Images of Galois representations

Leiden Repository

Images of Galois representations

Title: Images of Galois representations
Author: Anni, Samuele
Publisher: Mathematical Institute, Faculty of Science, Leiden University
Issue Date: 2013-10-24
Keywords: Galois representations
Elliptic curves over number fields
Modular forms
Local-global problem
Degeneracy maps
Abstract: 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 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.
Description: Promotores: S.J. Edixhoven, P.Parent
With Summary in French
Thèse Présentee à l'université Bordeaux I
Chapters 1,2 and 3 are going to appear in the Journal of the London Mathematical Society
Faculty: Faculteit der Wiskunde en Natuurwetenschappen
Citation: Anni, S., 2013, Doctoral Thesis, Leiden University

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application/pdf Part I: Chapter 1 342.4Kb View/Open
application/pdf Part I: Chapter 2 407.5Kb View/Open
application/pdf Part I: Chapter 3 401.6Kb View/Open
application/pdf Part II: Chapter 4 408.9Kb View/Open
application/pdf Part II: Chapter 5 416.2Kb View/Open
application/pdf Part II: Chapter 6 448.1Kb View/Open
application/pdf Part II: Chapter 7 400.8Kb View/Open
application/pdf Part II: Chapter 8 421.3Kb View/Open
application/pdf Part II: Chapter 9 376.9Kb View/Open
application/pdf Part II: Chapter 10 403.0Kb View/Open
application/pdf Part II: Chapter 11 347.3Kb View/Open
application/pdf Appendices_Index_Bibliography 394.0Kb View/Open
application/pdf Summary in English 263.5Kb View/Open
application/pdf Summary in Dutch 263.9Kb View/Open
application/pdf Summary in French 265.3Kb View/Open
application/pdf Acknowledgements_Curriculum Vitae 158.5Kb View/Open
application/pdf Propositions 187.0Kb View/Open

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