VIEWING MODULAR FORMS AS AUTOMORPHIC REPRESENTATIONS

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1 VIEWING MODULAR FORMS AS AUTOMORPHIC REPRESENTATIONS JEREMY BOOHER These notes answer the question How does the classical theory of modular forms connect with the theory of automorhic forms on GL 2? They are a more detailed version of talks given at a student reading grou based on Jacquet and Langland s book [JL70]. They were given after the first ten sections had been covered, which had discussed the local reresentation theory of GL 2 Q, GL 2 R, and GL 2 C, and then defined the global Hecke algebra and automorhic forms and reresentations. The aim was to connect the theory which had been develoed with something more familiar, and understand the motivation behind the theory which had been develoed. Given a cusidal holomorhic modular form fz of weight k, level N, and nebentyus character χ, we will describe how to associate an automorhic form F g on GL 2 R + and an automorhic form ϕ f g on GL 2 A Q. We attemt to collect all of the details necessary to understand what roerties F g and ϕ f g ossess, and make clear how these roerties of automorhic forms relate to classical roerties of modular forms. In the case when fz is an Hecke eigenform, we will also show there is an automorhic reresentation π f of GL 2 A Q associated to f containing ϕ f, and describe as much as ossible its local comonents. In the rocess of doing so, we will exlain how the data of the Hecke eigenvalues is encoded in the local reresentations. This will exlain the sense in which the Hecke eigenvalues are local information, a fact that is not obvious in the classical setting. It is ossible obtain information about most of the local comonents by ure thought. At rimes that divide the level N, the situation is more comlicated as suercusidal reresentations can arise. Loeffler and Weinstein describe the situation and an algorithm to deal with these cases [LW12]. Examle 0.1. For examle, one of the newforms of level 99 with trivial nebentyus and weight 2 has q-exansion fz = q q 2 q 4 4q 5 2q 7 + 3q 8 + 4q 10 q Using the algorithm as imlemented in SAGE, we find that The infinite comonent is a discrete series of weight 2. At = 2, the local comonent is an unramified rincial series. The two unramified characters send uniformizers to the roots of x 2 + x + 2. Note that the Hecke eigenvalue for T 2 is 1. At = 3, the local comonent is a suercusidal reresentation of conductor 2. Sage comutes a descrition in terms of two characters of extensions of Q. At = 11, the local comonent is a secial reresentation of conductor one. It is the twist of the Steinberg by the unramified character of Q 11 of order two. Note that the Hecke eigenvalue is 1. These notes assume the reader is familiar with the reresentation theory of GL 2 Q, GL 2 R, and GL 2 C over the comlex numbers, in articular the classification of irreducible smooth admissible reresentations. We also assume knowledge of the definition of automorhic forms and reresentations on GL 2 A Q, local and global Hecke algebras and the manner in which automorhic reresentations decomose as restricted tensor roducts. This is essentially the material in the first ten sections of Jacquet and Langland [JL70]. Other standard references for some or all of this material include books by Bum [Bum97], Bushnell and Henniart [BH06], and Gelbart [Gel75]. Date: Aril 14,

2 2 JEREMY BOOHER In Section 1, we relate quotients of the uer half lane to double coset saces for GL 2 R + and GL 2 A Q as a reliminary ste in transorting modular forms to GL 2 R and GL 2 A Q. Section 2 then constructs the automorhic forms F g and ϕ f g associated to a classical modular form f and establishes their imortant roerties. Using these automorhic forms, one can construct the automorhic reresentation associated to f. In order to do so, we first review some facts about the local reresentation theory of GL 2 R and GL 2 Q in Sections 3 and 4. Besides a general review of the classification results, we also include material that was not touched uon in the seminar or in [JL70]. In articular, Section 3 discusses g, K modules and the local Hecke algebra from a different ersective than [JL70], and connects the action of Ugl 2 C with exlicit differential oerators on C GL 2 R. Section 4 includes information about sherical reresentations and the sherical Hecke algebra. Finally in Section 5 we use the multilicity one theorem to construct an irreducible automorhic reresentation π f containing ϕ f and describe the local comonents as much as is ossible. 1. Preliminaries on the Uer Half Plane and Quotients In this section, we will relate quotients of the uer half lane H with quotients of GL 2 R and with quotients of GL 2 A Q. We consider the grou GL 2 with center Z. The oints over R with ositive determinant, GL 2 R + = {g GL 2 R : detg > 0}, act on the uer half lane H via fractional linear transformations. The connection between functions on the uer half lane and functions on GL 2 R boils down to the following observation: SL 2 R acts on H with stabilizer SO 2 R. It is technically better to work with GL 2 R + and consider searately K = SO 2 R and the center ZR + GL 2 R +. This idea and the Iwasawa decomosition give a convenient system of coordinates to use on GL 2 R +. Because GL 2 R + is the roduct of the maximal comact subgrou K and the Borel of uer triangular matrices, every element can be exressed in the form y 1/2 y 1/2 x cosθ sinθ 1.1 g = λ 0 0 λ 0 y 1/2 sinθ cosθ Letting this act on the uer half lane, we see that i is sent to x + iy. Therefore using x, y, θ, λ as coordinates on GL 2 R + subject to the obvious restrictions that λ > 0, y > 0, and θ [0, 2π gives a convenient coordinate system that is related to the standard coordinates on the uer half lane in the sense that g i = x + iy. Remark 1.1. This is a different coordinate system than is used in Bum s book. In articular, cosθ sinθ Bum uses the matrix for his rotation matrix. This is an unnatural direction sinθ cosθ to rotate, but simlifies certain formulas. In articular, actions of K by e ikθ become actions by e ikθ such as in 3.1 and the action of K on automorhic forms, and negative signs are removed in the differential oerators for and L in Section 3.1. We now turn to GL 2 A Q. Recall that the strong aroximation theorem states that for any comact oen subgrou K GL 2 A f such that detk = A f, we have GL 2A Q = GL 2 Q GL 2 RK. This relies on the class number of Q being one. A convenient comact oen subgrou to work with will be { } K 0 N = GL c d 2 Ẑ : c 0 mod N. The connection between the GL 2 R + or SL 2 R and GL 2 A Q is the following roosition. Proosition 1.2. For any ositive integer N, there are natural isomorhisms. Γ 0 N\ SL 2 R ZA Q GL 2 Q\ GL 2 A Q /K 0 N Γ 0 N\ GL 2 R + GL 2 Q\ GL 2 A Q /K 0 N.

3 VIEWING MODULAR FORMS AS AUTOMORPHIC REPRESENTATIONS 3 Remark 1.3. Adding in an archimedean comonent to K 0 N such as SO 2 R would give a more direct comarison with the uer half lane. Proof. As GL 2 Q contains elements with negative determinant, we may modify the strong aroximation statement to conclude that GL 2 A Q = GL 2 Q GL 2 R + K 0 N. Now consider the ma given by including GL 2 R + at the archimedean lace and assing to the quotient: GL 2 R + GL 2 A Q GL 2 Q\ GL 2 A Q /K 0 N. It is surjective by the strong aroximation theorem. Now let us analyze how close to being injective it is. Let g and g have the same image. This means there exist γ GL 2 Q and k 0 K 0 N such that g = γg k 0. The element γ is embedded diagonally, so searate the real and finite adelic arts, writing γ = γ γ f with γ GL 2 R and γ f GL 2 A f. As g and g are comletely in the archimedean lace, we see that g = γ g and γ f = k 1 0. The first says detγ > 0, and with the second it imlies γ f K 0 N GL 2 Q = Γ 0 N. Therefore g and g differ by an element of Γ 0 N. This roves the second isomorhism. For the first, we just add the centers. Strong aroximation for G m in this case says that Therefore we obtain an isomorhism ZA Q = ZR + ZQ ZA Q K 0 N. ZR + Γ 0 N\ GL 2 R + ZA GL 2 Q\ GL 2 A Q /K 0 N. But the left is isomorhic to Γ 0 N\ SL 2 R. 2. Constructing the Automorhic Form Associated to a Modular Form In this section, we will construct automorhic forms on GL 2 R + and GL 2 A Q associated to a holomorhic modular form. We are mainly interested in the adelic viewoint, but constructing the automorhic form adelically boils down to constructing it at the archimedean lace and then making a slight modification at the non-archimedean ones. The basic idea is to take the identification of saces in the revious section and set u a corresondence between functions on the uer half lane, GL 2 R +, and GL 2 A Q. Instead of actually working with the quotients, we will work with functions which satisfy transformation laws. In the course of the construction, we check the construction satisfies all of the requirements to be an automorhic form to illustrate how the classical roerties of modular forms aear in the automorhic language as conditions such as K-finiteness. Remark 2.1. The same story is true for Maass forms. For simlicity, we focus just on the case of holomorhic modular forms Classical Modular Forms. Let fz be a modular form of level N and weight k 2 with nebentyus character χ. Recall that this means: i f is a holomorhic function on the uer half lane H = {z : Imz > 0}. ii For Γ c d 0 N, we have az + b 2.1 f = χdcz + d k fz. cz + d

4 4 JEREMY BOOHER For γ = GL c d 2 R +, recall one defines jγ, z = cz + d and the slash oerator by g k γz = detγ k/2 jγ, z k gγ z. So this transformation rule can be rewritten as f k γ = χdf for γ Γ 0 N. iii f is holomorhic at the cuss. If the cus is infinity, this means that the q-exansion where q = e 2πiz is of the form fz = n 0 a n q n. For a general cus, ick γ SL 2 Z which takes infinity to that cus ask that f k γ be holomorhic at infinity. However, it is more convenient just to require that f k γ be holomorhic at infinity for all γ SL 2 Z. Recall that the coefficients a n are Fourier coefficients, so may be comuted for any z H via the integrals 2.2 a n = 1 0 fz + te 2πint dt. The theory of modular forms is discussed in many laces. Chater 1 of Bum [Bum97] resents the subject with a view towards connecting it with automorhic forms Automorhic Forms for GL 2 R. The classical theory of automorhic forms for GL 2 R is discussed in Chater 2 of Bum [Bum97]. It is imortant motivation and technical ingredient in the adelic theory for examle, sectral theory gives a decomosition of the sace of automorhic forms which is then translated to the adelic setting, but we will not assume knowledge of it. We will just discuss how a modular form is an examle of an automorhic form on GL 2 R before moving on the adelic situation. The observation that GL 2 R + acts on the uer half lane with stabilizer K = SO 2 R suggests how to connect modular forms on the uer half lane and automorhic forms on GL 2 R +. Given a cus form f, we consider the function defined on g = GL 2 R + by F g := f k gi = ad bc k/2 ci + d k f c d ai + b. ci + d This is the automorhic form for GL 2 R associated to f. It has many nice roerties. i For γ = Γ c d 0 N it satisfies F γg = f k γgi = χdf k gi = χdf g. cosθ sinθ ii For κ = K = SO sinθ cosθ 2 R, we see that detgκ k/2 jgκ, i k = detg k/2 jg, i k e ikθ as K is the stabilizer of i. Hence we see that Note that this imlies F is K-finite. λ 0 iii For γ = Z, we see 0 λ F gκ = detg k/2 jgκ, i k f g κ i = e ikθ F g. F γg = f k γgi = λ 2 k/2 detg k/2 λ k jg, i k fg i = ωγf g

5 VIEWING MODULAR FORMS AS AUTOMORPHIC REPRESENTATIONS 5 where ωγ is 1 when λ > 0 and is χ 1 when λ < 0. Note that χ 1 is arising because of subtleties arising from the choice of square root involved in defining detg k/2 : these are already resent in the classical theory of modular forms and are dealt with the same way. iv We show the function F is bounded if f was a modular form but not a cus form, it would be of moderate growth. First, a calculation shows that Im ai + b k/2 = ci + d detg ci + d 2 so it suffices to show that Imz k/2 fz is bounded. At infinity, using the q-exansion we see that for Imz > c k/2 fz = a 1 q + a 2 q C q where the constant C deends on c. But q = e 2π Imz, so Imz k/2 fz is bounded and goes to zero as Imz goes to infinity. Using the action of SL 2 Z, one can do a similar analysis for any cus. Then we use knowledge about fundamental domains in the uer half lane or the corresonding facts about Siegel domains for GL 2 to reduce to this case by covering by neighborhoods that is of this form around each cus. v Recall that the Lie algebra gl 2 R acts on the sace of smooth functions on GL 2 R via the derivative of the right translation action. So does its universal enveloing algebra Ugl 2 C. This action commutes with a Lalacian oerator which can be defined either by writing down an element in the center of the universal enveloing algebra or abstractly the negative of the Killing form gives a Riemannian metric on GL 2 R, and there is an associated Lalacian. We defer the extensive calculations to show that F is an eigenfunction of with eigenvalue k 2 1 k 2 to the analysis leading u to Corollary 3.3. vi We finally interret the condition of vanishing at the cuss. If a modular form fz vanishes at the cus at infinity, by 2.2 it follows that for all z H 1 0 fz + tdt = 0. In light of 1.1, elements of the form λ 0 y 1/2 y g = 1/2 x cosθ sinθ 0 λ 0 y 1/2 sinθ cosθ send i to x + iy. We calculate using ii and iii that 2.3 F g = f k gi = ωγe ikθ y k/2 fz. Furthermore, unwinding the left action of the uniotent matrix n t = F n t g = ωγe ikθ y k/2 fz + t. Therefore the cusidality condition is equivalent to 1 0 F n t gdt = 0 1 t we see for all g GL 2 R +. Using the action of SL 2 Z on the cuss, there is a similar statement for other cuss. This construction is discussed in [Bum97, Section 3.2]: in terms of the theory of automorhic forms on GL 2 R, this shows that F lies in the sace of cus forms A 0 Γ 0 N\ GL 2 R +, χ, ω.

6 6 JEREMY BOOHER 2.3. Automorhic Forms for GL 2 A Q. We now describe how to associate to f and F an automorhic form ϕ f on GL 2 A Q. The idea is similar to that of GL 2 R + : we use Proosition 1.2 to see a relation between the saces and then connect functions on GL 2 R with functions on GL 2 A Q that satisfy certain transformation roerties. The adelic ersective makes it much clearer that modular forms are arithmetic in nature and have local information at each rime. Given a cus form f, we define a function ϕ f on GL 2 A Q as follows. We use strong aroximation to write an element of GL 2 A as a roduct where γ GL 2 Q, g GL 2 R +, and k 0 K 0 N = { GL c d 2 Ẑ : c 0 mod N}. Then we define ϕ f γg k 0 := F g λk 0 = f k g iλk 0. The function λ is an adelization of the Dirichlet character χ. Since fz is not quite invariant under Γ 0 N, in light of Proosition 1.2 it is no surrise that ϕ f needs to incororate a correction from χ in order to end u K 0 N-invariant. We will first define λ and then check ϕ f is well defined. The character λ is defined to in two stes: first associate an adelic character ω to χ the grossencharacter, then convert this to a character of GL 2 A Q. Defining ω is an exercise in class field theory. The strong aroximation theorem states that A Q = Q R >0 Using the Chinese remainder theorem, we realize Z/NZ as a quotient of Z. Thus comosing with the inverse of χ gives us the grossencharacter ω = ω : A Q /Q C that is trivial on R >0. More details are found in [Bum97, Proosition 3.1.2]. Another way to exress this is to realize Z/NZ as the ray class grou over Q with modulus m = N. To get a character of K 0 N, we define λ := ωd = ω c d d N where d denotes the Q comonent of d. Let π A Q be the image of under the inclusion Q A Q. We record some roerties of ω and λ. Lemma 2.2. If N, we have ω Z = 1 and ωπ = χ. The archimedean comonent ω is trivial on R >0. Thus if d is a ositive integer rime to N, λ = χd c d 1. Proof. By construction ω Z = 1. Using the decomosition from the strong aroximation theorem, we can write π = 1 α where α Z with α v = 1/ for v and α = 1. Note that α reduces to 1/ in Z/NZ. 1 Then ωπ = χ 1 = χ. By construction ω is trivial on R >0. To rove the last claim, it suffices to show that χd = N Z ω 1 d. But as ω is trivial on Q, the right side is equal to N ω d. If d were a rational rime relatively rime to N, the first art of the lemma would imly this equals χd. Then extend by multilicativity. We now return to analyzing ϕ f. The roerties we established for the automorhic form F on GL 2 R associated to f contain most of the work.

7 VIEWING MODULAR FORMS AS AUTOMORPHIC REPRESENTATIONS 7 Proosition 2.3. The function ϕ f is well defined. It is an automorhic form with central character ω, and is a cus form. Proof. To check it is well defined, consider two decomositions γg k 0 = g = γ g k 0. Looking at the archimedean comonent of g gives γ g = γ g, so detγ 1 γ > 0. The finite comonents give that γ f k 0 = γ f k 0, so γ 1 f γ f K 0N. Hence γ 1 γ = = K c d 0 N GL 2 R + = Γ 0 N. Then the transformation law for F imlies that F g = F γ 1 γ g = χdf g. But by the lemma, χd 1 = λ = λγ c d 1 γ = λk 0 k 0 1. Hence F g λk 0 = F g λk 0. Thus ϕ f is well defined. To check that it is an automorhic form, there are a number of things to verify. It is smooth, because F is smooth in the archimedean sense and λ is locally constant. The following list of roerties is based on the roerties of F given in Section 2.2. i Left invariance under GL 2 Q follows by definition. But note well-definedness used the left invariance of F. ii Taking K = K 0 N SO 2 R, for k = k 0 k K we have that 2.4 ϕ f gk = F g k λkk 0 = e ikθ F g λkλk 0 = e ikθ λk 0 ϕ f g In articular, it is K-finite. iii For g GL 2 A Q and z A Q, we can check that ϕ f z 0 0 z g = ωzϕ f g. This is immediate for z Q, z R >0, and z Z, and follows in general using strong aroximation. iv ϕ f is bounded because we know that F is. v Letting Z be the center of the universal enveloing algebra of gl 2 R, ϕ f is Z-finite. This is because the action of Z is just in the archimedean comonent and we know that F is Z-finite and furthermore is an eigenfunction. vi The cusidality condition is that for all g GL 2 A Q, 1 x ϕ f g dx = 0. Q\A Q To evaluate the integral on the left, we use Q \A Q R >0 Ẑ and write the quotient measure as a roduct. But at the archimedean lace the integral is 0 because we know that F is cusidal. Remark 2.4. In fact, this construction gives an isomorhism between S k N, χ and the sace of functions on GL 2 A Q satisfying certain roerties: see [Gel75, Proosition 3.1] for a recise statement.

8 8 JEREMY BOOHER Thus from a cusidal holomorhic modular form f, we have constructed a cusidal automorhic form ϕ f A 0 GL 2 Q\ GL 2 A Q, ω. The next thing to do is to analyze the automorhic reresentation generated by ϕ f. We must first review the local archimedean and non-archimedean reresentation theory. 3. Background at the Archimedean Places In this section we will review material about reresentations of GL 2 R, starting with how the Lie algebra acts and what this action looks like in the coordinates of 1.1. We then review the classification of g, K-modules and the local Hecke algebra The Action of gl 2 R. Recall that via the adjoint reresentation, the Lie algebra gl 2 R acts on GL 2 R and hence gives an action of the sace of smooth functions on GL 2 R via right translation. For X gl 2 R and a smooth function f, the action is given by X fg = d dt fg extx t=0. In the case of GL 2, as I + tx is a ath in GL 2 R, for small t, tangent to the one arameter subgrou extx, an alternate exression is X fg = d dt fg1 + tx t=0. The following elements in gl 2 R are imortant: 0 0 ˆR =, ˆL =, Ĥ = 0 0, I = 0 1 Using these, we can define an element in the universal enveloing algebra = 1 Ĥ ˆR ˆL ˆR + 2ˆL. One shows that is in the center. In fact, the center is a olynomial algebra generated by and I. The action of gives the Lalace oerator. We will sketch how one derives an exlicit descrition of this oerator in terms of the x, y, θ, λ coordinate system on GL 2 R +. To do calculations, it is also convenient to work with related elements in the comlexification: R = 1 1 i, L = 1 1 i, H = i 2 i 1 2 i 1 Z =, = 1 4 H2 + 2RL + 2LR These are related via the Cayley transform to ˆR, ˆL.... Recall the Cayley transform is conjugation by C = 1 + i i 1. 2 i 1 It corresonds to the transformation taking the uer half lane into the unit disc. The actions of R and L are easier to comute. We will sketch a derivation of a formula for using this. Proosition 3.1. On the sace of smooth functions, the action of is given by y 2 2 x y 2 y 2 x θ.

9 VIEWING MODULAR FORMS AS AUTOMORPHIC REPRESENTATIONS 9 Proof. The idea is simle: write down the exonential for an element like ˆR or H and evaluate the derivative. Doing calculations is unavoidable, but necessary for what we wish to do. 0 1 For examle, let us do H. First consider W =. By definition, for a function f and g GL 2 R +, W fg = d dt fgk t t=0. By evaluating the matrix owers in the definition of the exonential, we see that cost sint extw = κ t = sint cost Exressing g in x, y, θ, λ coordinates using 1.1, we have λ 0 y 1/2 y g extw = 1/2 x 0 λ 0 y 1/2 κ θ+t This is x, y, θ +t, λ in that coordinate system, so the derivative evaluated at zero is θ fg. Using the action of W, one immediate gets the action of H. Likewise, we can easily deal with ˆR in the case of g having θ = 0. For then g ext ˆR λ 0 y 1/2 y = 1/2 x 1 t λ 0 y 1/2 y 0 λ 0 y 1/2 = 1/2 x + yt 0 λ 0 y 1/2. This has coordinates x + yt, y, 0, λ, so ˆR sends f to y f x g If θ is not zero, things are more comlicated because one needs to rearrange the roduct. The Cayley transform hels with this. Additional calculations can be found in [Bum97, Proosition 2.2.5]. Note that Bum uses θ for a coordinate in the rotation matrices which changes some signs. These calculations give the formula for. Remark 3.2. For future use, we also record that the action of L is given by e 2iθ iy x + y y + 1 2i θ If we had worked in more generality and identified Maass forms of weight k with automorhic forms, the oerators R and L would corresond to the raising and lowering oerators for Maass forms. Using this descrition, we can finally show that the automorhic form F on GL 2 R we associated to a cus form is an eigenfunction of. Corollary 3.3. Let f be a holomorhic modular form of weight k and F the associated automorhic form on GL 2 R +. Then F = k 1 k F 2 2 Proof. Unraveling the definition of F, we comute that F g = f k gi = e ikθ y k/2 fx + iy using the coordinates in 1.1. Now we comute that F = e ikθ y k/2+2 f xx + y k/2+2 f yy + ky k/2+1 f y + k k y k/2 f + y iky k/2 f x. Since f is holomorhic, f xx + f yy = 0 and if x = f y. Therefore we have that F = e ikθ k k y k/2 f = k 1 k F. 2 2

10 10 JEREMY BOOHER 3.2. g, K-Modules. Let g = gl 2 R and K = O 2 R. Recall that a g, K-module for GL 2 R is a vector sace V with reresentations π K and π g of K and g such that 1 The reresentations of g and K are comatible in the sense that an element X LieK acts the same way using π g action and through the infinitesimal action of K on V this refers to the action dπ K of LieK on V. 2 V is an algebraic direct sum of finite dimensional irreducible reresentations of K 3 For X g and k K we have π K kπ g Xπ K k 1 = π g AdkX as oerators on V. Such a module is admissible if each irreducible reresentation of K aears with finite multilicity. Let σ k be the reresentation of SO 2 R given by sending a matrix with angle θ to e ikθ. Then any g, K module V decomoses as a direct sum 3.1 V = k V k where V k is the isotyic comonent of V corresonding to σ k. Calculations with the Lie algebra show that ˆR and ˆL raise and lower k by two. Similarly, H acts by k on V k and Λ acts by k 2 1 k 2. For details, see [Bum97, Proosition 2.5.2]. There is also a classification of irreducible g, K modules in [Bum97, Theorem 2.5.5] or [JL70, Section 5]. If the reresentation is finite dimensional, it is a twist of the natural reresentation on degree n homogeneous olynomials in two variables. In this case V n 2, V n 4,..., V 2 n are one dimensional, and all the other V k are zero. This is the symmetric ower of the standard reresentation. Twisting means multilying by χ det where χ is a character of R. We could have V = V k k ɛ mod 2 where the V k aearing are one dimensional and ɛ = 0 or 1. This is a rincial series reresentation πχ 1, χ 2 where χ 1 and χ 2 are characters of R. Such a reresentation is irreducible rovided χ 1 χ 1 2 is not of the form signy ɛ y n 1 where n ɛ mod 2. Otherwise V is a discrete series or limit discrete series. In this case, there is a ositive integer k such that V l 0 recisely when l = ±k + 2n with n Z 0. Concretely, the discrete series can be realized as the K-finite vectors in the reresentation of GL 2 R + on the sace of holomorhic functions f : H C such that H fz 2 y k dxdy y 2 <. The action is given by sending f to f k g 1. Remark 3.4. This suggests the discrete series are connected with classical modular forms. We will see they are the comonent at infinity for the automorhic reresentation associated to a modular form Hecke Algebra. We will give a modern definition of the Hecke algebra in this context: this is slightly different aroach than in [JL70, Section 5]. A summary is found in [Bum97, Section 3.4] with references given to Kna and Vogan [KV95]. One definition of the Hecke algebra H G is as the algebra of comactly suorted distributions on GL 2 R that are suorted on K and are K-finite under left and right translation. However, there is a descrition more closely connected to g, K-modules. Let H K denote the sace of smooth functions on K which are K-finite under left and right translation by K. These can also be viewed as distributions by integrating over K. The universal enveloing algebra Ugl 2 C may be identified with distributions suorted at the identity by

11 VIEWING MODULAR FORMS AS AUTOMORPHIC REPRESENTATIONS 11 having Ugl 2 C act on functions and then evaluate at the identity. For X gl 2 C, this gives the distribution We record two imortant facts. f d dt fextx t=0. Proosition 3.5. The natural ma H K ULieK Ugl 2 C H G is an isomorhism. Proosition 3.6. Let V be a g, K-module. Then V is a smooth module for H G, and every smooth module arises in this way. 4. Background at the Non-Archimedean Places In this section, we review classification of irreducible admissible reresentations of GL 2 Q, and the theory of sherical reresentations Reresentations of GL 2 Q and Hecke Algebras. Recall that a reresentation of GL 2 Q on V is admissible if it is smooth and if for every comact oen subgrou K GL 2 Q we have dim V K <. There is a classification of irreducible admissible reresentations, found in [Bum97, Chater 4] or [JL70, Sections 2-4] or [BH06, Section 9]. All finite dimensional irreducible admissible reresentations are one dimensional, and factor through the determinant. There are rincial series reresentations πχ 1, χ 2 where χ 1 and χ 2 are characters of Q. When χ 1 χ 1 2 ±1, it is irreducible. There are secial reresentations, which are irreducible infinite dimensional subquotient of πχ 1, χ 2. These are all twists of the Steinberg reresentation, which occurs for χ 1 = 1/2 and χ 2 = 1/2. There are suer-cusidal reresentations. The Hecke algebra HG for G = GL 2 Q is the convolution algebra of locally constant comactly suorted functions. There is a natural action of HG on reresentations of GL 2 Q given by πφv = φgπgvdg. G Smooth reresentations of G are the same as HG-modules. Recall that idemotent elements of HG are given by normalized characteristic functions of comact oen subgrous K. The characteristic function of K = GL 2 Z is an idemotent for the usual measure. Definition 4.1. The sherical Hecke algebra HG, K is defined to be 1 K HG1 K, the sace left and right K-invariant elements of HG. An smooth admissible reresentation π, V is called sherical or unramified if V K 0. A non-zero element is called a sherical vector. The sherical vectors are a HG, K-module. We sketch some of the key facts about sherical vectors as these concets were not included in [JL70], 0 0 Let T and R be the characteristic functions of K K and K K. 0 Theorem 4.2. The sherical Hecke algebra HG, K is commutative. In articular, it is a olynomial algebra generated by T, R, and R 1. Proof. Using the transose ma, one defines an anti-involution of the sherical Hecke algebra. Then one checks that asis for HG, K, given by characteristic functions for double cosets of K, are invariant under transose [Bum97, Theorem 4.6.1]. Using the -adic Cartan decomosition, one

12 12 JEREMY BOOHER can describe all double cosets for K. Decomosing them as a union of left cosets, one can build all of them out of T, R, and R 1. For details, see Proositions and in [Bum97]. Remark 4.3. The identification of HG, K with the olynomial algebra is known as the Satake isomorhism. The other key fact says that the sherical vectors determine the reresentation. Theorem 4.4. For an irreducible unramified reresentations π, V, V K is one dimensional. There is an equivalence of categories between irreducible unramified reresentations and irreducible HG, K-modules sending V to V K. Proof. It is easy to see that V K is an irreducible HG, K-module, and it is automatically finite dimensional. As HG, K is commutative, it is one dimensional. [Bum97, Theorem 4.6.2] A quasi-inverse is defined in [BH06, 4.3 Proosition]. It sends a HG, K-module M to an irreducible submodule of U = HG HG,K M. The key content is using Zorn s lemma to construct a maximal subsace X U for which X K = 0. Anything outside X must generate U together with X. Hence the quotient U/X is an irreducible submodule with U/X K = M. One checks this is an equivalence of categories. Thus the way T and R act comletely determines an unramified reresentation. It is convenient to record their action using Satake arameters: if T and R act by λ and µ, the Satake arameters are the roots of x 2 k/2 1 λx + µ k Classification of Irreducible Sherical Reresentations. We now analyze which irreducible reresentations are sherical. If π, V is one dimensional, then GL 2 Q acts by χ det where χ is a quasi-character of Q. This is sherical if χ is unramified: in other words, if χ is trivial on Z. The rincial series πχ 1, χ 2 can be realized as the sace of smooth functions f : G C such that fbg = χ 1 χ 2 bδb 1/2 fg where b = is in the standard Borel B. Here χ 0 d 1 χ 2 b = χ 1 aχ 2 d and δb = a/d is the modular character. The Iwasawa decomosition gives that G = BK, and so a candidate for a sherical function is f sh bk = χ 1 χ 2 bδb 1/2. This function is well defined rovided it is trivial on K B. All such matrices are uer triangular and have units on the diagonal, so rovided χ 1 and χ 2 are unramified trivial on Z, f sh is well defined and gives a sherical vector. To determine the HG, K action on πχ 1, χ 2 K, we just need to understand how T and R act. The double coset for T decomoses as K 0 K = 1 b=0 b K K 0

13 Then we exand VIEWING MODULAR FORMS AS AUTOMORPHIC REPRESENTATIONS 13 T f sh = K 1 b f sh b = f sh b g + f sh 0 + f sh 0 = χ 1 1/2 + χ 2 1/2 = 1/2 χ 1 + χ 2. 0 The double coset for R is the single coset K, so 0 0 R f sh = f sh g dg K 0 0 = f sh 0 = χ 1 χ 2. g dg Proosition 4.5. The only sherical reresentations are χ det with χ unramified, and irreducible πχ 1, χ 2 with χ 1 and χ 2 unramified. Proof. Let V, π be sherical. Let T act on V K by λ and R act by µ. Note that µ is invertible as R is invertible in the Hecke algebra. The Satake arameters α 1 and α 2 are the roots of the quadratic x 2 k/2 1 λx + µ k 1. Let χ 1 and χ 2 be the unramified quasi-characters such that χ 1 = α 1 and χ k 1/2 2 = α 2. If πχ k 1/2 1, χ 2 is irreducible, its K-fixed vectors have the same Satake arameters. This forces V πχ 1, χ 2. So the only remaining ossible case is when πχ 1, χ 2 is not irreducible. This haens only when α 1 α2 1 = ±1. Without loss of generality, we may assume that α 1 α2 1 =, so by our knowledge of the rincial series we know πχ 1, χ 2 has a one dimensional invariant subsace. But then this must be the K-fixed vectors, so the reresentation is one dimensional. Remark 4.6. The conductor measures how far a reresentation is from being sherical. Let K 0 c denote matrices in GL 2 Z with lower left entry a multile of c, and let π, V be an infinite dimensional reresentation of GL 2 Q with central character ω. Then there exists a c 0 such that {v V : πgv = ωgv g K 0 c} is non-zero by a theorem of Casselman. The smallest such c for which this holds is called the conductor of π. In that case, the sace is one dimensional. A non-zero element is called a new vector. For the rincial series and Steinberg reresentation, it is relatively simle to find a new vector and comute the conductor by generalizing the construction of a sherical vector [Sch02]. In the case of the irreducible rincial series πχ 1, χ 2, the conductor is the sum of the conductors of χ 1 and χ 2. For the Steinberg, the conductor is 1. For a suercusidal reresentation, the conductor is least two. Furthermore, the central character of a suercusidal has conductor at most [ r 2 ] see [AL78, Theorem 4.3 ]. 5. The Automorhic Reresentation Associated to a Modular Form Let f be a holomorhic modular form of level N and nebentyus character character χ. Suose further that it is a cus form and an eigenfunction for the Hecke oerators T for N. We have

14 14 JEREMY BOOHER associated an adelic automorhic form ϕ f A 0 GL 2 Q\ GL 2 A Q, ω. In this section, we will rove the following result: Theorem 5.1. The automorhic form ϕ f lies in an unique irreducible admissible automorhic reresentation π f A 0 GL 2 Q\ GL 2 A Q, ω. The reresentation π f can be exressed as a restricted tensor roduct vπ f,v. We would also like to describe the comonents π f,v in terms of the roerties of f. After reviewing some facts about the Hecke algebra and connecting it to the classical Hecke oerators, we state the strong multilicity one theorem and then turn to roving Theorem 5.1 and discussing the local comonents The Hecke Algebra and Hecke Oerators. The global Hecke algebra H GL2 A Q is defined to be the restricted tensor roduct of the local Hecke algebras for the archimedean and nonarchimedean laces of Q reviewed in the revious two sections. For this to be defined, we needed to secify a sherical idemotent in all but finitely many of the local Hecke algebras. We use the characteristic function of GL 2 Z in H GL2 Q. The reresentation π f can be viewed as a H GL2 A Q -module. There is a factorization π f = vπ f,v where π f,v is a H GL2 Q v-module. This also means that there is a sherical vector in almost all of the π f,v a vector invariant under the action of GL 2 Z v. Such vectors are unique u to scaling. The first ste is to connect the classical Hecke oerators with the adelic ones. We studied the Hecke oerator T H GL2 Q given by convolution with the characteristic function of 0 H = GL 2 Z GL 2 Z in Section 4. In the following roosition, will denote an c d element of GL 2 A Q with secified matrix at and the identity elsewhere. Proosition 5.2. Let f be a modular form of level N, and suose is a rime such that N. Then T ϕ f = ϕ 1 k/2 T f. Proof. It suffices to check equality for g = g GL 2 R + because ϕ f is left GL 2 Q-invariant and because of Remark 2.4. Recall that the double coset decomoses as H = We calculate that 1 T ϕ f g = b=0 1 b=0 Z ϕ f b g GL 2 Z GL 0 2 Z b k dk + ϕ f g Z k dk. 0 As N, the comonent of K 0 N is GL 2 Z and λ is trivial on it Lemma 2.2. Remark 2.4, ϕ f is right invariant under GL 2 Z. Hence b b ϕ f g k dk = ϕ Z f g b=0 So by because the volume of Z is one. There is a similar exression for the other coset. Therefore we unwind to see that 1 b T ϕ f g = ϕ f g + ϕ f g. 0 Now let z = g i, and note that b g = b Q 1 b g α

15 where α v = VIEWING MODULAR FORMS AS AUTOMORPHIC REPRESENTATIONS 15 1 b for v, and is the identity otherwise. Observe that λα = 1. Therefore z b b ϕ f g = F 1 b g = k/2 detg k/2 jg, i k f Doing the same for the remaining coset gives ϕ f g = 0 k/2 detg k/2 jg, i k f z. Therefore we conclude that 1 k/2 1 1 z b T ϕ f g = detg k/2 jg, i k f + k 1 detg k/2 jg, i k fz b=0 = detg k/2 jg, i k T fz = ϕ Tf g using the standard normalizations of the classical Hecke oerators. Similarly, we interret the oerator R given by convolution with the characteristic function of 0 GL 0 2 Z to Proosition 5.3. For N, we have R ϕ f = χdϕ f. Proof. We simlify R ϕ f g = GL 2 Z ϕ f 0 g 0 k dk = ϕ f 0 g 0 But we can rearrange g = g α Q 1 0 where α v = for v, and is the identity elsewhere. We see that 0 ϕ f 0 g 0 = F g 0 λα. Now λα = χ by Lemma 2.2, and F is left invariant under ZR +. Thus R ϕ f = χϕ f Proof of Theorem 5.1. The roof will use the multilicity one theorem, a roof of which may be found in [JL70, Section 11] or [Bum97, Section 3.5]. Theorem 5.4 Multilicity One. Let π, V and π, V be two irreducible admissible subreresentations of A 0 GL 2 Q\ GL 2 A Q, ω. Assume that π v π v for all but finitely many laces. Then V = V. Using this, we now rove Theorem 5.1. Let f be a cus form and Hecke eigenform. By the general theory, the sace of automorhic forms decomoses as a direct sum. Let π, V be one irreducible factor such that the rojection of ϕ f is non-zero. We will show that all of the local comonents of π at laces not dividing N or infinity are determined by f. Let N be rime. We know that ϕ f is invariant under right translation by K 0 N, so in articular the local comonent π contains an..

16 16 JEREMY BOOHER element which is left and right GL 2 Z -translation invariant. It is an eigenvector for T and R with eigenvalues determined by f, and this action determines the action of HGL 2 Q, GL 2 Z. In other words, the local comonent π is an sherical reresentation of H that is comletely secified in terms of f. By the multilicity one theorem, this forces ϕ f to lie in a unique irreducible π, V, roving the theorem. Remark 5.5. It is also easy to analyze the comonent at infinity. The version of the multilicity one statement roven in Bum is slightly weaker in that it needs an isomorhism at the archimedean laces, so it is worth selling this out. From Section 3, we know quite it about the otential g, K-modules at infinity. In articular, the formula for the lowering oerator in Remark 3.2 allows us to calculate that for F g = e ikθ y k/2 fx + iy we have L ϕ f = e 2iθ iy F x + y F y + 1 F 2i θ = e ie 2πθ ikθ y k/2+1 f x + e ikθ k 2 yk/2 f + e ikθ y k/2+1 f y + 1 2i ike ikθ y k/2 f = 0 where the last ste uses that f is holomorhic so f y = if x. So any irreducible automorhic reresentation to which ϕ f rojects non-trivially contains a vector on which L acts by zero. We already know that acts by k 2 1 k 2. Using the classification of irreducible g, K-modules, we see the only one with this roerty is the discrete series of weight k. This should not be a surrise, as one realization of this reresentation includes a holomorhic modular form. It is harder to characterize the local comonent π f, for N. A recent aer by Loeffler and Weinstein gives an algorithm for comuting the local comonents [LW12]. The hard case is when the local comonent is suercusidal. We sketch a descrition of the non-suercusidal cases using the concet of conductor from Remark 4.6. We say that π f, is -rimitive if the conductor is minimal among the conductors of the - comonents of twists of f by characters. An irreducible rincial series πχ 1, χ 2 is rimitive if at least one of the characters is unramified. The conductor is the sum of the conductors for χ 1 and χ 2, so if both were unramified we could twist to reduce the conductor. A secial reresentation is a twist of a Steinberg, and the unramified twists the Steinberg reresentation have minimal conductor one. So suose π f, is -rimitive, and let r be the largest ower of which divides N. We know that r is the conductor of π f,. Recall ω is the -comonent of the central character, and was defined so that it is trivial rovided χ factors through Z/N/ r Z. We know that the central character of a suercusidal reresentation has conductor at most [ r 2 ]. Furthermore, we know suercusidals must have conductor at least 2. So if ω has conductor more than [r/2], π must be a rincial series reresentation. The only -rimitive one is πχ 1, χ 2 with χ 1 having conductor r and with χ 2 unramified. If r = 1 and ω is unramified, π must be a twist of a Steinberg by an unramified character. The remaining cases must then be suercusidal. The conclusion of this analysis is the following theorem. Theorem 5.6. Let fz = n 1 a n q n be a cusidal eigenform of weight k, level N and nebentyus character χ. Let π f be the associated irreducible automorhic reresentation, with π f = vπ f,v. For a rime N, ω is the -comonent of the central character. For v =, the local comonent π f, is a discrete series of weight k.

17 VIEWING MODULAR FORMS AS AUTOMORPHIC REPRESENTATIONS 17 For N, the local comonent π f, is an unramified rincial series. On the sherical vector, T and R act via multilication by a 1 k/2 and χ. The Satake arameters are therefore the roots of x 2 a x + χ k 1. Suose N = r N where N and that f is a new form at and that f is -rimitive. If the conductor of π f, is r, the local factor is the rincial series πχ 1, χ 2 where χ 1 is unramified and satisfies χ 1 = a / k 1/2 and χ 1 χ 2 = ω. With the same notation, if r = 1 and ω is unramified, then the local factor π f, is the twist of the Steinberg reresentation by an unramified character χ 1 such that χ 1 k 2/2 = a. Otherwise the local comonent at is suercusidal, and can be described by the algorithm in [LW12]. An examle of the automorhic reresentation associated to a weight 2 level 99 modular form is included in the introduction. References [AL78] A. O. L. Atkin and Wen Ch ing Winnie Li, Twists of newforms and seudo-eigenvalues of W -oerators, Invent. Math , no. 3, MR a:10040 [BH06] Colin J. Bushnell and Guy Henniart, The local Langlands conjecture for GL2, Grundlehren der Mathematischen Wissenschaften [Fundamental Princiles of Mathematical Sciences], vol. 335, Sringer-Verlag, Berlin, MR m:22013 [Bum97] Daniel Bum, Automorhic forms and reresentations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, MR k:11080 [Gel75] Stehen S. Gelbart, Automorhic forms on adèle grous, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975, Annals of Mathematics Studies, No. 83. MR #280 [JL70] H. Jacquet and R. P. Langlands, Automorhic forms on GL2, Lecture Notes in Mathematics, Vol. 114, Sringer-Verlag, Berlin-New York, MR #5481 [KV95] Anthony W. Kna and David A. Vogan, Jr., Cohomological induction and unitary reresentations, Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton, NJ, MR c:22023 [LW12] David Loeffler and Jared Weinstein, On the comutation of local comonents of a newform, Math. Com , no. 278, MR k:11064 [Sch02] Ralf Schmidt, Some remarks on local newforms for GL2, J. Ramanujan Math. Soc , no. 2, MR g:11056