Dessins, their delta-matroids and partial duals

Given a map $\mathcal M$ on a connected and closed orientable surface, the delta-matroid of $\mathcal M$ is a combinatorial object associated to $\mathcal M$ which captures some topological information of the embedding. We explore how delta-matroids associated to dessins d'enfants behave under the action of the absolute Galois group. Twists of delta-matroids are considered as well; they correspond to the recently introduced operation of partial duality of maps. Furthermore, we prove that every map has a partial dual defined over its field of moduli. A relationship between dessins, partial duals and tropical curves arising from the cartography groups of dessins is observed as well.Comment: 34 pages, 20 figures. Accepted for publication in the SIGMAP14 Conference Proceeding

Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.Comment: 20 pages. Some changes, the most important being on Conjecture 3.2, three references added ([Mas75], [MB90] and [Yu94]) and one reference updated [BS12]. Accepted in Bull. Brazil. Mat. So