Class numbers of real cyclotomic fields of composite conductor (Q2878850)

Let \(K\) denote a number field of degree n over \(\mathbb Q\). Let \(d(K)\) denote its discriminant. The \textit{root discriminant} \(\text{rd}(K)\) of \(K\) is defined by \(\text{rd}(K) := |d(K)|^{1/n}\). Let \(\varphi\) be the Euler phi-function. Exploiting Odlyzko's discriminant lower bounds, \textit{J. M. Masley} [Compos. Math. 37, 297--319 (1978; Zbl 0428.12003)] and \textit{F. J. van der Linden} [Math. Comput. 39, 693--707 (1982; Zbl 0505.12010)] established the class numbers of all real cyclotomic fields of composite conductor \(m\), provided that \(m \leq 200\), \(\varphi(m) \leq 72\) and \(m\neq 148, 152\). For fields of larger degree or conductor, the root discriminant becomes too large for their methods. To overcome this problem, the author establishes by an analytic approach a lower bound on some sums over prime ideals of the Hilbert class fields, which in turn allows to obtain an upper bound on the class number. NEWLINENEWLINENEWLINE The main result is:NEWLINENEWLINENEWLINE Theorem. Let \(m\) be a composite integer, \(m\not\equiv 2\bmod 4\), and let \(\mathbb Q(\zeta_m)^+\) the maximal real subfield of the \(m\)th cyclotomic field \(\mathbb Q(\zeta_m)\). Then the class number \(h^+\) of \(\mathbb Q(\zeta_m)^+\) is \(1\) if \(\varphi(m)\leq 116\) and \(m\not\in\{ 136,145,212\}\), \(2\) if \(m=136\), \(2\) if \(m=145\) and \(1\) if \(m=256\). Furthermore, under the generalized Riemann hypothesis (GRH), \(h^+_{212} = 5\) and \(h^+_{512} = 1\).NEWLINENEWLINEAs a consequence, the real cyclotomic field of conductor \(420\) has class number \(1\), which is the largest conductor for which the class number of a cyclotomic field has been calculated unconditionally. Note that this theorem on composite conductors complements the earlier results of the author on real cyclotomic fields of prime conductor [``Real cyclotomic fields of prime conductor and their class numbers'', Math. Comput. (to appear), \url{arXiv:1407.2373}].NEWLINENEWLINEThe result of the author in a previous article [Acta Arith. 164, No. 4, 381--397 (2014; Zbl 1305.11098)] dealing with some sums over prime ideals is essential for the proofs of this article:NEWLINENEWLINE Let \(K\) be a totally real field of degree \(n\) and let \(F(x)=\frac{e^{-(x/c)^2}}{\cosh(x/2)}\) for some positive constant \(c\). Suppose \(S\) is a subset of the prime integers which totally split into principal prime ideals of \(K\). Let NEWLINE\[NEWLINE\begin{split} &B=\frac{\pi}{2}+\gamma+\log (8\pi)- \log(\mathrm{rd}(K))-\int_0^\infty \frac{1-F(x)}{2} \Big(\frac{1}{\sinh((x/2)}+\frac{1}{\cosh(x/2)}\Big)\,dx\\ &+2\sum_{p\in S}\sum_{m=1}^\infty\frac{\log(p)}{p^{m/2}}F(m\log(p)). \end{split}NEWLINE\]NEWLINE If \(B>0\) (unconditionally under GRH), we have an upper bound for the class number \(h\) of \(K\) given by \(h<\frac{2c\pi}{nB}\).NEWLINENEWLINEThe author is currently investigating the application of the methods of this paper to certain nonabelian number fields.