MA 162B LECTURE NOTES: THURSDAY, JANUARY 15
1. Examples of Galois Representations: Complex Representations 1.1. Regular Representation. Consider a complex representation ρ : Gal Q/Q −−−−→ GLd(C) ker ρ with finite image. If we denote L = Q then we have a faithful map ρ : Gal(L/Q) ,→ GLd(C) where L/Q is a finite, Galois extension. We show explic- itly how to exhibit such a representation. Fix an irreducible polynomial q(x) ∈ Q[x] of degree d having roots xk, and set L = Q(x1, x2, . . . , xd) as its splitting field. The group of permutations of the roots Gal(L/Q) has a canonical representation as d × d matrices: write the d roots as d-dimensional unit vectors, such as 1 0 0 0 1 0 x = ; x = ; ... ; x = . 1 . 2 . d . . . . 0 0 1 Any permutation σ ∈ Gal(L/Q) on these roots may be represented as a d×d matrix ρ(σ) with entries being either 0 or 1. One easily checks that ρ is a multiplicative map i.e. ρ(σ1 σ2) = ρ(σ1) ρ(σ2). This is known as the regular representation of Gal(L/Q). The matrices ρ(σ) are usually called permutation matrices.
2 1.2. Quadratic√ Polynomial. As an explicit example, consider q(x) = a x +b x+ c. Then L = Q( b2 − 4 a c), and the only permutation of interest is √ √ −b + b2 − 4 a c −b − b2 − 4 a c 1 σ : 7→ =⇒ ρ(σ) = . 2 a 2 a 1 Of course, if we choose instead the basis 1 1 1 1 1 x 0 = x 0 = =⇒ ρ0(σ) = . 1 2 1 2 2 −1 −1 √ 0 b 0 b2−4ac This corresponds to setting x1 = − 2a and x2 = 2a . Hence we can break the Galois representation ρ into a series of smaller irreducible representations. Let’s also explain this using group rings. Consider V = C[Gal(L/Q)] = C ⊕ C σ as a complex vector space of dimension 2. If we consider the elements 1 + σ 1 − σ e = , e = =⇒ e2 = e , e2 = e , e e = 0, e + e = 1. 1 2 2 2 1 1 2 2 1 2 1 2 These are idempotents, so consider the 1-dimensional subspaces Vi = C[Gal(L/Q)] ei for i = 1, 2. These spaces are 1-dimensional because 1 ± σ 1 ± σ C[Gal(L/Q)] ei = (a + b σ) a, b ∈ C = (a ± b) a, b ∈ C ' C. 2 2 1 2 MA 162B LECTURE NOTES: THURSDAY, JANUARY 15
Clearly V = V1 ⊕ V2, and we have
σ e1 = e1, σ e2 = −e2 so that V1 corresponds to the trivial representation while V2 corresponds to the alternating representation.
1.3. Cubic Polynomial. Now consider an irreducible cubic polynomial q(x) with roots x1, x2, and x3. The Galois group of the field L = Q (x1, x2, x3) is generated by two automorphisms: 0 1 0 0 0 1 σ2 = (1 2), σ3 = (1 2 3) =⇒ ρ(σ2) = 1 0 0 , ρ(σ3) = 1 0 0 . 0 0 1 0 1 0 If we choose a different basis then the matrices can be block diagonalized, but let’s consider this instead using group rings. The vector space V = C[Gal(L/Q)] has dimension 6, so consider the following idempotents: [1] + [σ ] + [σ ] e = 2 3 1 6 [1] = 1 [1] − [σ2] + [σ3] e2 = where [σ2] = σ2 + σ2 σ3 + σ3 σ2 6 −1 2[1] − [σ ] [σ3] = σ3 + σ3 e = 3 3 3 (The notation [σ] denotes the sum of the elements in the conjugacy class of σ.) 2 One checks that e1 + e2 + e3 = 1, ei = ei while ei ej = 0 for i 6= j. Consider the subspaces Vi = C[Gal(L/Q)] ei for i = 1, 2, 3. Clearly V = V1 ⊕ V2 ⊕ V3, and we have
σ2 e1 = σ3 e1 = e1, σ2 e2 = −e2, σ3 e2 = e2 so that V1 and V2 are 1-dimensional subspaces of V . Hence V3 must be a 2- dimensional subspace of V . Note that V1 corresponds to the trivial representa- tion, V2 corresponds to the alternating representation, while V3 corresponds to the irreducible 2-dimensional representation of the symmetric group of three letters. 1.4. 2-Dimensional Representations. Let ρ : Gal Q/Q → GLd(C) be a com- plex representation with finite image. As mentioned in the previous lecture, there are field extensions L/Q and F/Q such that the following diagram commutes: 1 −−−−→ Gal (L/F ) −−−−→ Gal (L/Q) −−−−→ Gal (F/Q) −−−−→ 1 ρ y y yρe × 1 −−−−→ C −−−−→ GLd(C) −−−−→ P GLd(C) −−−−→ 1 Felix Klein classified all the possible finite images in P GL2(C) i.e. the possible types of finite groups which have a projective 2-dimensional representation: cyclic dihedral im ρ ' Gal (F/Q) ' A4 (tetrahedral) S4 (octahedral) A5 (icosahedral) MA 162B LECTURE NOTES: THURSDAY, JANUARY 15 3
Note that in these cases in order to construct a 2-dimensional Galois representation it suffices to find a polynomial q(x) ∈ Q[x] such that the splitting field F/Q has Galois group as above. Note that quadratic polynomials yield cyclic representations, while cubic polynomials yield dihedral representations. Polynomials of degree 4 and 5 yield the latter several types.
1.5. Dirichlet Characters. We explain how one can define a 1-dimensional com- plex Galois representation using a Dirichlet character. Fix a positive integer N, and consider a group homomorphism
× χ :(Z/N Z) −−−−→ C×. This is a Dirichler character modulo N. Recall that we have an isomorphism
ϕ :( /N )× −−−−→∼ Gal ( (ζ )/ ), a 7→ {σ : ζ 7→ ζa } . Q(ζN )/Q Z Z Q N Q a N N This is simply the Artin map. Hence the composition gives a Galois representation:
ϕ −1 Q(ζN )/Q × χ × ρ : Gal Q/Q −−−−→ Gal (Q(ζN )/Q) −−−−−−−→ (Z/N Z) −−−−→ C . Of course one can create a d-dimensional representation by placing d Dirichlet characters along the diagonal. One checks that when p - N is a prime then the Frobenius automorphism σp ∈ Gal Q/Q maps to ρ(σp) = χ(p). In this case the field cut out by ρ is L ⊆ Q (ζN ), and the field cut out by ρe is F = Q. In particular, if χ is a quadratic character then L is a quadratic field. Note that L/Q is always finite.
2. Examples of Galois Representations: `-adic Representations 2.1. Cyclotomic Character. Now we explain how one can define a 1-dimensional `-adic representation. Fix a prime `, and say N = `n is a prime power. We have a series of isomorphisms
n × ∼ (Z/` Z) −−−−→ Gal (Q(ζ`n )/Q) coming from the Artin map so this gives an isomorphism
× ∼ n × ∼ ϕ : −−−−→ proj lim ( /` ) −−−−→ Gal ( (ζ ∞ )/ ) . Q(ζ`∞ )/Q Z` n Z Z Q ` Q S We have denoted Q(ζ`∞ ) = n≥1 Q(ζ`n ) as the field generated by the `-power roots of unity. The inverse of this map is known as the `-adic cyclotomic character:
ϕ −1 Q(ζ`∞ )/Q × × ` : Gal Q/Q −−−−→ Gal (Q(ζ`∞ )/Q) −−−−−−−−→ Z` −−−−→ Q` . As before, one can create a d-dimensional representation by placing d cyclotomic characters – or powers thereof – along the diagonal. One checks that when p 6= ` is a prime then the Frobenius automorphism σp ∈ Gal Q/Q maps to `(σp) = p. In this case the field cut out by ρ is L = Q (ζ`∞ ), and the field cut out by ρe is F = Q. Note that L/Q is not finite. 4 MA 162B LECTURE NOTES: THURSDAY, JANUARY 15
2.2. Tate Twists. Given a nonnegative integer m and an `-adic representation m ρ : GQ → GL(V ), we define V (m) = V ⊗ ` as the mth Tate twist as follows. We know that for each σ ∈ GQ we have a linear transformation ρ(σ): V → V . We m then have the twist ρ ⊗ ` : V (m) → V (m) with action m v 7→ `(σ) ρ(σ) · v. As an example, fix a prime power `n, and consider the ring Z/`n Z. The elements in the Galois group GQ act trivially because this is simply a ring. Hence the `-adic vector space n V = proj lim Z/` Z ⊗Z` Q` = Q` n has trivial action by GQ, so a representation GQ → GL(V ) is trivial. On the other hand, we have the isomorphism
n ∼ × `n Z/` Z −−−−→ µ`n = ζ ∈ C | ζ = 1 where the latter group does have an action by GQ. Hence the Tate twist is Z`(1) = proj limn µ`n = µ`∞ . Of course, V (1) ' Q` as vector spaces, but this is isomorphism does not preserve the Galois structure.
2.3. Elliptic Curves. An elliptic curve E is a curve of dimension 1 with a specified rational point. If the curve can be defined over Q, then there is a lattice Λ ⊆ C such that E(C) ' C/Λ. Fix a positive integer N and consider the points of order 1 N i.e. E[N] = N Λ/Λ. Since Λ ' Z × Z, we have the (noncanonical) isomorphism E[N] ' (Z/N Z) × (Z/N Z) . In particular, when N = `n is a prime power, we have the projective limit n T`(E) = proj lim E[` ] ' Z` × Z`. n This is called the `-adic Tate module of E. This has no Galois action, but one can choose things so that the Galois action yields a continuous 2-dimensional homo- morphism ρE,` : Gal Q/Q −−−−→ GL (V`(E)) −−−−→ GL2(Q`) where we have defined V`(E) = T`(E)⊗Z` Q` as a 2-dimensional `-adic vector space. This representation has the folowing properties: (1) The composition
ρE,` × ` : Gal Q/Q −−−−→ GL (V`(E)) −−−−→ Q` is just the `-adic cyclotomic character. This follows directly from the prop- erties of the `-adic Weil pairing, which is a perfect alternating, bilinear pairing
T`(E) × T`(E) −−−−→ Z`(1). (2) For almost all rational primes p, we have
trace ρE,`(σp) = p + 1 − #E(Fp), det ρE,`(σp) = `(σp) = p.
The trace is usually denoted by ap(E). MA 162B LECTURE NOTES: THURSDAY, JANUARY 15 5
2.4. Abelian Varieties of GL2-Type. Let A be an abelian variety over Q of dimension d. (If d = 1 then A is an elliptic curve.) There is a lattice Λ ⊆ Cd such that A(C) ' Cd/Λ. As a generalization of elliptic curves, we have the noncanonical isomorphism 1 A[N] = Λ/Λ ' ( /N )2d . N Z Z so that when N = `n is a prime power, we have the projective limit n 2d T`(A) = proj lim A[` ] ' Z` . n
If we choose the 2d-dimensional `-adic vector space V`(A) = T`(A) ⊗Z` Q` then we have the 2d-dimensional Galois representation ρA,` : Gal Q/Q −−−−→ GL (V`(A)) −−−−→ GL2d(Q`).
We say that A is of GL2-type if the endomorphism ring over Q, namely EndQ(A), contains an order O in an extension K/Q of degree d. This is equivalent to saying there is a field K of degree d = dim(A) such that K,→ EndQ(A) ⊗Z Q. As the name suggests, there exists a 2-dimensional Galois representation ρA,λ : Gal Q/Q −−−−→ GL (Vλ(A)) −−−−→ GL2(Kλ). We’ll discuss more properties of this representation in the next lecture.