The Iwasawa invariants and the higher \(K\)-groups associated to real quadratic fields. (English) Zbl 1082.11071

Sumida-Takahashi, Hiroki

Exp. Math. 14, No. 3, 307-316 (2005).

Before Wiles’ proof of Fermat Last Theorem (FLT), it was known that FLT was true for regular primes. A rational prime number \(p\) is regular if \(p\) does not divide the class number of the cyclotomic number field \({\mathbb Q} (\zeta_p)\) and irregular otherwise. Kummer proved that an odd prime \(p\) is irregular if and only if \(p\) divides the numerator of some Bernoulli number \(B_k\), \(k=2,4,\ldots, p-3\). If \(p\) divides the numerator of \(B_k\), we say that \((p,k)\) is an irregular pair.
However, even after Wiles’ proof of FLT, the computation of irregular primes and irregular pairs are still important in order to obtain some knowledge of class groups of cyclotomic fields.
Let \(p\) be an odd prime and \(A_n\) the \(p\)-Sylow subgroup of the ideal class group of the cyclotomic field \(K_n:= {\mathbb Q}(\zeta_{p^{n+1}})\). Let \(A_n^+\) and \(A_n^-\) be the even and odd part of \(A_n\), that is, \(A_n^+\) is the \(p\)-Sylow subgroup of \(K_n^+:= K_n \cap {\mathbb R}= {\mathbb Q}(\zeta_{p^{n+1}} + \zeta_{p^{n+1}}^{-1})\) and \(A_n^-:= A_n/A_n^+\). Vandiver’s conjecture states that \(A_n^+ = \{0\}\). It has been verified that \(A_n^+=\{0\}\) and \(A_n^- \cong ({\mathbb Z} /p^{n+1} {\mathbb Z})^{r_p}\) for all \(n\geq 0\) and \(p< 12 \times 10^6\), where \(r_p\) is the number of irregular pairs \((p,k)\).
Let \(\omega= \omega_p\) be the Teichmüller character \({\mathbb Z}/p{\mathbb Z} \to {\mathbb Z}_p\) such that \(\omega(a)\equiv a \bmod p\) and let \(\chi\) be an even quadratic character. The paper under review has two main purposes. The first one is to effectively find exceptional pairs \((p,\chi \omega^k)\), where \((p,\chi\omega^k)\) is an exceptional pair if \(\chi \omega^k(p)\neq 1\), \(\chi \omega^{1-k}(p)\neq 1\) and one of the following conditions is satisfied: \(\nu_p(\xi\omega^k)>0\), \(v_p (L_p(1,\chi \omega^k)) >1\), \(v_p (L_p(0,\chi \omega^k)) >1\) or \(\widetilde{\lambda}(\chi \omega^k) >1\). Here \(L_p\) denotes the Kubota-Leopoldt \(p\)-adic \(L\)-function; \(\Delta'= \text{Gal} ({\mathbb Q}(\sqrt {f_{\chi\omega^k}}, \zeta_p)/{\mathbb Q})\); \(e'_{\chi\omega^k} = 1/| \Delta'| \sum_{\delta \in \Delta'} \chi\omega^k(\delta)\delta^{-1}\); \(\nu_p(\chi \omega^k)\) denotes the \(\nu_p\)-Iwasawa invariant associated to \(e'_{\chi\omega^k} A_n\), that is, \(| e'_{\chi\omega^k} A_n| = p^{ \lambda_p(\chi\omega^k) n+\mu_p ( \chi\omega^k) p^n + \nu_p (\chi\omega^k)}\) for \(n\) sufficiently large; \(v_p\) denotes the \(p\)-adic valuation such that \(v_p(p)=1\) and \(\widetilde {\lambda}_p\) denotes the degree of \(g\) where \(g\) is the Iwasawa polynomial associated to \(L_p\).
The author computes the Iwasawa invariants of \({\mathbb Q} (\sqrt {f_\chi},\zeta_p)\) for \(1<f_\chi<200\) and \(3\leq p < 100,000\). In particular \(\lambda_p ({\mathbb Q} (\sqrt {f_\chi}, \zeta_p + \zeta_p^{-1}) = 0\) for this range and for \(f_\chi=5\) and \(p<2,000,000\) there is no exceptional pair.
The second main purpose of the paper is to provide information on the higher \(K\)-groups of the ring of integers \(\vartheta_F\) where \(F= F_\chi\) is the real quadratic field associated to \(\chi\). For example, as a consequence, under the Quillen-Lichtenbaum’s conjecture, it is obtained that for \(3\leq p< 100,000\), \(p\) divides the order of \(K_{68372} (\vartheta_{{\mathbb Q}({\sqrt {8}})})\) if and only if \(p=34,301\).

Reviewer: Gabriel D. Villa-Salvador (México D.F.)

MSC:

11R23	Iwasawa theory
11R70	\(K\)-theory of global fields
11R29	Class numbers, class groups, discriminants

Keywords:

Iwasawa invariants; \(K\)-groups, Iwasawa theory, regular and irregular primes, Vandiver’s conjecture, Greenberg’s conjecture, Quillen-Lichtenbaum’s conjecture, Bernoulli numbers.