Abelian Varieties of GLn-type and Galois Representations
[eng] In this thesis, the artimetic properties of abelian manifolds are considered in relation to their algebras of endomorphisms. More precisely, we study the good reductions of an abelian manifold defined over a field of numbers, as well as those of associated Galois representations. We also give...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/227592 |
| Acceso en línea: | https://hdl.handle.net/2445/227592 http://hdl.handle.net/10803/696831 |
| Access Level: | acceso abierto |
| Palabra clave: | Teoria algebraica de nombres Varietats abelianes Teoria de Galois Algebraic number theory Abelian varieties Galois theory |
| Sumario: | [eng] In this thesis, the artimetic properties of abelian manifolds are considered in relation to their algebras of endomorphisms. More precisely, we study the good reductions of an abelian manifold defined over a field of numbers, as well as those of associated Galois representations. We also give some results of modularity of abelian manifolds with respect to modular Siegel forms. Chapter 1 gives some definitions and preliminary results on simple algebras and abelian manifolds. The work itself begins with Chapter 2, where we study the immersions of simple algebras. We have placed special emphasis on characterizing the existence of an immersion between two simple algebras. We give a specialization of a Chia-Fu Yu criterion for algebras on global and local bodies, which plays an important role in Chapters 3, 5 and 7. Chapter 3 studies the properties of abelian manifolds defined on finite fields under the hypothesis that the algebra of endomorphisms is noncommutative. We focus on classifying the 4-abelian manifolds with quaternionic multiplication. We prove the following result: an abelian manifold over a field of numbers with noncommutative endomorphisms is a non-simple modulus all first out of a finite set. This statement generalizes the analogous result for the so-called "false elliptical curves", and makes one of the directions of the Murty and Patankar Conjecture more precise. On the other hand, we also give an example of a 4-abelian manifold with exactly two geometrically simple reductions. In Chapter 4, we begin the description of Galois representations associated with abelian manifolds with some non-integer endomorphism. We describe the irreducible components of the Tate modulus in terms of the algebra of endomorphisms, and prove that Galois representations take values in a form of the algebraic group GLn. We also explain the nature of Weil's pairing in these irreducible components, which depends on the Albert type of the algebra of endomorphisms. Chapter 5 is based on a joint work with F. Fité and X. Guitart. We define the notion of genuinely GLn-type abelian manifold, generalizing the GL2-type abelian manifolds without potential complex multiplication considered by Ribet. The system of representations associated with these manifolds has the property of being absolutely irreducible when making a change of basis to a finite extension. We give a theory of building blocks for these varieties. Under a certain technical hypothesis, we also define the inner twists and the character of an abelian variety, called Nebentipus. We characterize the manifolds with symplectic or orthogonal Galois representation, which we genuinely call GSpn or GOn type, respectively. In this way, we extend the class of abelian varieties with such representations given earlier by Banaszak, Gajda and Krasoń. Finally, we show that a genuinely GL4 type abelian manifold is Siegel's modular if and only if it is genuinely GSp4 type. In Chapter 6 we construct a GL4 family of building blocks. These are given by the Jacobian women of certain curves of genus 2 with a Richelot isogeny in their Galois conjugates. Under certain conditions, Weil's constraint gives us examples of 4-abelian manifolds genuinely of type GSp4. The family includes examples of images of Galois representations in GSp4 and non-trivial Nebentype. Chapter 7 is based on a joint work with A. Pacetti. It deals with Galois representations associated with abelian k-manifolds. One of the main points is that we do not assume that all endomorphisms of the manifold are defined on the base field. We give a procedure to construct a representation of the absolute Galois group of k associated with a k-abelian manifold. In addition, we give some results on the induced Weil pairing to these representations. As an application, we demonstrate that abelian surfaces over Q with potential quaternionic multiplication are Siegel modulars. |
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