Galois Representations¶
Summary
Galois representations translate the symmetries of algebraic equations into the language of linear algebra: homomorphisms from Galois groups into matrix groups. Every elliptic curve yields a natural 2-dimensional representation, and Wiles' proof establishes modularity at precisely this level.
Prerequisites¶
- Groups and Symmetry – homomorphisms, normal subgroups, quotient groups
- Rings and Fields – field extensions, finite fields \(\mathbb{F}_p\), \(p\)-adic numbers \(\mathbb{Z}_p\)
- Galois Theory – Galois groups, Frobenius elements, ramification
- Elliptic Curves – group structure, torsion points, reduction modulo \(p\)
1. From Galois Groups to Matrices¶
What is a representation?¶
A representation of a group \(G\) is a homomorphism
where \(\text{GL}_n(K)\) is the group of invertible \(n \times n\) matrices over a field (or ring) \(K\). The representation "translates" the abstract group structure into the concrete language of linear algebra.
Why representations?¶
Galois groups – in particular the absolute Galois group \(G_{\mathbb{Q}}\) – are infinite and highly complex. Working with them directly is often impossible. Representations provide a tractable tool: instead of studying the group itself, one studies its action on vector spaces.
The central insight of Wiles' proof is: modularity of an elliptic curve can be formulated as a property of its Galois representation – and proved at that level.
2. The Absolute Galois Group¶
Definition¶
The absolute Galois group of \(\mathbb{Q}\) is
where \(\overline{\mathbb{Q}}\) is the algebraic closure of \(\mathbb{Q}\) (the set of all algebraic numbers). \(G_{\mathbb{Q}}\) consists of all field automorphisms of \(\overline{\mathbb{Q}}\) that fix \(\mathbb{Q}\) element-wise.
Profinite structure¶
\(G_{\mathbb{Q}}\) is a profinite group – the inverse limit of all finite Galois groups \(\text{Gal}(K/\mathbb{Q})\):
It is uncountable and carries a natural topology (the Krull topology), under which it is compact and totally disconnected. Every open subgroup has finite index.
Decomposition groups and Frobenius¶
For every prime \(p\) there is a decomposition group \(D_p \subset G_{\mathbb{Q}}\) and an inertia group \(I_p \subset D_p\). The Frobenius element
is the "signature" of the prime \(p\) in \(G_{\mathbb{Q}}\). It acts on reductions modulo \(p\) as \(x \mapsto x^p\).
For a representation \(\rho: G_{\mathbb{Q}} \to \text{GL}_n(K)\), one can compute the trace of the Frobenius
– and this number encodes the arithmetic information of the representation at \(p\).
3. \(p\)-Torsion of Elliptic Curves¶
The Galois module \(E[p]\)¶
Let \(E\) be an elliptic curve over \(\mathbb{Q}\) and \(p\) a prime. The \(p\)-torsion points are:
where \(\mathcal{O}\) is the point at infinity (the identity element of the group structure).
As an abelian group, \(E[p]\) is isomorphic to
It is a two-dimensional vector space over \(\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}\).
The Galois action¶
The points in \(E[p]\) have coordinates in \(\overline{\mathbb{Q}}\), and the absolute Galois group acts on them through its action on the coordinates:
This action respects the group structure: \(\sigma(P + Q) = \sigma(P) + \sigma(Q)\). Thus \(E[p]\) is a Galois module – an \(\mathbb{F}_p\)-vector space with a linear action of \(G_{\mathbb{Q}}\).
4. The Residual Representation¶
Definition¶
Choosing a basis \(\{P_1, P_2\}\) of \(E[p]\) over \(\mathbb{F}_p\), the Galois action is described by a matrix:
This defines the residual Galois representation:
It is a continuous group homomorphism (with respect to the Krull topology on \(G_{\mathbb{Q}}\) and the discrete topology on \(\text{GL}_2(\mathbb{F}_p)\)). Up to conjugation, it is independent of the choice of basis.
Irreducibility¶
The representation \(\bar{\rho}_{E,p}\) is called irreducible if \(E[p]\) has no non-trivial \(G_{\mathbb{Q}}\)-invariant subgroup (i.e., no \(\mathbb{F}_p\)-subspace stable under the Galois action).
Irreducibility is a crucial hypothesis for Wiles' proof. For the Frey curve, \(\bar{\rho}_{E,p}\) is irreducible for \(p \geq 5\) – a consequence of Mazur's groundbreaking work on isogenous elliptic curves.
Ramification and conductor¶
The representation \(\bar{\rho}_{E,p}\) is unramified at a prime \(q \neq p\) if the inertia group \(I_q\) acts trivially on \(E[p]\). This happens precisely when \(E\) has good reduction at \(q\).
The Artin conductor \(N(\bar{\rho}_{E,p})\) measures the ramification of the representation and is a divisor of the conductor \(N_E\) of the curve.
5. The \(p\)-adic Representation¶
The Tate module¶
Instead of considering only the \(p\)-torsion, one can capture all \(p^n\)-torsion points simultaneously. The Tate module is the inverse limit:
where the transition maps are the multiplication-by-\(p\) maps \(E[p^{n+1}] \to E[p^n]\).
As a \(\mathbb{Z}_p\)-module, \(T_p(E)\) is free of rank 2:
The \(p\)-adic representation¶
The Galois action on \(T_p(E)\) yields the \(p\)-adic Galois representation:
This is a continuous representation with respect to the \(p\)-adic topology. It is a "lift" of the residual representation: reduction modulo \(p\) gives
The connection to \(L\)-series¶
The \(p\)-adic representation encodes the arithmetic information of the curve completely:
for every prime \(q\) of good reduction (with \(q \neq p\)). Thus the representation determines the \(L\)-series \(L(E, s)\) – and vice versa.
6. Modular Representations¶
Representations from modular forms¶
Not only elliptic curves yield Galois representations – modular forms do as well. For every newform \(f\) of weight 2 and level \(N\), Eichler and Shimura constructed an associated Galois representation:
with the property
where \(b_q\) is the \(q\)-th Fourier coefficient of \(f\).
Modularity as a property of representations¶
Now the connection becomes clear: an elliptic curve \(E\) is modular if and only if its Galois representation \(\rho_{E,p}\) agrees with the representation \(\rho_f\) of a newform \(f\):
The modularity conjecture (TSC) thus becomes a statement about Galois representations: every representation coming from an elliptic curve also comes from a modular form.
Residual modularity¶
Analogously, \(\bar{\rho}_{E,p}\) is called modular if it is isomorphic to the reduction modulo \(p\) of a modular representation:
for some newform \(f\). This is a weaker condition than full modularity – and precisely the starting point of Wiles' proof strategy.
7. Wiles' Strategy¶
The two steps¶
Wiles' proof of the modularity of semistable elliptic curves breaks into two major steps:
Step 1: Establish residual modularity. One must prove that \(\bar{\rho}_{E,p}\) is modular – i.e., comes from a newform. For \(p = 3\), this follows from a famous result of Langlands and Tunnell: since \(\text{GL}_2(\mathbb{F}_3)\) is solvable, Langlands' base change techniques can be applied. For \(p = 5\), Wiles uses the so-called 3-5 switch (see Article 07).
Step 2: "Lift" from residual to full modularity. This is the heart of the proof: given that \(\bar{\rho}_{E,p}\) is modular, one must show that the full representation \(\rho_{E,p}\) is also modular. For this, Wiles introduces the language of deformation theory (see Article 04).
Why representations are the right framework¶
The reformulation of the TSC in the language of Galois representations has decisive advantages:
- Algebraic tools: Representation theory, cohomology, and commutative algebra become applicable.
- Local-global principles: One can study representations "locally" (at each prime) and "globally" (over \(\mathbb{Q}\)).
- Deformations: The residual representation \(\bar{\rho}\) has a "space of all lifts" – the deformation space, which can be analysed with algebraic methods.
- Reduction: One can prove modularity step by step – first residually, then fully.
This perspective – modularity as a property of Galois representations – was Wiles' decisive conceptual innovation and has profoundly shaped number theory since 1995.
Outlook¶
With Galois representations, we have the language in which Wiles' proof is formulated. The next step is the central question: how does one show that a residually modular representation can be lifted to a fully modular representation?
| Article | Topic |
|---|---|
| 04 – Deformation Theory | The universal deformation ring \(R\) and Mazur's theory |
| 05 – R = T | Why \(R = T\) proves modularity |
Sources¶
- Andrew Wiles: Modular elliptic curves and Fermat's Last Theorem, Annals of Mathematics 141 (1995), §1
- Nigel Boston: The Proof of Fermat's Last Theorem (2003), Chapter 10 – Galois representations
- Jean-Pierre Serre: Abelian \(\ell\)-adic representations and elliptic curves, W.A. Benjamin (1968) – Classical reference for \(\ell\)-adic representations
- Barry Mazur: Deforming Galois representations, in: Galois Groups over \(\mathbb{Q}\), MSRI Publications 16 (1989) – Foundational for the deformation approach