Some Families of Quaternion Fields and the Second Chinburg Conjecture

Abstract

<p>Let N/K be a finite Galois extension of number fields with Galois group G. The second Chinburg conjecture asserts the equivalence in the locally free class group CL(Z[G]) of the classes corresponding to two arithmetic invariants attached to the extension, namely the Frōhlich-Cassou-Noguès class Wɴ/k and the second Chinburg invariant Ω(ɴ/k,2). Frōhlich originally formulated a conjectural equivalence of the class Wɴ/k and the class [Oɴ] determined by the ring of integers in N, and this was subsequently verified by M. Taylor. The second Chinburg invariant Ω(N/K,2) may be interpreted as a generalisation of the class [Oɴ] to wild extensions, and Chinburg showed that Frōhlich's conjecture for tame extensions implied Chinburg's conjecture for tame extensions.</p> <p>More generally, the second Chinburg conjecture has been verified when G is absolutely abelian of odd conductor (C. Greither) and for several infinite families of wildly ramified quaternion fields (S. Kim). When N/K is wildly ramified and G is nonabelian, little further evidence for the conjecture is known.</p> <p>In this thesis we consider some families of extensions N/Q with G ≅ Q₈, the quaternion group of order 8, in which the prime 2 is totally ramified. In particular we consider those quatermon extensions containing a biquadratic subfield of the form Q(√a, √b), where a ≡ 2 mod 16 and b ≡ mod 8. We verify the second Chinburg conjecture for all such extensions.</p> <p>We begin by localizing at the prime 2 and make use of a cohomological classification of a large class of cohomologically trivial 2-primary Z[Q₈]-modules to explicitly compute the local second Chinburg invariant. Globally, we construct a projective Z[Q₈]-module, X, that lies inside ON, the ring of integers of N. We then work with Frōhlich's Hom-description of the class group and show that, in fact, [X] = Wɴ/Q in CL(Z[Q]), where [X] denotes the class of X. Combining this with the local information, and using congruence methods, we conclude that in CL(Z[Q₈]),</p> <p>Ω(ɴ/Q,2) = Wɴ/Q,</p> <p>i.e., the second Chinburg conjecture holds for these extensions.</p> <p>Combining this result with work of S. Kim, V. Snaith and M. Tran, this establishes the second Chinburg conjecture for all extensions ɴ/Q having Galois group G ≅ Q₈.</p>