Diffeological, Fr\"{o}licher, and Differential Spaces

Differential calculus on Euclidean spaces has many generalisations. In particular, on a set $X$, a diffeological structure is given by maps from open subsets of Euclidean spaces to $X$, a differential structure is given by maps from $X$ to $\mathbb{R}$, and a Fr\"{o}licher structure is given by maps from $\mathbb{R}$ to $X$ as well as maps from $X$ to $\mathbb{R}$. We illustrate the relations between these structures through examples.Comment: 21 page

Explicit computation of Galois representations occurring in families of curves

We extend our method to compute division polynomials of Jacobians of curves over Q to curves over Q(t), in view of computing mod ell Galois representations occurring in the \'etale cohomology of surfaces over Q. Although the division polynomials which we obtain are unfortunately too complicated to achieve this last goal, we still obtain explicit families of Galois representations over P^1_Q, and we study their degeneration at places of bad reduction of the corresponding curve.Comment: Comments welcom

The primitive solutions to x^3+y^9=z^2

We determine the rational integers x,y,z such that x^3+y^9=z^2 and gcd(x,y,z)=1. First we determine a finite set of curves of genus 10 such that any primitive solution to x^3+y^9=z^2 corresponds to a rational point on one of those curves. We observe that each of these genus 10 curves covers an elliptic curve over some extension of Q. We use this cover to apply a Chabauty-like method to an embedding of the curve in the Weil restriction of the elliptic curve. This enables us to find all rational points and therefore deduce the primitive solutions to the original equation.Comment: 8 page

A new two--parameter family of isomonodromic deformations over the five punctured sphere

The object of this paper is to describe an explicit two--parameter family of logarithmic flat connections over the complex projective plane. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of a circle and three tangent lines. By restricting them to generic lines we get an algebraic family of isomonodromic deformations of the five--punctured sphere. This yields new algebraic solutions of a Garnier system. Finally, we use the associated Riccati one--forms to construct an interesting non--generic family of transversally projective Lotka--Volterra foliations.Comment: English text, 30 page

Explicit Solution By Radicals, Gonal Maps and Plane Models of Algebraic Curves of Genus 5 or 6

We give explicit computational algorithms to construct minimal degree (always $\le 4$) ramified covers of \Prj^1 for algebraic curves of genus 5 and 6. This completes the work of Schicho and Sevilla (who dealt with the $g \le 4$ case) on constructing radical parametrisations of arbitrary genus $g$ curves. Zariski showed that this is impossible for the general curve of genus $\ge 7$. We also construct minimal degree birational plane models and show how the existence of degree 6 plane models for genus 6 curves is related to the gonality and geometric type of a certain auxiliary surface.Comment: v3: full version of the pape