility of the polynomials \(p^4r^2X^2+r\) and \(p^4X^2 +p\) guarantees the existence of infinitely many \(m\) for which \(d^{(i)}_m\) is squarefree.
scientific article; zbMATH DE number 897709