Lifting Galois Representations: A Detailed Guide

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Hey everyone! Today, we're diving deep into the fascinating world of Galois representations and the question of lifting them. This is a topic that sits at the intersection of abstract algebra, representation theory, Galois theory, and number theory, so buckle up – it's going to be a fun ride! We'll start by setting the stage with some background and then delve into the specifics of the lifting problem.

Understanding Galois Representations

Let's begin by understanding Galois representations. A Galois representation is essentially a way to "visualize" a Galois group, which encodes the symmetries of field extensions, as a group of matrices. More formally, given a Galois extension K/FK/F with Galois group G=Gal(K/F)G = \text{Gal}(K/F), a Galois representation is a homomorphism:

ρ:Gβ†’GLn(L)\rho: G \to \text{GL}_n(L)

where LL is a field (often a pp-adic field Qp\mathbb{Q}_p or a finite field Fq\mathbb{F}_q) and GLn(L)\text{GL}_n(L) is the general linear group of nΓ—nn \times n invertible matrices with entries in LL. The dimension nn is called the dimension (or degree) of the representation. Galois representations are incredibly powerful tools because they allow us to use linear algebra to study Galois groups, which can be quite abstract. Think of it as translating a difficult problem in group theory into a (hopefully) more manageable problem in linear algebra.

Now, why are these representations so important? Well, they pop up all over the place in number theory and arithmetic geometry. For example, the Γ©tale cohomology groups of algebraic varieties carry natural Galois actions, giving rise to Galois representations. Moreover, the study of modular forms and elliptic curves is deeply intertwined with Galois representations. The celebrated modularity theorem (formerly known as the Taniyama-Shimura conjecture) states that every elliptic curve over Q\mathbb{Q} is modular, which roughly means that its associated Galois representation comes from a modular form. This theorem, a cornerstone of modern number theory, highlights the profound connection between Galois representations and other areas of mathematics.

Why do we care about GLn/Β±En\text{GL}_n/{\pm E_n}? When studying Galois representations, it's sometimes useful to consider projectivized representations, where we quotient out by scalar matrices. In the specific case mentioned, we're looking at GLn/Β±En\text{GL}_n/{\pm E_n}, which means we're identifying a matrix AA with its negative βˆ’A-A. This can arise naturally in certain contexts, such as when dealing with projective representations or when the specific scalar multiple of a matrix is not important for the application at hand. For instance, in some geometric situations, the action of a matrix and its negative might have the same geometric effect. Also, the representation may naturally take values in GLn/Β±En\text{GL}_n/{\pm E_n} due to the structure of the problem.

The Lifting Problem: Overcoming Obstacles

The lifting problem, in the context of Galois representations, asks whether we can "lift" a representation with values in a quotient group to a representation with values in the original group. More precisely, suppose we have a Galois representation:

ρ‾:GalQpβ†’(GLn/Β±En)(Qp)\overline{\rho}: \text{Gal}_{\mathbb{Q}_p} \to (\text{GL}_n/{\pm E_n})(\mathbb{Q}_p)

The lifting problem asks: Does there exist a Galois representation

ρ:GalQpβ†’GLn(Qp)\rho: \text{Gal}_{\mathbb{Q}_p} \to \text{GL}_n(\mathbb{Q}_p)

such that the composition of ρ\rho with the projection map GLn(Qp)β†’(GLn/Β±En)(Qp)\text{GL}_n(\mathbb{Q}_p) \to (\text{GL}_n/{\pm E_n})(\mathbb{Q}_p) gives us ρ‾\overline{\rho}? In other words, can we find a "true" representation ρ\rho that, when we project down to (GLn/Β±En)(Qp)(\text{GL}_n/{\pm E_n})(\mathbb{Q}_p), gives us the representation ρ‾\overline{\rho} that we started with? This is not always possible, and the obstruction to lifting lies in cohomology groups. To find an obstruction we need to find the inflation-restriction sequence.

Why is lifting difficult? Lifting is not always possible because of topological and algebraic obstructions. Galois groups like GalQp\text{Gal}_{\mathbb{Q}_p} have complicated topological structures, and the algebraic structure of GLn(Qp)\text{GL}_n(\mathbb{Q}_p) and its quotients can create difficulties. The existence of a lift depends on the specific representation ρ‾\overline{\rho} and the structure of the groups involved. The lifting problem can often be formulated in terms of group cohomology, where the obstruction to lifting is an element in a certain cohomology group. If this element is non-zero, then a lift does not exist.

Let's try to build a p-adic Galois representation step by step. For instance, consider a scenario where we have an elliptic curve EE defined over Qp\mathbb{Q}_p. The Tate module Tp(E)T_p(E) is a free Zp\mathbb{Z}_p-module of rank 2, and the Galois group GalQp\text{Gal}_{\mathbb{Q}_p} acts on Tp(E)T_p(E). This action gives us a Galois representation:

ρE:GalQpβ†’GL2(Zp)\rho_E: \text{Gal}_{\mathbb{Q}_p} \to \text{GL}_2(\mathbb{Z}_p)

This is a pp-adic Galois representation because its image lies in GL2(Zp)\text{GL}_2(\mathbb{Z}_p), which is a subgroup of GL2(Qp)\text{GL}_2(\mathbb{Q}_p). This representation encodes a lot of information about the arithmetic of the elliptic curve EE. For example, the reduction of ρE\rho_E modulo pp gives us a representation into GL2(Fp)\text{GL}_2(\mathbb{F}_p), which is related to the number of points on the reduced elliptic curve over the finite field Fp\mathbb{F}_p.

The Inertia Subgroup IpI_p: A Key Player

The inertia subgroup IpI_p of GalQp\text{Gal}_{\mathbb{Q}_p} plays a crucial role in understanding the local behavior of Galois representations. The inertia subgroup essentially measures the ramification of the extension Qpunr/Qp\mathbb{Q}_p^{unr}/\mathbb{Q}_p, where Qpunr\mathbb{Q}_p^{unr} is the maximal unramified extension of Qp\mathbb{Q}_p. In other words, IpI_p consists of the elements of GalQp\text{Gal}_{\mathbb{Q}_p} that act trivially on the residue field. Understanding the restriction of a Galois representation to the inertia subgroup is essential for studying its local properties.

Why is IpI_p so important? The inertia subgroup helps us understand how primes split in field extensions. It tells us whether a prime pp remains prime, splits into multiple primes, or becomes ramified in the extension. The structure of IpI_p is intimately related to the ramification index and the residue degree of the extension. So, when studying a pp-adic Galois representation, we often focus on its restriction to IpI_p to gain insight into its local behavior. The wild inertia subgroup PP is the Sylow pro-pp subgroup of II, and is thus normal in II. The quotient I/PI/P is isomorphic to ∏lβ‰ pZl\prod_{l \neq p} \mathbb{Z}_l.

Now, you might be asking: Guys, what if we're dealing with Q\mathbb{Q} instead of Qp\mathbb{Q}_p? Well, the Galois group GalQ\text{Gal}_{\mathbb{Q}} is much more complicated than GalQp\text{Gal}_{\mathbb{Q}_p}. However, we can still use local information to study global Galois representations. For each prime pp, we can consider the decomposition group DpβŠ‚GalQD_p \subset \text{Gal}_{\mathbb{Q}}, which is the subgroup of elements that fix a prime ideal above pp in a fixed algebraic closure of Q\mathbb{Q}. The decomposition group DpD_p is isomorphic to GalQp\text{Gal}_{\mathbb{Q}_p}, and we can use the local Galois representation ρ∣Dp\rho|_{D_p} to study the global representation ρ\rho.

Lifting with Inertia: A Tricky Task

Now, let's get back to the original question. Suppose we have a pp-adic Galois representation:

ρ:GalQpβ†’(GLn/Β±En)(Qp)\rho: \text{Gal}_{\mathbb{Q}_p} \to (\text{GL}_n/{\pm E_n})(\mathbb{Q}_p)

The question is: What can we say about the possibility of lifting ρ\rho to a representation with values in GLn(Qp)\text{GL}_n(\mathbb{Q}_p), especially considering the role of the inertia subgroup IpI_p? The answer, as you might expect, is not always straightforward. The existence of a lift depends on the specific representation ρ\rho and the properties of its restriction to IpI_p. For example, if the restriction of ρ\rho to IpI_p is trivial, then lifting might be easier. However, if the restriction to IpI_p is non-trivial and has a complicated structure, then lifting can be much more challenging.

To summarize: The lift of a Galois representation from GalQp\text{Gal}_{\mathbb{Q}_p} to (GLn/Β±En)(Qp)\left(\text{GL}_n/{\pm E_n}\right)(\mathbb{Q}_p) to GLn(Qp)\text{GL}_n({\mathbb{Q}_p}) is related to the inertia subgroup IpI_p, and is a complex question. It depends on the specific representation and structure of the group. The obstruction of lifting lies in cohomology groups, and the inertia subgroup is a key player.

Conclusion

So, there you have it – a glimpse into the world of lifting Galois representations. This is a rich and active area of research, with many open questions and connections to other parts of mathematics. Whether you're interested in abstract algebra, representation theory, number theory, or arithmetic geometry, the study of Galois representations offers a fascinating and rewarding journey. Keep exploring, keep questioning, and keep pushing the boundaries of our mathematical understanding!