Local constancy for reductions of two-dimensional crystalline representations

We prove the existence of local constancy phenomena for reductions in a general prime power setting of two-dimensional irreducible crystalline representations. Up to twist, these representations depend on two parameters: a trace $a_p$ and a weight $k$. We find an (explicit) local constancy result with respect to $a_p$ using Fontaine's theory of $(\varphi, \Gamma)$-modules and its crystalline refinement due to Berger via Wach modules and their continuity properties. The local constancy result with respect to $k$ (for $a_p\not=0$) will follow from a local study of Colmez's rigid analytic space parametrizing trianguline representations. This work extends some results of Berger obtained in the semi-simple residual case.Comment: Comments are welcome

Are Houses Too Big or In the Wrong Place? Tax Benefits to Housing and Inefficiencies in Location and Consumption

Tax benefits to owner-occupied housing provide incentives to consume housing, offsetting weaker disincentives of the property tax. These benefits also help counter the penalty federal taxes impose on households who work in productive high-wage areas, but reinforce incentives to consume local amenities. We simulate the effects of these benefits in a parameterized model, and determine the consequences of various tax reforms. Reductions in housing tax benefits generally increase efficiency in consumption, but reduce efficiency in location decisions, unless they are accompanied by tax rate reductions. The most efficient policy would eliminate most tax benefits to housing and index taxes to local wage levels

Reduction of Galois Representations of slope 1

We compute the reductions of irreducible crystalline two-dimensional representations of $G_{\mathbf{Q}_p}$ of slope 1, for primes $p \geq 5$, and all weights. We describe the semisimplification of the reductions completely. In particular, we show that the reduction is often reducible. We also investigate whether the extension obtained is peu or tr\`es ramifi\'ee, in the relevant reducible non-semisimple cases. The proof uses the compatibility between the $p$-adic and mod $p$ Local Langlands Correspondences, and involves a detailed study of the reductions of both the standard and non-standard lattices in certain $p$-adic Banach spaces.Comment: Refereed version. Some arguments have been simplified in Section

Supersymmetric Kaluza-Klein reductions of M-waves and MKK-monopoles

We investigate the Kaluza-Klein reductions to ten dimensions of the purely gravitational half-BPS M-theory backgrounds: the M-wave and the Kaluza-Klein monopole. We determine the moduli space of smooth (supersymmetric) Kaluza-Klein reductions by classifying the freely-acting spacelike Killing vectors which preserve some Killing spinor. As a consequence we find a wealth of new supersymmetric IIA configurations involving composite and/or bound-state configurations of waves, D0 and D6-branes, Kaluza-Klein monopoles in type IIA and flux/nullbranes, and some other new configurations. Some new features raised by the geometry of the Taub-NUT space are discussed, namely the existence of reductions with no continuous moduli. We also propose an interpretation of the flux 5-brane in terms of the local description (close to the branes) of a bound state of D6-branes and ten-dimensional Kaluza-Klein monopoles.Comment: 36 pages (v2: Reference added, "draft" mode disabled; v3: two singular reductions discarded, appendix on spin structures added, references updated