Adelic Modular Forms

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George Chapman
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1 Aelic Moular Form October 3, 20 Motivation Hecke theory i concerne with a family of finite-imenional vector pace S k (N, χ), inexe by weight, level, an character. The Hecke operator on uch pace alreay provie a very rich theory. It will be very avantageou to pa to the aelic etting, however, for the ame reaon that Hecke character on number fiel houl be tuie in the aelic etting (rather than a homomorphim out of the group of fractional ieal). In Tate thei, we learne that once we view Hecke character a iele cla character, we can apply the tool of harmonic analyi on locally compact group. We re going to o omething very imilar with moular form: we ll view them a pecial function (calle automorphic form) on an aelic group GL(2). Some avantage of thi point of view are (at leat): For a Hecke eigenform f, it will become clear what the local component of f are uppoe to mean. Furthermore, the local component of f etermine the Euler factor of L(f, ), an a well the local factor appearing in it functional equation, jut a in Tate thei. In orer to generalize the notion of moular form to other number fiel (or function fiel) in a uniform way, there i little alternative to the aelic theory. A for the lat point, one can urvive for a time tuying Hilbert moular form, Maa waveform, Bianchi moular form, etc, a thee all can till be realize a function on a ymmetric pace. But that will never quite reveal

2 the unity of thee contruction: they are all avatar of automorphic form (on GL(2) over a totally real fiel, on GL(2) over Q but with nonzero eigenvalue for the Laplacian, an on GL(2) over an imaginary quaratic fiel). To ay nothing of other group! The fiel mot important number-theoretic reult (the Langlan-Tunnell theorem, the Taniyama-Shimura conjecture, the Sato-Tate conjecture) really o rely on the theory of automorphic form on (at leat) the group GL(n) for n > 2. In the en, automorphic form on GL(2) over Q are uppoe to be function on the quotient GL 2 (Q)\ GL 2 (A Q ). But at the moment it will be convenient to take a ifferent tack, which avoi ome ifficultie at the infinite place. The goal i to create a large vector pace M k, amitting an action of GL 2 (A fin Q ), which alo contain every M k(n, χ) for every level N an character χ. 2 Smooth Repreentation Firt we will nee to make a etour into repreentation theory. Let G be a topological group, certainly not aume compact. We will want to tuy the category of repreentation ρ: G GL(V ) which ha ome of the nice propertie of the theory in the compact cae. There are a few natural choice for how to o thi. One i to give V an inner prouct which make V into a Hilbert pace, an to require that ρ(g) be unitary for each g. Such a repreentation of G i calle unitary. Thi i certainly an important concept. Another (perhap impler) thing to o i to give V the icrete topology, an require that G V V be a continuou map. Thi i equivalent to aying that for each v V, the tabilizer of v in G i an open ubgroup. Such a repreentation i calle mooth. If ρ: G GL(V ) i any repreentation, an v V, then v i mooth if it tabilizer in G i open. Thu V i mooth when all of it vector are. Let V m be the et of mooth vector in V ; it i eay to check that V m i a vector ubpace of V which i preerve by G. Smooth repreentation are going to be few an far between if G oen t have many open ubgroup. For intance if G = SL 2 (R), there aren t any nontrivial mooth repreentation! (GL 2 (R) ha a mooth one-imenional repreentation, gn et.) But p-aic group uch a GL 2 (Q p ) have many intereting mooth repreentation. One way to contruct mooth repreentation i through mooth inuction. If H G i a ubgroup, an ρ: H GL(W ) i a repreentation, let 2

3 V = m-in G HW be the moothly inuce repreentation: that i, V i the pace of function f : G W which atify the propertie f(hg) = f(g) for all h H, g G, There exit an open ubgroup K G (epening on f) uch that f(gk) = f(g) for all h H. Then V ha an action of G, call it η, efine by η(g)f(g ) = f(g g). Note that m-in G HW = (In G H W ) m. Thi i how one contruct the principal erie repreentation of G = GL 2 (Q p ). Chooe two character χ, χ 2 of Q p, let χ = χ χ 2 be the correponing character of the Borel ubgroup B G, an let π(χ, χ 2 ) = m-in G Bχ. Irreucible repreentation of G appearing in the π(χ, χ 2 ) are known a the principal erie repreentation. (In practice, however, it i more convenient to normalize thing a bit ifferently, o that ual work the right way, but more on thi later.) 3 The pace M k of aelic moular form Let W k be the pace of holomorphic function on the upper half-plane H with the property that, for all γ GL + 2 (Q), f γ,k i boune on the region Iz > y 0 for every y 0 > 0. The weight k i going to be fixe for now, o we ll jut write f γ for f γ,k. Give thi pace a (left) action r k : GL + 2 (Q) W k by r k (γ)f = f γ Now efine a mooth repreentation ρ k : GL 2 (A fin Q ) GL(M k) by M k = m-in GL 2(A fin Q ) W GL + 2 (Q) k A we will ee, thi pace contain all the moular form of weight k at once! 3. Strong approximation for GL(2) Jut a (A fin Q ) = Q + >0Ẑ, 3

4 there i a ecompoition GL 2 (A fin Q ) = GL + 2 (Q) GL 2 (Ẑ). (The analogy in t perfect, becaue the firt ecompoition i a irect prouct, while the econ one i far from being a irect prouct: GL + 2 (Q) GL 2 (Ẑ) = SL 2(Z).) Better yet, we have the more general Theorem 3.. Let K GL 2 (Ẑ) be an open ubgroup uch that et : K Ẑ i urjective. Then GL 2 (A fin Q ) = GL+ 2 (Q)K The retriction on K i neceary, becaue the image of GL + 2 (Q)K uner et i Q >0 et K, an thi will not equal (A fin Q ) unle et K = Ẑ. We ll prove the reult for K = GL 2 (Ẑ). The key obervation we nee i that SL 2 (Z) SL 2 (Z/NZ) i urjective, which in t ifficult to prove. Suppoe g GL 2 (A fin Q ). After tranlating g by GL+ 2 (Q) it i clear we can aume that g entrie lie in Ẑ. Let L = Afin A fin. Since g p i almot alway in GL 2 (Z p ), g(l) mut have finite inex in L. Apply the tructure theorem of finite abelian group to L/g(L): there exit r uch that L/g(L) = Z/rZ Z/Z. Thi mean exactly that g GL 2 (Ẑ) GL 2 (Ẑ). Write g = u w with u, w GL 2 (Ẑ). Let N = r. Let > 0 be an ( ) integer with et u (mo N). Then the reuction of u moulo N lie in SL 2 (Z/NZ). Lift it to an integral matrix γ SL 2 (Z), o that ( ) ( ) γ u (mo N). Write z = γ u, o that z ˆΓ(N) 4

5 (that i, z i congruent to moulo N). Then g = u w ( ) ( ( ) r = γγ u w ) GL + 2 (Q)z GL 2 (Ẑ) ( ) ( ) r r GL + 2 (Q) z GL 2 (Ẑ). The theorem i prove with the calculation that conjugation by move ˆΓ(N) into a ubgroup of GL 2 (Ẑ). The theorem applie to a few aelic ubgroup of interet: the group K 0 (N), K (N), ( an) K(N), ( which ) conit ( ) of matrice in GL 2 (Ẑ) congruent 0 moulo N to,, an, repectively. The interection of thee group with GL + 2 (Q) are Γ 0 (N), Γ (N), an Γ(N), repectively. Let K be one of K 0 (N), K (N), K(N), an let Γ = K GL + 2 (Q). We will now how how to aociate to f M k (Γ) an element φ f M k. Given g GL 2 (A fin ), fin γ GL + 2 (Q) an h K with g = γh. Then efine φ f (g) = f γ Thi i well-efine becaue if γ h i another ecompoition, then γ γ = h h GL + 2 (Q) K = Γ, o that f γ = f (γ γ )(γ ) = f (γ ). Alo, φ f i mooth becaue it i K-invariant. Therefore there i a map M k (Γ) M k f φ f which i injective. The map are compatible in an obviou ene: for intance when N N, the map M k (Γ(N)) M k factor through M k (Γ(N)) 5

6 M k (Γ(N )). Thu there i an injective map lim M k (Γ(N)) M k. N Note that each M k (Γ(N)) ha an action of SL 2 (Z/NZ), o that a priori the limit M k = lim M k (Γ(N)) ha an action of lim SL 2 (Z/NZ) = SL 2 (Ẑ). The action of the larger group GL 2 (A fin ) make M k omewhat eaier to work with. One benefit i that the appearance of moular form with character can now be explaine very neatly: the center (A fin Q ) of GL 2 (A fin Q ) act on M k, an the ubgroup Q >0 (A fin Q ) act trivially on M k, o the action of the center factor through an action of (A fin Q ) /Q >0 = Ẑ. Thu M k plit up a a irect um of ubpace M k (χ), where χ run over the et of character of Ẑ ; i.e. Dirichlet character. One mut check that if χ i a Dirichlet character moulo N, then the invere image of M k (χ) uner M k (Γ (N)) M k i M k (N, χ). Of coure there i alo a cupial verion of M k, call it S k, in which the original pace W k i reefine o that every f γ (x + iy) i not jut boune a y, but approache the limit 0 uniformly in x. Our goal in the coming lecture i to prove the following theorem, which elegantly encapulate Hecke theory in the language of repreentation theory. By a cupial newform of weight k, we mean a normalize Hecke eigenform f S k (N, χ) + (for ome N an χ). A we have een, a newform i etermine completely by it Hecke eigenvalue. For uch a newform f, let π f M k be the mallet GL 2 (A fin Q )-table ubpace of S k containing φ f. Theorem For each newform f of weight k, π f i irreucible. 2. S k i the irect um of the π f a f run through all newform of weight k. Recall that a Hecke character χ i necearily the prouct of local character χ v. In a imilar fahion, our repreentation π f of GL 2 (A fin Q ) i the retricte tenor prouct of repreentation π f,p of GL 2 (Q p ). We now turn our attention to thee local repreentation. 6

Lecture 3. January 9, 2018

Lecture 3. January 9, 2018 Lecture 3 January 9, 208 Some complex analyi Although you might have never taken a complex analyi coure, you perhap till know what a complex number i. It i a number of the form z = x + iy, where x and