Serre’s uniformity problem asks whether there exists a bound $k$ such that for any , the Galois representation associated to the $p$-torsion of an elliptic curve $E/\mathbb{Q}$ is surjective independent of the choice of $E$. Serre showed that if this representation is not surjective, then it has to be contained in either a Borel subgroup, the normalizer of a split Cartan subgroup, the normalizer of a non-split Cartan subgroup, or one of a finite list of “exceptional” subgroups. We will focus on the case when the image is contained in the normalizer of a split Cartan subgroup. In particular, we will show that the only elliptic curves whose Galois representation at $11$ is contained in the normalizer of a split Cartan have complex multiplication. To prove this we compute $X_s^{+}(11)$ using modular units, use the methods of Poonen and Schaefer to compute its Jacobian, and then use the method of Chabauty and Coleman to show that the only points on this curve correspond to CM elliptic curves.

This paper is the first of two papers whose combined goal is to explore the dessins d’enfant and symmetries of quasi-platonic actions of $PS{L}_2(q)$. A quasi-platonic action of a group $G$ on a closed Riemann $S$ surface is a conformal action for which $S/G$ is a sphere and $S\to S/G$ is branched over $\{0,1,\infty \}$. The unit interval in $S/G$ may be lifted to a dessin d’enfant $\mathcal{D}$, an embedded bipartite graph in $S$. The dessin forms the edges and vertices of a tiling on $S$ by dihedrally symmetric polygons, generalizing the idea of a platonic solid. Each automorphism $\psi$ in the absolute Galois group determines a transform $S^{\psi }$ by transforming the coefficients of the defining equations of $S$. The transform defines a possibly new quasi-platonic action and a transformed dessin ${\mathcal{D}}^{\psi }$.

Here, in this paper, we describe the quasi-platonic actions of $PS{L}_2(q)$ and the action of the absolute Galois group on $PS{L}_2(q)$ actions. The second paper discusses the quasi-platonic actions constructed from symmetries (reflections) and the resulting triangular tiling that refines the dessin d’enfant. In particular, the number of ovals and the separation properties of the mirrors of a symmetry are determined.

Matrix multivariate Pearson type II-Riesz distribution is defined and some of its properties are studied. In particular, the associated matrix multivariate beta distribution type I is derived. Also the singular values and eigenvalues distributions are obtained.