MA 162B LECTURE NOTES: THURSDAY, JANUARY 15

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1 MA 6B LECTURE NOTES: THURSDAY, JANUARY 5 Examples of Galois Represetatios: Complex Represetatios Regular Represetatio Cosider a complex represetatio ρ : Gal ( Q/Q ) GL d (C) with fiite image If we deote L = Q ker ρ the we have a faithful map ρ : Gal(L/Q) GL d (C) where L/Q is a fiite, Galois extesio We show explicitly how to exhibit such a represetatio Fix a irreducible polyomial q(x) Q[x] of degree d havig roots x k, ad set L = Q(x, x,, x d ) as its splittig field The group of permutatios of the roots Gal(L/Q) has a caoical represetatio as d d matrices: write the d roots as d-dimesioal uit vectors, such as x = ; x = ; ; x d = Ay permutatio σ Gal(L/Q) o these roots may be represeted as a d d matrix ρ(σ) with etries beig either or Oe easily checks that ρ is a multiplicative map ie ρ(σ σ ) = ρ(σ ) ρ(σ ) This is kow as the regular represetatio of Gal(L/Q) The matrices ρ(σ) are usually called permutatio matrices Quadratic Polyomial As a explicit example, cosider q(x) = a x +b x+ c The L = Q( b 4 a c), ad the oly permutatio of iterest is b + b σ : 4 a c b ( ) b 4 a c = ρ(σ) = a a Of course, if we choose istead the basis x = ( x ) = ( ) = ρ (σ) = ( ) This correspods to settig x = b a ad x = b 4ac a Hece we ca break the Galois represetatio ρ ito a series of smaller irreducible represetatios Let s also explai this usig group rigs Cosider V = C[Gal(L/Q)] = C C σ as a complex vector space of dimesio If we cosider the elemets e = + σ, e = σ = e = e, e = e, e e =, e + e = These are idempotets, so cosider the -dimesioal subspaces V i = C[Gal(L/Q)] e i for i =, These spaces are -dimesioal because { C[Gal(L/Q)] e i = (a + b σ) ± σ } a, b C = { (a ± b) ± σ } a, b C C

2 MA 6B LECTURE NOTES: THURSDAY, JANUARY 5 Clearly V = V V, ad we have σ e = e, σ e = e so that V correspods to the trivial represetatio while V correspods to the alteratig represetatio 3 Cubic Polyomial Now cosider a irreducible cubic polyomial q(x) with roots x, x, ad x 3 The Galois group of the field L = Q (x, x, x 3 ) is geerated by two automorphisms: σ = ( ), σ 3 = ( 3) = ρ(σ ) =, ρ(σ 3 ) = If we choose a differet basis the the matrices ca be block diagoalized, but let s cosider this istead usig group rigs The vector space V = C[Gal(L/Q)] has dimesio 6, so cosider the followig idempotets: e = [] + [σ ] + [σ 3 ] 6 [] = e = [] [σ ] + [σ 3 ] where [σ ] = σ + σ σ 3 + σ 3 σ 6 e 3 = [] [σ [σ 3] 3 ] = σ 3 + σ 3 3 (The otatio [σ] deotes the sum of the elemets i the cojugacy class of σ) Oe checks that e + e + e 3 =, e i = e i while e i e j = for i j Cosider the subspaces V i = C[Gal(L/Q)] e i for i =,, 3 Clearly V = V V V 3, ad we have σ e = σ 3 e = e, σ e = e, σ 3 e = e so that V ad V are -dimesioal subspaces of V Hece V 3 must be a - dimesioal subspace of V Note that V correspods to the trivial represetatio, V correspods to the alteratig represetatio, while V 3 correspods to the irreducible -dimesioal represetatio of the symmetric group of three letters 4 -Dimesioal Represetatios Let ρ : Gal ( Q/Q ) GL d (C) be a complex represetatio with fiite image As metioed i the previous lecture, there are field extesios L/Q ad F/Q such that the followig diagram commutes: Gal (L/F ) Gal (L/Q) Gal (F/Q) ρ eρ C GL d (C) P GL d (C) Felix Klei classified all the possible fiite images i P GL (C) ie the possible types of fiite groups which have a projective -dimesioal represetatio: cyclic dihedral im ρ Gal (F/Q) A 4 (tetrahedral) S 4 (octahedral) A 5 (icosahedral)

3 MA 6B LECTURE NOTES: THURSDAY, JANUARY 5 3 Note that i these cases i order to costruct a -dimesioal Galois represetatio it suffices to fid a polyomial q(x) Q[x] such that the splittig field F/Q has Galois group as above Note that quadratic polyomials yield cyclic represetatios, while cubic polyomials yield dihedral represetatios Polyomials of degree 4 ad 5 yield the latter several types 5 Dirichlet Characters We explai how oe ca defie a -dimesioal complex Galois represetatio usig a Dirichlet character Fix a positive iteger N, ad cosider a group homomorphism χ : (Z/N Z) C This is a Dirichler character modulo N Recall that we have a isomorphism ϕ Q(ζN )/Q : (Z/N Z) Gal (Q(ζ N )/Q), a {σ a : ζ N ζ a N} This is simply the Arti map Hece the compositio gives a Galois represetatio: ρ : Gal ( Q/Q ) Gal (Q(ζ N )/Q) ϕ Q(ζN )/Q (Z/N Z) χ C Of course oe ca create a d-dimesioal represetatio by placig d Dirichlet characters alog the diagoal Oe checks that whe p N is a prime the the Frobeius automorphism σ p Gal ( Q/Q ) maps to ρ(σ p ) = χ(p) I this case the field cut out by ρ is L Q (ζ N ), ad the field cut out by ρ is F = Q I particular, if χ is a quadratic character the L is a quadratic field Note that L/Q is always fiite Examples of Galois Represetatios: l-adic Represetatios Cyclotomic Character Now we explai how oe ca defie a -dimesioal l-adic represetatio Fix a prime l, ad say N = l is a prime power We have a series of isomorphisms (Z/l Z) Gal (Q(ζ l )/Q) comig from the Arti map so this gives a isomorphism ϕ Q(ζl )/Q : Z l proj lim (Z/l Z) Gal (Q(ζ l )/Q) We have deoted Q(ζ l ) = Q(ζ l) as the field geerated by the l-power roots of uity The iverse of this map is kow as the l-adic cyclotomic character: ɛ l : Gal ( Q/Q ) Gal (Q(ζ l )/Q) ϕ Q(ζl )/Q Z l Q l As before, oe ca create a d-dimesioal represetatio by placig d cyclotomic characters or powers thereof alog the diagoal Oe checks that whe p l is a prime the the Frobeius automorphism σ p Gal ( Q/Q ) maps to ɛ l (σ p ) = p I this case the field cut out by ρ is L = Q (ζ l ), ad the field cut out by ρ is F = Q Note that L/Q is ot fiite

4 4 MA 6B LECTURE NOTES: THURSDAY, JANUARY 5 Tate Twists Give a oegative iteger m ad a l-adic represetatio ρ : G Q GL(V ), we defie V (m) = V ɛ l m as the mth Tate twist as follows We kow that for each σ G Q we have a liear trasformatio ρ(σ) : V V We the have the twist ρ ɛ l m : V (m) V (m) with actio v ɛ l (σ) m ρ(σ) v As a example, fix a prime power l, ad cosider the rig Z/l Z The elemets i the Galois group G Q act trivially because this is simply a rig Hece the l-adic vector space ( ) V = proj lim Z/l Z Zl Q l = Q l has trivial actio by G Q, so a represetatio G Q GL(V ) is trivial O the other had, we have the isomorphism Z/l Z µ l = { ζ C ζ l = } where the latter group does have a actio by G Q Hece the Tate twist is Z l () = proj lim µ l = µ l Of course, V () Q l as vector spaces, but this is isomorphism does ot preserve the Galois structure 3 Elliptic Curves A elliptic curve E is a curve of dimesio with a specified ratioal poit If the curve ca be defied over Q, the there is a lattice Λ C such that E(C) C/Λ Fix a positive iteger N ad cosider the poits of order N ie E[N] = N Λ/Λ Sice Λ Z Z, we have the (ocaoical) isomorphism E[N] (Z/N Z) (Z/N Z) I particular, whe N = l is a prime power, we have the projective limit T l (E) = proj lim E[l ] Z l Z l This is called the l-adic Tate module of E This has o Galois actio, but oe ca choose thigs so that the Galois actio yields a cotiuous -dimesioal homomorphism ρ E,l : Gal ( Q/Q ) GL (V l (E)) GL (Q l ) where we have defied V l (E) = T l (E) Zl Q l as a -dimesioal l-adic vector space This represetatio has the folowig properties: () The compositio ɛ l : Gal ( Q/Q ) ρ E,l GL (Vl (E)) Q l is just the l-adic cyclotomic character This follows directly from the properties of the l-adic Weil pairig, which is a perfect alteratig, biliear pairig T l (E) T l (E) Z l () () For almost all ratioal primes p, we have trace ρ E,l (σ p ) = p + #E(F p ), det ρ E,l (σ p ) = ɛ l (σ p ) = p The trace is usually deoted by a p (E)

5 MA 6B LECTURE NOTES: THURSDAY, JANUARY Abelia Varieties of GL -Type Let A be a abelia variety over Q of dimesio d (If d = the A is a elliptic curve) There is a lattice Λ C d such that A(C) C d /Λ As a geeralizatio of elliptic curves, we have the ocaoical isomorphism A[N] = N Λ/Λ (Z/N Z)d so that whe N = l is a prime power, we have the projective limit T l (A) = proj lim A[l ] Z d l If we choose the d-dimesioal l-adic vector space V l (A) = T l (A) Zl Q l the we have the d-dimesioal Galois represetatio ρ A,l : Gal ( Q/Q ) GL (V l (A)) GL d (Q l ) We say that A is of GL -type if the edomorphism rig over Q, amely Ed Q (A), cotais a order O i a extesio K/Q of degree d This is equivalet to sayig there is a field K of degree d = dim(a) such that K Ed Q (A) Z Q As the ame suggests, there exists a -dimesioal Galois represetatio ρ A,λ : Gal ( Q/Q ) GL (V λ (A)) GL (K λ ) We ll discuss more properties of this represetatio i the ext lecture