On the \(p\)-rank of the ideal class group of a normal extension with simple Galois group (Q2199512)

Let \(L\) be a finite normal extension of the algebraic number field \(K\), let \(G=\) Gal\((L/K)\) be the Galois group of \(L\) over \(K\) and assume \(G\) to be a simple and non-abelian group. Let \(p\) be a prime. Write \(\mathrm{Cl}_p(L)\) for the \(p\)-part of the ideal class group of \(L\). Let the ambiguous \( p\)-class group of \(L \) with respect to \(K\), noted \(\mathrm{Amb}_p(L/K)\), be the set of elements \(J \) of \(\mathrm{Cl}_p(L)\) such that \(\sigma(J)=J\) for each \(\sigma\) of \(G.\) Let \(h_p(L)\) be the cardinality of \(\mathrm{Cl}_p(L)\) and \(a_p(L/K)\) be the cardinality of \(\mathrm{Amb}_p(L/K)\). The goal of this nice paper is to come up with some lower bounds of the quotient \(h_p(L)/a_p(L/K)\). The interest of the problem is emphasized by the fact that \(\mathrm{Amb}_p(L/K)=\mathrm{Cl}_p(K)\) when \(p\) does not divide the order of \(G\). When \(h_p(L) > a_p(L/K)\), the author gives some explicit lower bounds of the \(p\)-rank of \(\mathrm{Cl}_p(L)/\mathrm{Amb}_p(L/K)\). One lower bound is the maximum of all \(f_\ell (p)\) among all the primes \(\ell \) dividing the order of \(G\), with \(\ell \not =p\), where \(f_\ell(p)\) is the minimal \(i\) such that \(p^i \equiv 1 \pmod\ell\). When \(\ell\) is a prime divisor of the order of \(G\), another lower bound for any \(p\not=\ell\) is given by 1 plus the maximal value of the dimensions of the elementary abelian \(\ell\)-subgroups of \(G\) over the field \(F_\ell\). When \(\ell\) is an odd prime dividing the order of \(G\), the author proves that a lower bound for the \(\ell\)-rank of \(\mathrm{Cl}_\ell(L)/\mathrm{Amb}_\ell (L/K)\) is given by \(2\sqrt{\nu_\ell}\), where \(\nu_\ell\) is the \(\ell\)-adic order of the maximal order of an abelian \(\ell\)-subgroup of \(G\). Some comments are also made about these bounds.