Notes for GL 2 (A Q )-reps representations

Transcription

1 Note for GL 2 (A Q )-re rereentation Motivation In thi lat note we exlained what tye of rereentation of GL 2 (Q ) were imortant to u (the irreducible+mooth/irreducible+admiible) one. We aid a a motivating factor for thi that we were intereted in tudying rereentation of GL 2 (A Q ) (where, again, we remark that thee are not honet rereentation of GL 2 (A Q ) but ((gl 2 ) C, O 2 (R)) GL 2 (A Q )-module) and, in articular, how thee (conjecturally) related to 2-dimenional Galoi rereentation of G Q. Now, going from rereentation of GL 2 (Q ) traight to GL 2 (A Q )-re i a large lea, eecially becaue the comlicating advent of ((gl 2 ) C, O 2 (R))-module. So, intead of juming traight from the urely local icture to the global icture we intead take an intermediate route and tudy rereentation of GL 2 (A Q ). Thee have the advantage that, formally, they are much like the cae of rereentation of GL 2 (Q ) thi i becaue GL 2 (A Q ) i a TD grou. That aid, thi intermediary te i not one urely of metered convenience it not jut o we don t get overwhelmed with the full definition of a GL 2 (A Q )-rereentation there i eriou, valuable intuition about the claic theory of modular form contained in the tudy of GL 2 (A Q )-rereentation. Particularly, one can ue thi rereentation theoretic erective to illuminate the definition of Hecke oerator (which, uon a firt encounter, eem omewhat... unmotivated) and exlain what reciely i the intrinic ignificance of hrae like Hecke eigenform and newform. 2 Hecke algebra 2. Baic definition One thing that we neglected to dicu in the lat note i the notion of Hecke algebra which are an invaluable tool in the tudy of mooth rereentation of TD grou. The idea i imle through the following analogy: Hecke algebra are to TD grou a grou algebra are to finite grou. Namely, the Hecke algebra H (G) of a TD grou G i made o that, eentially Re m (G) = Mod m (H (G)) where the uercrit m mean mooth (we ll exlain what a mooth H (G)-module i hortly). So, without further adieu, let G be a TD grou. We can then glibly define H (G) to be the convolution algebra of bi-invariant (under ome comact oen K G) meaure on G. But, once we fix a Haar meaure µ on G (which we do!) we can write it down in more down-to-earth term. Namely, we define the Hecke algebra of G, denoted H (G), to be the C-ace Cc (G) of comactly uorted locally contant function f : G C with the obviou addition and multilication given by convolution: (f f 2 )(g) = f (x)f 2 (x g)dµ for f, f 2 C c (G). G Remark 2.: The identification of H (G) with bi-invariant meaure on G follow, eentially, from the Radon-Nikodym theorem although there i robably a imler roof here. We note that if G i not comact then H (G) i not necearily unital. Indeed, if G i comact then a unit i certainly given by µ(g) G. Now while thi element obviouly won t work for general G (one reaon, amongt many, i that G i then not comactly uorted) it maybe lightly non-obviou whether H (G)

2 i unital. That aid, one can confirm one uicion H (G) i indeed unital if and only if G i comact. Thi non-unitalne in general make module theory over H (G) lightly more nuanced. Now, if K i a comact oen ubet of G we denote by H (G, K) H (G) the ubalgebra of thoe comactly uorted locally contant function f : G C which are bi-invariant under K (i.e. invariant under right or left multilication by K). Note that H (G, K) i a unital algebra with unit µ(k) K. Indeed, if f H (G, K) then ( ) µ(k) K f (g) = G µ(k) K(x)f(x g)dµ = f(x g)dµ µ(k) K = f(g)dµ µ(k) = f(g) and imilarly for right multilication by µ(k) K. Now what make the tudy of H (G)-module reaonable i that while it i not unital (in general) it i a colimit of unital thing. Namely, we claim the following: Theorem 2.2: Let G be a TD grou. Then H (G) = lim H (G, K) where, of coure, if K K then H (G, K ) H (G, K). Proof: Indeed, thi require jut howing that if f : G C i comactly uorted and locally contant then f i bi-invariant by ome comact oen of G. To ee thi, note that for each oint x Su(f) w we can find a neighborhood of f where f i contant. But, a neighborhood bai of x i (ince G i TD) given by right tranlation by comact oen ubgrou. Thu, for each x there i ome comact oen K x uch that f xkx. Since Su(f) i comact we can cover it by finitely many xk x and thu, evidently, the interection over thi et of finitely many K x i a comact oen ubgrou for which f i right invariant. Do the ame thing for left invariance and interect them thi how the claim! In fact, we can even decribe H (G, K) H (G) in a urely algebraic fahion. Namely, for notational convenience, let u et e K := µ(k) K for every comact oen K G. We then have the following imle rooition that we leave to the reader: Theorem 2.3: Let f H (G). Then, f i right K-invariant if and only if f e K = f and f i left K-invariant if and only if e K f = f. From thi we deduce that H (G, K) i reciely e K H (G)e K. Thu, H (G) i a (oibly) non-unital algebra with a family of idemotent {e K } uch that K H (G) = K e K H (G)e K making H (G) a o-called idemotented algebra. The algebra of idemotented algebra, in articular their module theory, i largely tranlatable from the claical theory. 2.2 Relationhi to rereentation theory Now that we have a bare-bone undertanding of Hecke algebra we can exlain why they re of interet to u how they are related to rereentation theory. We intuited at the beginning of thi ection that they hould lay a role analogou to that layed by the grou algebra in the cae of finite grou. One reaon that thi hould not be urriing in the lightet i the following: 2

3 Examle 2.4: Let G be a finite grou (conidered a a TD grou with the dicrete toology). H (G) = C[G]. Then, u But, beyond thi literal connection we can actually jutify the claim Re m (G) = Mod m (H (G)) in general. In articular, let u call a H (G)-module M mooth if H (G)M = H (G). Note that while thi i immediate in the module theory of unital algebra (ince M = M!) thi i not at all a vacuou condition here. That aid, one of the key oint of noticing that H (G) i a idemotented algebra i to conclude that, in fact, H (G) itelf i mooth. Namely, if f H (G) then f H (G, K) for ome K and thu e K f = f o that f H (G)H (G). Thu, with thi etu we can tate the deired reult a follow: Theorem 2.5: Let G be a TD grou. Then, there i a natural equivalence of categorie Re m (G) = Mod m (H (G)). Here, of coure, Re m (G) denote the category of mooth G-rereentation of G (a la the lat note) and Mod m (H (G)) denote the category of mooth H (G)-module. Of coure, it not overwhelmingly ueful to know thi equivalence in the abtract, and o we d like to make more exlicit reciely what the equivalence i. So, let uoe that V i a mooth G-rereentation. We then define the tructure of H (G)-module on V, decribed a a homomorhim π : H (G) End(V ), a follow: π(f)(v) := f(g)g(v)dµ () G While thi eem a bit cary looking it really quite tame. Namely, ince V wa aumed mooth we know that tab(v) G i oen and o contain a comact oen ubgrou K. Moreover, ince f H (G) we know that f i bi-invariant for ome comact oen ubgrou K 2 of G. Let K := K K 2. Then, one can check that π(f)(v) = µ(k) f(g i )g i (v) (2) i if g i K i a dijoint oen cover of Su(f). Converely, uoe that V i a mooth H (G)-module (again reented a a homomorhim π : H (G) End(V )) we then want to roduce a mooth G-module V. Set, not hockingly, V (a an underlying C-ace) to be jut itelf and let u define the G-module tructure on V a follow. So, if v V then the fact that V i a mooth H (G)-module imlie that v = π(f)(w) for ome w V and f H (G). Suoe that K G i comact oen uch that f H (G, K). Then we define gv := π ( µ(k) KgK ) (v) which one can quickly check i, indeed, well-defined. Let u jut quickly verify that thee oeration are invere to one another. Namely, uoe that V i a G-module. Then, we want to verify that π( KgK )(v) = g(v). That aid, by (2) we have that π ( µ(k) KgK ) )(v) = µ(k) µ(k) KgK(g)g(v) = g(v) a deired. Checking the other ide of the invere i exactly the ame. From thi more exlicit decrition of the equivalence we can eaily u out the following: Theorem 2.6: Let G be a TD grou, V a mooth G-re, and K G a comact oen ubgrou. Then, the following hold:. The oerator π(e K ) i the rojection oerator V V K. 2. The rereentation V i admiible if and only if for all f H (G) the endomorhim π(f) of V ha finite rank. 3. The G-table ubace of V are the H (G)-ubmodule of V. 4. The C-ubace V K V i a H (G, K)-module. 3

4 Eentially what we ee from thi theorem i that ince V = V K, V K i a H (G, K)-module, and K H (G) = lim H (G, K) that a mooth G-module V i nothing more than a comatible family of H (G, K)- module, and that admiibility i that each of thee H (G, K)-module are finite dimenional (over C). In articular, the fact that H (G) = lim H (G, K) i eentially the equality V = V K with V = H (G) K acting on itelf (the regular rereentation). Smooth GL 2 (A Q )-re and a baby Flath theorem We would now like to begin our tudy, in earnet, of the admiible irreducible rereentation of the TD grou GL 2 (A Q ). The key obervation i that uch rereentation are built from local iece. To make thi recie we need to make a brief detour to dicu retricted tenor roduct. Of coure, a mentioned in the lat note, the fact that GL 2 (A Q ) = GL 2 (Q ) hould mean that it mooth irreducible rereentation hould decomoe a V = V with V a mooth irreducible rereentation of GL 2 (Q ). Of coure, te one to undertanding why thi i true i to undertand what even mean. So, let u uoe that S i ome et of indice and uoe that for every S we have a C-ace V. Let u define for a finite ubet T S the ace V T := V t. What we eentially want to define V t T to be i lim V T a T run over the finite ubet of S. Of coure, for thi work then we need for T T a T ma V T V T. So, let u uoe that for all S we ve fixed vector 0 v 0 V. Then, let define the ma V T V T to be v v v 0 v 0 m if {,..., m } = T T. We then et, rigorouly, V to be lim V T. We call it the retricted tenor roduct of the family {V } S. T Remark 2.7: The above can be made lightly nicer by chooing an ordering on T for, a of now, we ve not ecified the order of the tenor roduct in V T. Thi i not a detail we ll concern ourelve with. Since thi will alo be ueful for u let u conider how we might define retricted tenor roduct of algebra. Namely, let u now aume that for each S we re given intead of jut a C-ace V a C-algebra A. We then want to define A in the exact ame way but, of coure, we d like to get an algebra out of it. The key i that we can t arbitrarily chooe v 0 A ele the ma A T A T won t be multilicative. What we need i that each v 0 i actually an idemotent (not zero of coure). So, let uoe that we are given idemotent e 0 A for all. Then, we can define the retricted tenor roduct algebra A a lim A T which i, indeed, a C-algebra ince each A T A T i a ma of C-algebra. T Before we give an examle, let u oint out the obviou what the univeral roerty of thi retricted tenor roduct algebra/vector ace i. A C-algebra homomorhim A B (a written on the tin of being a tenor roduct/colimit) i, for every finite T, a et of ma f t : A t B, which i comatible a T get larger. In eence, the retricted roduct of algebra i omething like an infinite coroduct (where a ma from it i jut a collection of ma) but even though all the algebra A are taken into account, one only ee finitely many algebra at a given time one cannot ecify the image of a A for infinitely many. So, what i an examle of an algebra that i like utting together a bunch of algebra, but only finitely many variable are involved at a given oint : Examle 2.8: The olynomial algebra C[T, T 2,...] in infinitely many variable i n C[T n ] where the idemotent we take are, not hockingly, the identity element. In general the retricted tenor roduct of 4

5 A n := C[T i,n ]/(f i,j ) hould be omething like C[{T i,n } i,n ]/({f i,j } i,n ). u Now we d like to undertand how one can ue retricted tenor roduct to undertand rereentation of TD grou obtained a retricted direct roduct. The connection i omewhat clear. Namely, let uoe that G i a TD grou and we can write G = G with TD grou G with reect to ome comact oen ubgrou H G. Note then that if we tart with rereentation V of each G we d like to ay that V i a rereentation of G. Now, of coure, for each finite T S we have that V T i a rereentation of G T := G T H t and ince G = lim G T we eem golden. Of coure, thi i only if under the tranition T t T t/ T ma G T G T we have that the aociated ma V T V T are intertwining. A little thought how that thi i true reciely when the vector v0 we ve choen are H table for all S. In thi cae we ee that i a G-re. Note that ince we need only need to have H G given for all but finitely many one ee that the dicuion of the reviou dicuion extend to the cae when we ignore finitely many S, and thu we can ignore icking v 0 V H for finitely many. We now come to tatement of Flath theorem: Theorem 2.9 (Baby Flath): Let V be an irreducible admiible rereentation of GL 2 (A Q ). Then, there exit a unique decomoition V = V where V i an admiible irreducible rereentation of GL 2 (Q ) and V i unramified for almot all (recall that thi mean that V GL2(Z) 0). One hould note that, while we have been loy ureing the deendence on the vector {v} 0 in the notation V, it i now actually imortant. Namely, Theorem 2.9 i tated in uch a way that it eem to not deend on a chocie of vector (u to iomorhim) and thi i, indeed, the cae. That aid, the jutification of thi reult require a bit of work. The key i that we mut take our vector v 0 V GL2(Z), and we do o only when V GL2(Z) 0 (which by the tatement of Flath theorem i non-zero for all but finitely many ). Thu, any ambiguity in the tatement of Flath theorem come from the non-ecification of how we chooe thee v 0 and whether thi retricted tenor roduct deend on thi choice. Thankfully, it doe not deend on uch a choice. The key i the following: Theorem 2.0 (Baby Satake iomorhim): Let T := K and S := K2 where ( ) 0 K = GL 2 (Z ) GL 0 2 (Z ) V and K 2 = GL 2 (Z ) ( ) 0 GL 0 2 (Z ) Then H (GL 2 (Q ), GL 2 (Z )) = C[T, S, S ] In articular, we ee that H (GL 2 (Q ), GL 2 (Z )) i commutative. Now, if V GL2(Z) 0 then, ince V i irreducible, thi H (GL 2 (Q ), GL 2 (Z ))-module (by Schur lemma) mut be -dimenional. Thu any two choice of v 0 differ by a contant, and one can then how that the iomorhim cla of the retricted tenor roduct i unaffected by thi choice. We will not rove Flath theorem here referring the intereted reader to The Corvalli Proceeding and Flath orginal article. The reult, while dee, i fairly eay to rove relying motly on baic algebra. 5

6 Remark 2.: Note that the irreducibility of V in Flath theorem i ivotal. Namely, one cannot attemt to decomoe a general rereentation of GL 2 (A Q ) into local factor. In articular, one cannot aly thi to the ace A 0 (GL 2 ) of GL 2 -automorhic form to obtain local automorhic form. Thi i omewhat dee roblem, to find what a local analogue of automorhic form hould be. Let u end thi ection by tating what i, morally, an equivalent tatement of Theorem 2.9 but on the level of Hecke algebra (but it i eaier it actually a firt te into roving Flath theorem): Theorem 2.2: There i an iomorhim of C-algebra H (GL 2 (A Q )) = H (GL 2 (Q )) here the idemotent we take each H (GL 2 (Z )) i GL2(Z ) (where, imlicitly, we ve normalized the Haar meaure on GL 2 (Q ) uch that GL 2 (Z ) ha meaure ). Moreover, uoe that U GL 2 (A Q ) i a comact oen with factorization U = U. Then H (GL 2 (A Q ), U) = H (GL 2 (Q ), U ) 3 Connection to modular form 3. Baic etu Let u now give a non-trivial examle of mooth GL 2 (A Q )-rereentation and, along the way, clarify the baic theory of modular form. The firt te i to generalize the notion of a modular form of weight k and level Γ (for Γ a congruence ubgrou) to work with level any comact oen ubgrou U GL 2 (A Q ). The reaon why thi hould eem like a generalization (and we ll ee later i literally a generalization) i that the congruence ubgrou Γ SL 2 (Z) are reciely ubgrou of SL 2 (Z) of the form U SL 2 (Z) with U GL 2 (A Q ) comact oen. So, without further delay let u define uch a modular form. Namely, let u ay that a function f : h ± GL 2 (A Q ) C (where h ± i the uer and lower half-lane) i a modular form of weight k and level U, where U GL 2 (A Q ) i comact oen, if it atifie the following condition: MF. For each fixed g GL 2 (A Q ) we have that f(h, g) i a holomorhic function h± C. MF.2 For each γ GL 2 (Q) we have that f(γh, γg) = deg(γ) j(γ, z) k f(h, g) where GL 2 (Q) act on h ± by fractional linear tranformation and on GL 2 (A Q ) diagonally, and (( ) ) a b where j, z = cz + d. c d MF.3 For all u U we have that f(h, gu) = f(h, g). MF.4 For all fixed g GL 2 (A Q ) we have that lim f(h, g) exit. h i Condition MF.4 can be rehraed in two ueful way. Firt, it i equivalent to ay that (again for each fixed g) for every A 0 there i a C A > 0 and N N uch that f(h, g) C A Im(h) N when ( Re(x) ) A N and Im(h) 0 in other word, that f(g, h) i of moderate growth. Second, note that U for 0 N 0 which how that f(h, g) = f(h + N, g) o that f till ha a Fourier exanion f(h, g) = ( ) 2πih a n ex (3) N n Z 6

7 and we require that a n = 0 for n < 0. If, in addition, we aume that for each g we have that lim f(h, g) = 0 then we ay that f i a cuform h i of weight k and level U. Equivalently, we can ay that, for each fixed g, the function f(h, g) i of raid decay meaning that for each fixed A and k N, there i a C A,k uch that f(h, g) C A Im(h) k for x A and Im(h) 0. Finally, we can require that for each fixed g the coefficient a 0 in (3) i 0. One can think about what we ve done above a follow. It a common firt exercie in the tudy of automorhic form to exlain how one can inflate function f : h C to function f : SL 2 (R) C which begin one on the journey of undertanding automorhic form for SL 2 (R) and how they generalize claic modular form. Here we have, intead, inflated our modular form to live on non-archimedean grou (or rather, thoe that have a large non-archimedean comonent) and left the archimedean factor alone. Thi i why, for intance, recrition of how O 2 (R) act are not neceary we ve done no inflating on the Archimdean factor. One of the ower of thi aroach i that it allow u to unite the tudy of modular form and cu form acro different weight together. Namely, let u denote for each comact oen U the C-ace M k (U) of modular form of weight k and level U and imilarly we denote cuform by S k (U). Note that if U U then evidently M k (U ) M k (U) and S k (U ) S k (U). So, let u define the following: M k := lim M k (U) U S k := lim S k (U) U And note that GL 2 (A Q ) now actually act on thee ace. Secifically, if g 0 GL 2 (A Q ) and we define, a er uual, the function g 0 f by the rule (g 0 f)(h, g) = f(h, gg 0 ) then for any f M k (U) we have that g 0 f M k (g0 Ug 0) and if f S k (U) then g 0 f S k (g0 Ug 0). Thu, while we can t ee omething like a GL 2 (A Q )-action at a fixed level (we ll ee hortly what tye of action we really get there) once we unify the weight k form acro all level the GL 2 (A Q ) action become aarent. Before we continue, let u rove a mall lemma that will how in many cae that M k (U) and S k (U) are familiar object: Theorem 3.: Let U GL 2 (Ẑ) be a comact oen. Suoe moreover that det(u) = Ẑ. Then, if Γ := SL 2 (Z) U then Γ i a congruence ubgrou and the ma f f(h, ) define a bijection M k (U) M k (Γ) and S k (U) S k (Γ). Thi theorem i fairly imle, but let u remark on the incluion of the condition det(u) = Ẑ. Recall that the algebraic grou GL 2 doe not atify trong aroxmation. Thu, to rove uch a theorem we need to boottra from trong aroximation for SL 2 and the condition that det(u) = Ẑ eentially allow u to make the equality which i what make thi oible. GL 2 (A Q )/U = SL 2(A Q )/(U SL 2(A Q )) Remark 3.2: Of coure, thi then wonder why we re not working with jut SL 2 intead of GL 2 where the above theorem wouldn t need uch qualification. There are two anwer. Firt i, well, we re intereted in GL 2 (A Q ) (becaue it on the eminar organizer qualifying exam). Secondly, it GL n that much of what we ay extend to. Namely, we ll ue below heavily the fact that GL 2 ha the trong multilicity one roerty. Thi alo hold for GL n. That aid, while it hold for SL 2 it doe not hold for SL n for n > 2. In articular, let aly Theorem 3. to ome familiar ituation. Namely, let u conider the following oen ubgrou of GL 2 (Ẑ): 7

8 {( ) } a b U(N) := GL c d 2 (Ẑ) : a mod N, b c 0 mod N {( ) } a b U (N) := GL c d 2 (Ẑ) : a mod N, c 0 mod N {( ) } a b U 0 (N) := GL c d 2 (Ẑ) : c 0 mod N Then thee all atify det(u) = Ẑ. Moreover, one can check that and thu we have equalitie M k (U(N)) = M k (Γ(N)) M k (U (N)) = M k (Γ (N)) M k (U 0 (N)) = M k (Γ 0 (N)) U(N) SL 2 (Z) = Γ(N) U (N) SL 2 (Z) = Γ (N) U 0 (N) SL 2 (Z) = Γ 0 (N) S k (U(N)) = S k (Γ(N)) S k (U (N)) = S k (Γ (N)) S k (U 0 (N)) = S k (Γ 0 (N)) thu connecting thi theory back to the mot baic cae of modular form. Remark 3.3: Be careful! Note that a more natural choice for comact oen ubgrou of GL 2 (A Q ) giving the Γ i (N) would have been Γ i (N) their rofinite comletion. Thee are certainly erfectly fine comact oen ubgrou of GL 2 (A Q ) but they do not atify the urjectivity condition for the determinant, thu one cannot directly relate modular form of their level to modular form (in the claical ene) of level Γ i (N). In articular, ince the canonical neighborhood bai of the identity i Γ(N) note that we cannot, a riori, uniformize all generalized modular form (with level comact oen of GL(A Q )) in term of claical modular form. Namely, it not true that we can decribe M k and M K a a colimit of claical modular form ace. Thi hould be clear ince uch a ace would not have a GL 2 (A Q ) action (jut a GL2(Ẑ) action!). So, we have alo really gained omething by aing to thi more general level. So let u begin by making the following two obervation which, while not overly dee, are alo not extremely obviou: Theorem 3.4: For all U GL 2 (A Q ) comact oen the C-ace M k(u), and thu S k (U), are finitedimenional. Proof (idea): Jut like in the claical cae one can write down a (oibly diconnected) comact Riemann urface for which M k (U) and S k (U) are ection of a line bundle, thu finite-dimenional. We alo need the following: Theorem 3.5: There i a natural inner roduct on S k by which GL 2 (A Q ) act unitarily (it, u to a character twit, omething like the Peteron inner roduct). So, uing Theorem 3.4 we can actual ay omething omewhat urriing/dee. Namely, the rereentation M k and S k of GL 2 (A Q ) are admiible! Indeed, the fact that they are mooth i clear. If f M k (U) M k then f i fixed by U and thu tab(f) i oen (the ame goe for S k ). And, to ee it admiible we note that for each comact oen U we have that M U k = M k(u), S U k = S k(u) which are finite-dimenional by Theorem 3.4. Now that we know that M k and S k are reaonable rereentation of GL 2 (A Q ) (they are admiible) we can hoe to try and undertand their admiible irreducible ubrereentation/ubquotient. To thi end, let u make the following definition. A modular rereentation of weight k (for GL 2 ) i an irreducible admiible ubquotient of M k. A cuidal rereentation of weight k (for GL 2 ) i an admiible ubrereentation of S k. 8

9 Remark 3.6: Note that in our definition of cuidal rereentation of weight k we aid ubrereentation intead of ubquotient. One can how that the two notion are equivalent. Thu, the ret of thi note will be devoted to undertanding reciely the cuidal rereentation of weight k. 3.2 Hecke oerator Before we do thi directly, it helful (and extremely illuminating!) to firt undertand how the Hecke oerator in the claical etting relate to the Hecke algebra in thi etting of mooth GL 2 (A Q )-rereentation. So, we aid earlier that the individual level M k (U) and S k (U) did not oe a GL 2 (A Q )-action. So, then, what action do they have? Thi i a little cumberome to tate in the language of rereentation but if we think of M k and S k a mooth H (GL 2 (A Q ))-module the icture uddenly become much more unny. Namely, M k (U) = M U k i reciely a H (GL 2(A Q ), U)-module and imilarly for S k(u). Thu, the action that actually exit at a finite level i jut the action of the U bi-invariant Hecke algebra H (GL 2 (A Q ), U). So, in articular, let u think about the cae of M k (Γ (N)) = M k (U (n)) = M U(N) k and imilarly for S k (Γ (N)). Namely, by the above analyi thi hould carry an action of H (GL 2 (A Q ), U (N)). So, for tarter, what reciely doe thi algebra look like? Well, uing Theorem 2.2 and Theorem 2.0 we can give a retty atifactory anwer to thi quetion. Namely, let u begin by noting that we can write U (N) = N GL 2 (Z ) N U, (N) where, here, {( ) } a b U, (N) := GL c d 2 (Z ) : c 0 mod v(n), a mod v(n) Then, from Theorem 2.2 we deduce that H (GL 2 (A Q ), U (N)) = N H (GL 2 (Q ), GL 2 (Z )) N H (GL 2 (Q ), U, (N)) That aid, we know from Theorem 2.0 that H (GL 2 (Q ), GL 2 (Z )) = C [ T, S, S and o combining thi with Examle 2.8 we deduce that H (GL 2 (A Q ), U (n)) = C[ { T, S, S } ] H (GL N 2 (Q ), U, (N)) N ] Thu, we ee that for each N we get oerator T and S on S k (Γ (N)) = S U(N) k. The thing we d hoe i, in fact, true: Theorem 3.7: Under the identification S k (Γ (N)) = S Γ(N) k the oerator T act a k 2 T c where T c the claical Hecke oerator and S act a multilication by χ() if χ i the Nebentyu for f. Of coure, one can extend the above reult to the Hecke oerator at rime N but that require more work (in articular, Baby Satake [or Paa Satake] don t aly). Let ell out the hiloohical imlication of all of thi. When one i firt introduced to Hecke oerator, unle they had a articularly good teacher they robably eemed unmotivated and forced. One uually trie to jutify them a averaging oerator ued to rove thing about Fourier coefficient. But, the above tell u that if we ay attention to the GL 2 (A Q )-action on modular form (the thing that we care about from a rereentation theoretic erective) then the Hecke oerator are forced on u a being the real imortant art of thi GL 2 (A Q ) action. They don t eem random anymore. 9

10 Remark 3.8: One can alo undertand the Hecke oerator naturally from a geometric oint of view. Namely, the ingular cohomology of any ymmetric ace ha Hecke oerator coming from the natural Hecke correondence of the family of ace obtained by quotienting thi ymmetric ace by congruence ubgrou. Thi i then connected to modular form via the Matuhima formula and then one obtain, again, the claical Hecke oerator. 3.3 Decomoition S k So, in thi ection we d like to undertand how reciely rereentation of S k look what cuidal rereentation of weight k look like. The firt key obervation if the following: Theorem 3.9: We have an (algebraic) decomoition S k = V cu. re. wght k V n V (4) Thi i eentially due to Theorem 3.5. Alo, note that ince S k i admiible we know that n V for all V. Now, the firt big reult that we will need (and blackbox heavily) i the following: i finite Theorem 3.0 (Strong multilicity for GL 2 ): Suoe that V and W are cuidal rereentation of weight k embedded into S k. Then, if V = W for almot all (where V and W are a in Theorem 2.9) then V = W. In other word, for all V we have that n V = in (4) and for V and W occuring in (4) we have that V = W if and only if V = W for almot all. Thi i an extremely owerful and urriing reult and will eentially be the key to all of what we ay after. Remark 3.: Note that trong multilicity i a truly global henomemon. For examle, doing the Archimedean analogue of the decomoition in (4) give arbitrarily high multilicitie. So, we now come to what i, in my etimation, one of the mot beautiful theorem in the baic theory of automorhic form/rereentation which clarifie much of the baic theory. But, before we come to the theorem directly, let u firt notate for f S k (U) the orbit GL 2 (A Q )f S k by π f. We then have the following: Theorem 3.2: For f S k (Γ (N)) the ubrereentation π f S k i irreducible if and only if f i an eigenform. Moreover, for every cuidal rereentation V of weight k we have that V = π f for ome eigenform f S k (Γ (N)) (for ome N). Proof (Idea): If π f i irreducible then we can, by Theorem 2.9, decomoe it a V and for almot all we have that T act by a character, and thu f mut be an eigenvalue for T. To ee that π f i irreducible if f i an eigenform we decomoe π f into irreducible and note that for each irreducible factor it th Flath comonent need to have k 2 λ (f) (where λ (f) i the eigenvalue of T c on f) a the eigenvalue for T. By trong multilicity thi imlie that all the irreducible factor are equal and, again, imlie that there mut only be one irreducible factor. The fact that all how u a π f i omewhat obviou. Namley, if V i a cuidal rereentation of weight k we know that V = GL 2 (A Q )f for any non-zero f V. If we can how that f S k(γ (N)) then necearily f i an eigenform ince the T in the local Hecke algebra mut act a a character. Thu, the oomh i howing that we can take f S k (Γ (N)). Thi we don t do here. Thu, we ee that the reaon we care about eigenform i that, well, they are the irreducible! They are the imortant art the generating et of the whole of cuform. 0

11 We end thi ection by exlaining how newform (which i hort for new normalized Hecke eigencuform) enter the icture. The baic idea i imle. Namely, we now know that in the decomoition (4) each V i ome π f. Thu, we d like rewrite thi decomoition that form. But, the quetion i the following: what indexing et of f hould it run over? Namely, there are many eigenform with the ame eigenvalue (e.g. (z) and (2z) for the uual dicriminant form of weight 2 and level SL 2 (Z)). The key i that two newform which have the ame Hecke eigenvalue are equal. Thu, the oint i that each π f ha a canonical rereentative given by a newform (and all other Hecke eigenform are obtained from thi on by a GL 2 (A Q )-action). Thu, we can rewrite the decomoition (4) a follow: S k = π f (5) f newform which I find exceedingly beautiful and illuminating, exlaining why eigenform, and in articular newform, are o imortant. 3.4 Local factor and L-function We end thi note by trying to tie the rereentation π f, with f a newform for Γ (N) (for ome N), together with the material from the lat note. Namely, by Theorem 2.9 we know that π f = where V i an admiible GL 2 (Q )-rereentation, and o one might begin to wonder what thee are. Well, one cae thi i clear i in the unramified cae. Namely if N then it fairly clear from our analyi above that V,f mut be unramified and thu, from our dicuion lat time, an irreducible rincile erie P (χ, χ 2 ) and we know that, u to reordering, χ i (in the arlance of lat time) the character ( k 2 a (f), 0, ) and χ 2 the character (χ(), 0, ). What haen for rime N i more comlicated. Namely, one can how that in thi cae one never get a character and thu one i left with either an irreducible (ramified) rincial erie, a ecial rereentation St GL2, or a uercuidal. Loeffler and Weintein have a aer detailing an algorithm to decide which i which. Regardle, one thing that one can rove abtractly (without knowing the ecific identity of the local rereentation V f, ) i that for every rime one ha that L (f, ) = L (π f, ) where, by definition, L (π f, ) := L(V,f, ) where we defined lat time the definition of the L-function of (non-character) irreducible admiible rereentation of GL 2 (Q ). Thu, not only do newform intimately factor into the tudy of rereentation of GL 2 (A Q ) but their native L-function agree with the automorhic L-function obtained by tudying their local rereentation. V,f