An elementary approach to dessins d'enfants and the Grothendieck-Teichm\"uller group

We give an account of the theory of dessins d'enfants which is both elementary and self-contained. We describe the equivalence of many categories (graphs embedded nicely on surfaces, finite sets with certain permutations, certain field extensions, and some classes of algebraic curves), some of which are naturally endowed with an action of the absolute Galois group of the rational field. We prove that the action is faithful. Eventually we prove that this absolute Galois group embeds into the Grothendieck-Teichm\"uller group $GT_0$ introduced by Drinfel'd. There are explicit approximations of $GT_0$ by finite groups, and we hope to encourage computations in this area. Our treatment includes a result which has not appeared in the literature yet: the Galois action on the subset of regular dessins - that is, those exhibiting maximal symmetry -- is also faithful.Comment: 58 pages, about 30 figures. Corrected a few typos. This version should match the published paper in L'enseignement Mathematiqu

Examples of Sweedler cohomology in positive characteristic

In this paper we provide a detailed calculation of the Sweedler cohomology of the algebra of functions on (Z/2)^r, in all degrees, over a field of characteristic 2. The result is strikingly different from the characteristic 0 analog. Then we show that there is a variant in characteristic p of the result obtained by Kassel and the author in characteristic zero, which provides a near-complete calculation of the second lazy cohomology group in the case of function algebras over a finite group.Comment: With this update we pretty much revert to version 2 (including the old title), for the good reason that this version has been accepted for publication in Comm. Alg. (without part II which appeared in version 3). So this final paper is identical with the published on

The multiple planets transiting Kepler-9 I. Inferring stellar properties and planetary compositions

The discovery of multiple transiting planetary systems offers new possibilities for characterising exoplanets and understanding their formation. The Kepler-9 system contains two Saturn-mass planets, Kepler-9b and 9c. Using evolution models of gas giants that reproduce the sizes of known transiting planets and accounting for all sources of uncertainties, we show that Kepler-9b (respectively 9c) contains $45^{+17}_{-12}$\,\mearth\ (resp. $31^{+13}_{-10}$\,\mearth) of hydrogen and helium and $35^{+10}_{-15}$\,\mearth (resp. $24^{+10}_{-12}$\,\mearth) of heavy elements. More accurate constraints are obtained when comparing planets 9b and 9c: the ratio of the total mass fractions of heavy elements are $Z_{\rm b}/Z_{\rm c}=1.02\pm 0.14$, indicating that, although the masses of the planets differ, their global composition is very similar, an unexpected result for formation models. Using evolution models for super-Earths, we find that Kepler-9d must contain less than 0.1% of its mass in hydrogen and helium and predict a mostly rocky structure with a total mass between 4 and 16\,\mearth.Comment: 5 pages + 7 pages of online material ; revised article submitted to A\&A and accepted on March 3

Vector fields, separatrices and Kato surfaces

We prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is compact). We also prove that, in a singular Stein surface endowed with a complete holomorphic vector field, a singular point of the surface where the zeroes of the vector field do not accumulate is either a quasihomogeneous or a cyclic quotient singularity. The proofs rely in a combinatorial description of the vector field on a resolution of the singular point based on previous work of Rebelo and the author. With the same tools, we reprove some facts about the classification of compact complex surfaces admitting holomorphic vector fields.Comment: post-referee versio

Chow rings and cobordism of some Chevalley groups

We compute the cobordism ring $MU^*(BG)\otimes \mathbb{F}_p$ when $G$ is a Chevalley group. In the particular case of the general linear group, we prove that it agrees with the Chow ring (as defined by Totaro).Comment: 18 pages, uses psp.cls, to appear shortly in Math. Proc. Cam. Phil. So

Evolution of "51Peg b-like" Planets

About one-quarter of the extrasolar giant planets discovered so far have orbital distances smaller than 0.1 AU. These ``51Peg b-like'' planets can now be directly characterized, as shown by the planet transiting in front the star HD209458. We review the processes that affect their evolution. We apply our work to the case of HD209458b, whose radius has been recently measured. We argue that its radius can be reproduced only when the deep atmosphere is assumed to be unrealistically hot. When using more realistic atmospheric temperatures, an energy source appears to be missing in order to explain HD209458b's large size. The most likely source of energy available is not in the planet's spin or orbit, but in the intense radiation received from the parent star. We show that the radius of HD209458b can be reproduced if a small fraction (~1%) of the stellar flux is transformed into kinetic energy in the planetary atmosphere and subsequently converted to thermal energy by dynamical processes at pressures of tens of bars.Comment: 11 pages including 9 figures. A&A, in press. Also available at http://www.obs-nice.fr/guillot/pegasi-planets