Affine surfaces with trivial Makar-Limanov invariant

We study the class of 2-dimensional affine k-domains R satisfying ML(R) = k, where k is an arbitrary field of characteristic zero. In particular, we obtain the following result: Let R be a localization of a polynomial ring in finitely many variables over a field of characteristic zero. If ML(R) = K for some field K included in R and such that R has transcendence degree 2 over K, then R is K-isomorphic to K[X,Y,Z]/(XY-P(Z)) for some nonconstant polynomial P(Z) in K[Z].Comment: 12 pages. See also http://aix1.uottawa.ca/~ddaigle/index.htm

On the local Tamagawa number conjecture for Tate motives over tamely ramified fields

The local Tamagawa number conjecure, first formulated by Fontaine and Perrin-Riou, expresses the compatibility of the (global) Tamagawa number conjecture on motivic $L$-functions with the functional equation. The local conjecture was proven for Tate motives over finite unramified extensions $K/\mathbb{Q}_p$ by Bloch and Kato. We use the theory of $(\phi, \Gamma_K)$-modules and a reciprocity law due to Cherbonnier and Colmez to provide a new proof in the case of unramified extensions, and to prove the conjecture for the motive $\mathbb{Q}_p(2)$ over certain tamely ramified extensions.Comment: 45 pages, LaTeX; extensive revisions and clarifications based on feedback; to appear in Algebra & Number Theor