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GALOISREPRESENTATIONS

Rihard Taylor

http://www.math.harvard.edu/~rtaylor

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RATIONAL NUMBERS:Q = fa=b : a; b integers; b 6= 0g

FUNCTION FIELD:(Z=pZ)(X) rational funtions in one vari-able over a �nite �eld.LOCAL FIELDS:e.g. Qp

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+ +The Riemann Zeta Funtion:

�(s) = P1n=1 n�s= Qp(1� p�s)�1 (Re s > 1)

Riemann (1860): �(s) has analyti ontin-uation to C exept for one simple pole ats = 1 and satis�es a funtional equation��s=2�(s=2)�(s) = �(s�1)=2�((1�s)=2)�(1�s)Herbrand(1932)-Ribet(1976): If n is a pos-itive even integer then

�(1� n) 2 Qand

pj�(1� n)() Cl(Z[e2�i=p℄)[p℄1�n 6= (0)+ 3

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Ellipti Curves:

E : y2 = x3+ ax+ bwhere a; b 2 Q and 4a3 6= 27b2.Mordell (1921):

E(Q) �= ZrE � �nite abelian groupwhere rE 2 Z�0 is the RANK of E.

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The L-funtion:L(E; s) =Yp (1 + ap(E)p�s+ p1�2s)�1

for Re s > 3=2, where

p+ap(E) = #f(x; y) 2 Z=pZ : y2 � x3+ax+b mod pgFaltings (1983): L(E; s) = L(E0; s) if andonly if E and E0 are isogenous.

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Breuil, Conrad, Diamond, Taylor (2000):L(E; s) has analyti ontinuation to C andsatis�es a funtional equationNs=2(2�)�s�(s)L(E; s) =WN(2�s)=2(2�)s�2�(2� s)L(E;2� s)

where W = �1 and N 2 Z>0 are onstantsdepending on E.Birh-Swinnerton-Dyer Conjeture (1963,now worth $1,000,000):The rank rE of E equals the order of van-ishing of L(E; s) at s = 1.Gross-Zagier (1986), Kolyvagin (1989):True if the order of vanishing is � 1.

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X=Q a smooth projetive variety (i.e. anbe de�ned in projetive spae by polyno-mials with rational number oeÆients)The ZETA FUNCTION of X is

�(X; s) =YpY

x2X�Z=pZ(1� p�sdegx)�1

Examples:

�(point; s) = �(s)

�(E; s) = �(s)�(s� 1)=L(E; s)

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In general we expet �(X; s) to� have meromorphi ontinuation to C,� satisfy a funtional equation relatingthe value at s to the value at 1+dimX�s, and� enode important arithmeti informa-tion about X.

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Grothendiek (1960's):

Hi(X(C);Ql) = Hi(X(C);Q)QQlhas a ontinuous ation of GQ = Gal(Q=Q)and

�(X; s) =Yi L(Hi(X(C);Ql); s)(�1)i

Example:

�(E; s) = �(s)�(s� 1)=L(E; s)

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RATIONAL NUMBERS:Q = fa=b : a; b integers; b 6= 0g

ALGEBRAIC NUMBERS:Q = f� 2 C : � satis�es a polynomialequation with rational oeÆientsg

ABSOLUTE GALOIS GROUP OF Q:GQ = Aut(Q)= fbijetions Q! Q preserving +;�gwith weakest topology for whih the sta-biliser of every algebrai number is open.

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+ +The usual (arhimedean) absolute valuej j1 = j j indues a metri on Q. Com-pleting Q with this metri gives the �eldR of real numbers.

Q ,! R = CGQ - GR = Autts(C) = f1; gFor a prime p we have the p-adi absolutevalue on Q:

j�jp = p�r if � = pra=b with p6 jabp-adi numbers Qp = ompletion of Q forj jp.

Q ,! QpGQ - GQp = Autts(Qp)+ 11

+ +p-adi integers Zp = elements � 2 Qp withj�pjp � 1.

Zp=pZp = Z=pZ

GQp !! GZ=pZ = hFrobpikernel = Ip = inertia group at p.Frobp = (geometri) Frobenius element:(Frobp �)p = �.

GQp � GQ � f1; gMOTIVATING ALGEBRAIC PROBLEM:Desribe GQ along with GQp, Ip, Frobp et.inside it.+ 12

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Grothendiek (1960's):

Hi(X(C);Ql) = Hi(X(C);Q)QQlhas a ontinuous ation of GQ and

�(X; s) =Yi L(Hi(X(C);Ql); s)(�1)i

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1. (Grothendiek, 1960's) For all but �nitelymany (aa) p the inertia group Ip atstrivially on Hi(X(C);Ql) (i.e. is `un-rami�ed' at p).2. (Fontaine, Messing, Faltings, Kato,Tsuji, de Jong; ~1980-1995)Hi(X(C);Ql) is a de Rham represen-tation of GQl (and for aa l it is rys-talline).3. (Deligne, 1974) For aa p the hara-teristi polynomial of Frobp on Hi(X(C);Ql)(for l 6= p) has oeÆients in Q and allits roots in C have absolute value pi=2(i.e. is `pure' of weight i).

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If V=Ql is a �nite dimensional vetor spaeand ifr : GQ �! GL(V )

is a ontinuous representation satisfyingthese three properties de�ne an L-funtionL(V; s) asQp6=l det(1V � p�sFrobp)j�1V Ip�(similar fator at l)

in Re s > 1+ i=2.(We �x one and for all

C � Q � Ql:)

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CONJECTURE (TATE):Suppose X=Q is a smooth projetive vari-ety. Then there is a deomposition

Hi(X(C);Q) =Mj Mjsuh that for eah prime l and eah em-bedding � : Q ,! Ql, Mj Q;� Ql is an irre-duible subrepresentation of Hi(X(C);Ql).Moreover

L(Mj Q;�Ql; s)is independent of l and �.

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AsL(V1 � V2; s) = L(V1; s)L(V2; s);

this would give a fatorisation�(X; s) =Yj L(Vj; s)

�1with Vj irreduible representations of GQ.

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+ +FONTAINE-MAZUR CONJECTURE (1988)Suppose that

r : GQ �! GL(V )is a ontinuous irreduible representationsatisfying properties 1. and 2. Then:� (Up to Tate twist) V o

urs in someHi(X(C);Ql).� V also satis�es property 3.� L(V; s) is has analyti ontinuation toC (exept possibly for one simple poleif dimV = 1) and satis�es an (expliit)funtional equation relating L(V; s) toL(V �;1� s).

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1. (Grothendiek, 1960's) For all but �nitelymany (aa) p the inertia group Ip atstrivially on Hi(X(C);Ql) (i.e. is `un-rami�ed' at p).2. (Fontaine, Messing, Faltings, Kato,Tsuji, de Jong; ~1980-1995)Hi(X(C);Ql) is a de Rham represen-tation of GQl (and for aa l it is rys-talline).3. (Deligne, 1974) For aa p the hara-teristi polynomial of Frobp on Hi(X(C);Ql)(for l 6= p) has oeÆients in Q and allits roots in C have absolute value pi=2(i.e. is `pure' of weight i).

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(piees of)varietiesover Q (i.e.motives)

Galois repre-sentationssatisfying 1.and 2.

entireL-funtionswith FE

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Topologial ring of adeles:

A = R� (QZYp Zp) (� R�Yp Qp)

Q � A - disrete and o-ompatCLASS FIELD THEORY:Artp : Q�p �! GabQp injetive, dense image

Art1 : R�=R�>0 ��! GR

Art =Yx Artx : Q�R�>0nA���! GabQ

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+ +LANGLANDS:Algebrai uspi-dal automorphirepresentationsof GLn(A)

() irreduible n-dimensional rep-resentations ofGQIrreduible representations

� =Ox0�x

are CUSPIDAL AUTOMORPHIC if theyo

ur inL2�;0(GLn(Q)nGLn(A));

where (gf)(h) = f(hg).Note thatGLn(Q)nGLn(A)=Yp GLn(Zp) = GLn(Z)nGLn(R)+ 22

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� �x: �p (resp. �1) is a representationof GLn(Qp) (resp. GLn(R)).� �: f(zg) = �(z)g(g) for z 2 R�>0.� 0: RN(Q)=N(A) f(ng)dn = 0 for N a sub-group Im �0 In�m

!� GLn:

� is ALGEBRAIC or REGULAR if �1 is.

L(�; s) =Yp L(�p; s)

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EXAMPLES:GL1: Cuspidal automorphi representations� Dirihlet haraters

(Z=NZ)�! C�

GL2: Regular algebrai uspidal automor-phi forms �uspidal holomorphi modular forms whihare newforms.

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(piees of)varietiesover Q (i.e.motives)

Galois repre-sentationssatisfying 1.and 2.

algebrai

uspidal au-tomorphirepresentations

entireL-funtionswith FE

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Tehniques for GALOIS REPRESENTA-TIONS =) AUTOMORPHIC REPRESEN-TATIONS:� Class �eld theory (1920's)� Base hange (Langlands 1980, Arthur-Clozel 1989)� Converse theorems (Weil 1967, ... ,Cogdell-Piatetski-Shapiro 1999)� Lifting theorems (Wiles, Taylor-Wiles1995, ...)� Brauer's theorem (1947)

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CLASS FIELD THEORYArt : Q�R�>0nA� ��! GabQ

Everything follows for GL1.

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BASE CHANGE is the analogue for au-tomorphi forms of restrition of a Ga-lois representation to an open normal sub-group with �nite yli quotient. Uses thetrae formula.CONVERSE THEOREMS assert that L-funtions with good properties arise fromautomorphi representations.BASE CHANGE + CONVERSE THEO-REMS =) two dimensional representa-tions r of GQ with FINITE SOLUBLE IM-AGE are assoiated to automorphi rep-resentations of GL2(A) and L(r; s) is holo-morphi.(Langlands, Tunnell 1981)

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LIFTING THEOREMS (general form):If r; r0 : GQ ! GLn(Zl), if r mod l = r0 mod l,if r0 arises from an automorphi represen-tation and if ... then r also arises from anautomorphi representation.

+ (Langlands-Tunnell) =) TWO dimen-sional representations of GQ with PROSOL-UBLE image +... arise from automorphirepresentations.=) (Wiles, Breuil-Conrad-Diamond-Taylor)Shimura-Taniyama onjeture.(GL2(Z3) is pro-soluble)=) (Frey, Ribet, Wiles) Fermat's last the-orem.+ 29

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NEED MANY OTHER ASSUMPTIONS:� r should be odd and self-dual (i.e. thereis a perfet pairing with (r(�)x; r(�)y) =�(�)(x; y) and (x; y) = �()(y; x)).� r should be de Rham with distint HTnumbers.� r should be ordinary or rystalline withHT numbers small relative to l.� r mod l should have large image?� dim r = 2?

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BRAUER'S THEOREMIf r is a representation of a �nite groupG then there are nilpotent subgroups Hi,haraters �i of Hi and integers ni suhthat

r =Xi niIndGHi�i:

FIRST APPLICATION: If r is a represen-tation of GQ with FINITE IMAGE thenL(r; s) has meromorphi ontinuation andFE.L(r; s) = QiL(IndGHi�i; s)ni= QiL(�i; s)ni

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THEOREM: If dim r = 2, r(Ip) = f1g foraa p and r is rystalline with Hodge-Tatenumbers n1; n2 satisfying0 < jn1 � n2j < (l� 1)=2

then r is pure and L(r; s) has meromor-phi ontinuation to C and satis�es theexpeted funtional equation.COROLLARY: The L-funtion of any reg-ular (distint Hodge numbers) rank 2 mo-tive has meromorphi ontinuation to Cand satis�es the expeted funtional equa-tion.

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EXAMPLE:If X=Q is a rigid (i.e. H2;1(X(C);C) = (0))Calabi-Yau 3-fold then �(X; s) has mero-morphi ontinuation to C and satis�esthe expeted F.E. relating �(X; s) and �(X;4�s).

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PROOF OF MAIN THEOREMSTEP 1: Show that r mod l omes froman automorphi form over some Galois to-tally real �eld. To this end one uses thefollowing theorem.Proposition (Moret-Bailly, Pop): Supposethat X=Q is smooth and geometrially on-neted and that X(R) 6= ;. Then there ispoint x 2 X(Q) all whose onjugates lie inX(R).QUESTION: Can x to be hosen in a sol-uble extension of Q?

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STEP 2: Show that r omes from an au-tomorphi form over some Galois totallyreal �eld F=Q (lifting theorem).STEP 3: Dedue from base hange thatr omes from an automorphi form on anyintermediate �eld F=Ki=Q with F=Ki solu-ble.STEP 4: Use Brauer to write

1 =Xi niIndGal(F=Q)Gal(F=Ki)�i

so thatr =Xi niInd

GQGKi(rjGKi �i)and

L(r; s) =Yi L(rjGKi �i; s)ni:

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GLn(Q) � GLn(A)disrete �nite o-volume

GLn(Q)nGLn(A)=Yp GLn(Zp) = GLn(Z)nGLn(R)

If � : R�>0 ! C� then

L2�;0(GLn(Q)nGLn(A)) =M�is a semi-simple representation

(gf)(h) = f(hg)of GLn(A), whose irreduible onstituents

� =Ox0�x

are alled CUSPIDAL AUTOMORPHICREPRESENTATIONS.+ 36