アルシュ ノ アーベルタイ ノ キンテイ ニツイテ

Let K be an abeilian field over the rationals Q and let Z_K be the ring of integers of K.K is said to be monogenic when there exists an element θ of Z_K with Z_K = Z[θ]. In thiscase θ is said to be a generator of Z_K. Hasse proposed for any given field,

Effective Methods for Norm-Form Equations

While effective resolution of Thue equations has been well understood since the work of Baker in the 1960s, similar results for norm-form equations in more than two variables have proven difficult to achieve. In 1983, Vojta was able to address the case of three variables over totally complex and Galois number fields. In this paper, we extend his results to effectively resolve several new classes of norm-form equations. In particular, we completely and effectively settle the question of norm-form equations over totally complex Galois sextic fields.Comment: Final version, accepted by Math Annalen. A few changes from the previous version-- in particular there is a new result that also applies over non-Galois extensions. The explicit example was removed and will appear elsewher

Arithmetic of singular Enriques Surfaces

We study the arithmetic of Enriques surfaces whose universal covers are singular K3 surfaces. If a singular K3 surface X has discriminant d, then it has a model over the ring class field d. Our main theorem is that the same holds true for any Enriques quotient of X. It is based on a study on Neron-Severi groups of singular K3 surfaces. We also comment on Galois actions on divisors of Enriques surfaces.Comment: 32 pages; v2: Section 2 expanded, minor additions and edit