Quaternion \(L\)-value congruences and governing fields of \(S\)-class groups (Q810580)

Field extensions \(N/\mathbb{Q}\) of Galois group the quaternion group of order eight are the first technically interesting case for T. Chinburg's conjectures on the Galois module structure of \(S\)-units. We give congruences for \(L\)-values associated to certain such extensions which show that certain sets of primes generate the 2-Sylow subgroup of the ideal class groups discussed. The congruences are strong enough to determine governing fields for the variation of the Galois module structure of the \(S\)-class group for twists through infinite families of quaternion extensions. The techniques are based upon those of T. Chinburg, with a new application of G. Gras's analytic genus theory to obtain certain Dirichlet \(L\)-value congruences. Thus, all of the techniques are founded upon Deligne and Ribet's deep results on values of \(L\)-functions.