We constructed a G-zip over the special fiber of Kisin's integral canonical model. This induces a morphism from the special fiber to the stack of G-zips. Fibers of this morphism are Ekedahl-Oort... Show moreWe constructed a G-zip over the special fiber of Kisin's integral canonical model. This induces a morphism from the special fiber to the stack of G-zips. Fibers of this morphism are Ekedahl-Oort strata. We proved that this morphism is smooth, and hence we can transfer results about geometry of the stack of G-zips to results about Ekedahl-Oort strata. In particular, each stratum is smooth, and we know its dimension and closure. Show less
In this thesis we investigate $2$-dimensional, continuous, odd, residual Galois representations and their images. This manuscript consists of two parts. In the first part of this thesis we analyse... Show moreIn this thesis we investigate $2$-dimensional, continuous, odd, residual Galois representations and their images. This manuscript consists of two parts. In the first part of this thesis we analyse a local\--global problem for elliptic curves over number fields. Let $E$ be an elliptic curve over a number field $K$, and let $\ell$ be a prime number. If $E$ admits an $\ell$-isogeny locally at a set of primes with density one then does $E$ admit an $\ell$-isogeny over $K$? The study of the Galois representation associated to the $\ell$-torsion subgroup of $E$ is the crucial ingredient used to solve the problem. We characterize completely the cases where the local\--global principle fails, obtaining an upper bound for the possible values of $\ell$ for which this can happen. In the second part of this thesis, we outline an algorithm for computing the image of a residual modular $2$-dimensional semi-simple Galois representation. This algorithm determines the image as a finite subgroup of $\GL_2(\overline{\F}_\ell)$, up to conjugation, as well as certain local properties of the representation and tabulate the result in a database. In this part of the thesis we show that, in almost all cases, in order to compute the image of such a representation it is sufficient to know the images of the Hecke operators up to the Sturm bound at the given level $n$. In addition, almost all the computations are performed in positive characteristic. In order to obtain such an algorithm, we study the local description of the representation at primes dividing the level and the characteristic: this leads to a complete description of the eigenforms in the old-space. Moreover, we investigate the conductor of the twist of a representation by characters and the coefficients of the form of minimal level and weight associated to it in order to optimize the computation of the projective image. The algorithm is designed using results of Dickson, Khare\--Wintenberger and Faber on the classification, up to conjugation, of the finite subgroups of $\PGL_2(\overline{\F}_\ell)$. We characterize each possible case giving a precise description and algorithms to deal with it. In particular, we give a new approach and a construction to deal with irreducible representations with projective image isomorphic to either the symmetric group on $4$ elements or the alternating group on $4$ or $5$ elements. Show less
In this thesis, the existence and stability of pulse solutions in two-component, singularly perturbed reaction-diffusion systems is analysed using dynamical systems techniques. New phenomena in... Show moreIn this thesis, the existence and stability of pulse solutions in two-component, singularly perturbed reaction-diffusion systems is analysed using dynamical systems techniques. New phenomena in very general types of systems emerge when geometrical techniques are applied. Show less
In this thesis we prove several properties of the Galois closure of commutative algebras defined by Manjul Bhargava and Matthew Satriano. We also define some related constructions, and study their... Show moreIn this thesis we prove several properties of the Galois closure of commutative algebras defined by Manjul Bhargava and Matthew Satriano. We also define some related constructions, and study their properties. The properties of these constructions are compared with those of classical constructions in Galois theory of fields. Some examples are computed. Show less
We bound Arakelov invariants of curves in terms of their Belyi degree. We give three applications which are algorithmic, geometric and Diophantine of nature, respectively
