This dissertation is a collection of four research articles devoted tothe study of Kummer theory for commutative algebraic groups. In numbertheory, Kummer theory refers to the study of field... Show moreThis dissertation is a collection of four research articles devoted tothe study of Kummer theory for commutative algebraic groups. In numbertheory, Kummer theory refers to the study of field extensions generatedby n-th roots of some base field. Its generalization to commutativealgebraic groups involves fields generated by the division points of afixed algebraic group, such as an elliptic curve or a higher dimensionalabelian variety. Of particular interest in this dissertation is the degreeof such field extensions. In the first two chapter, classical results forelliptic curves are improved by providing explicitly computable bounds anduniform and explicit bounds over the field of rational numbers. In thelast two chapters a general framework for the study of similar problemsis developed. Show less
This thesis deals primarily with the study of Galois representations attached to torsion points on elliptic curves. In the first chapter we consider the problem of determining the image of the... Show moreThis thesis deals primarily with the study of Galois representations attached to torsion points on elliptic curves. In the first chapter we consider the problem of determining the image of the Galois representation flE attached to a non-CM elliptic curve over the rational number field Q. We give a deterministic algorithm that determines the image of flE as a subgroup of GL2(__Z), where the output is given as an integer m together with a finite subgroup G(m) _ GL2(Z/mZ). The image of flE is then the subgroup of all elements of GL2(__Z) whose reduction modulo m belongs to G(m). In the second part we develop a method using character sums that uses the image of flE to describe densities of sets of primes p for which _E (Fp) has certain prescribed properties. If E is an elliptic curve over Q, then it follows by work of Serre and Hooley that, under the assumption of the Generalized Riemann Hypothesis, the density of primes p such that the group of Fprational points of the reduced curve _E(Fp) is cyclic can be written as an infinite product r ___ of local factors ___ reflecting the degree of the _-torsion fields, multiplied by a factor that corrects for the entanglements between the various torsion fields. We show that this correction factor can be interpreted as a character sum, and the resulting description allows us to easily determine nonvanishing criteria for it. We apply our character sum method to a variety of other settings. Among these, we consider the aforementioned problem with the additional condition that the primes p lie in a given arithmetic progression. We also study the conjectural constants appearing in Koblitz__s conjecture, a conjecture which relates to the density of primes p for which the cardinality of the group of Fp-points of E is prime. The unifying theme in all these settings is that the constants we are interested in are completely determined by the image of flE. The final chapter deals with the classification of non-Serre curves. An elliptic curve over Q is a Serre curve if its attached Galois representation is as large as possible, and it is known that most elliptic curves over Q are of this type. We exhibit a modular curve of level 6 that completes a set of modular curves which parametrise non-Serre curves. This modular curve also gives an infinite family of elliptic curves with non-abelian "entanglement fields". Exhibiting such a family is naturally motivated by questions arising in the previous chapter regarding the classification of elliptic curves to which we can apply the character sum method described above. Show less
The 1st chapter is of an introductory nature. It discusses the basic invariants of algebraic number fields and asks whether or to which extent such invariants characterize the number field. It... Show moreThe 1st chapter is of an introductory nature. It discusses the basic invariants of algebraic number fields and asks whether or to which extent such invariants characterize the number field. It surveys some of the older results in the area before focusing on the case of absolute abelian Galois groups that occurs center stage in the next two chapters, and on a question for elliptic curves that can be attacked with the techniques from those two chapters. Chapters 2 & 3 include also the non-trivial proof of the fact that the key criterion to find imaginary quadratic fields with `minimal__ absolute abelian Galois groups can also be used to find Galois groups that are "provably" non-minimal. Chapter 4 moves in a different direction. It explicitly computes adelic point groups of elliptic curves over the field of rational numbers, and shows that the outcome can be made as explicit as in the case of the minimal absolute abelian Galois groups, and, in an even stronger sense than in that case, barely depends on the particular elliptic curve. The results obtained do generalize to arbitrary number fields, and it is this generalization that we plan to deal with in a forthcoming paper. 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
This thesis is about arithmetic, analytic and algorithmic aspects of modular curves and modular forms. The arithmetic and analytic aspects are linked by the viewpoint that modular curves are... Show moreThis thesis is about arithmetic, analytic and algorithmic aspects of modular curves and modular forms. The arithmetic and analytic aspects are linked by the viewpoint that modular curves are examples of arithmetic surfaces. Therefore, Arakelov theory (intersection theory on arithmetic surfaces) occupies a prominent place in this thesis. Apart from this, a substantial part of it is devoted to studying modular curves over finite fields, and their Jacobian varieties, from an algorithmic viewpoint. The end product of this thesis is an algorithm for computing modular Galois representations. These are certain two-dimensional representations of the absolute Galois group of the rational numbers that are attached to Hecke eigenforms over finite fields. The running time of our algorithm is (under minor restrictions) polynomial in the length of the input. This main result is a generalisation of that of work of Jean-Marc Couveignes, Bas Edixhoven et al. Several intermediate results are developed in sufficient generality to make them of interest to the study of modular curves and modular forms in a wider sense. Show less
