THE MAASS SPACE FOR U(2, 2) AND THE BLOCH-KATO CONJECTURE FOR THE ADJOINT MOTIVE OF A MODULAR FORM

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1 THE MAASS SPACE FOR U(2, 2) AND THE BLOCH-KATO CONJECTURE FOR THE ADJOINT MOTIVE OF A MODULAR FORM KRZYSZTOF KLOSIN Abstract. Let K be an imaginary quadratic field of discriminant D K. We introduce a notion of an adelic Maass sace Sk, k/2 M for automorhic forms on the quasi-slit unitary grou U(2, 2) associated with K and rove that it is stable under the action of all Hecke oerators. When D K is rime we obtain a Hecke-equivariant descent from Sk, k/2 M to the sace of ellitic cus forms S k 1 (D K, χ K ), where χ K is the quadratic character of K. For a given φ S k 1 (D K, χ K ), a rime l > k, we then construct (mod l) congruences between the Maass form corresonding to φ and hermitian modular forms orthogonal to Sk, k/2 M whenever val l(l alg (Symm 2 φ, k)) > 0. This gives a roof of the holomorhic case of the unitary analogue of Harder s conjecture. Finally, we use these congruences to rovide evidence for the Bloch-Kato conjecture for the motives ad 0 ρ φ ( 1) and ad 0 ρ φ (2), where ρ φ denotes the Galois reresentation attached to φ. 1. Introduction In 1990 Bloch and Kato [6] formulated a conjecture whose version relates the order of a Selmer grou of a motive M to a secial value of an L-function of M. This is a very far-reaching conjecture which is currently known only in a handful of cases, mostly concerning the situations when M arises from a one-dimensional Galois reresentation. However, in 2004 Diamond, Flach and Guo roved a very strong result in a three-dimensional case [11]. Indeed, they roved the Bloch-Kato conjecture for the adjoint motive ad 0 ρ φ (and its Tate twist ad 0 ρ φ (1)) of the l- adic Galois reresentation ρ φ attached to a classical modular form φ without any restrictions on the weight ( 2) or the level of φ. Their roof was highly influenced by the ideas that were first alied by Taylor and Wiles in their roof of Fermat s Last Theorem ([43], [48]). In 2009 the author roved a (weaker) result roviding evidence for the conjecture for a different Tate twist of ad 0 ρ φ (namely ad 0 ρ φ ( 1) and ad 0 ρ φ (2)) for modular forms φ of any weight k 1 (with k divisible by 4), level 4 and non-trivial character [27]. The method was different from that of [11] (but similar to the one used by Brown [8] who worked with Saito-Kurokawa lifts). It relied on constructing congruences between a certain lift of φ (called the Maass lift) to the unitary grou U(2, 2) defined with resect to the field Q(i) and hermitian modular forms (i.e., forms on U(2, 2)) which were orthogonal to the Maass sace (the san of such lifts). The elements in the relevant Selmer grous were then constructed using ideas of Urban [44]. Unfortunately, some of the methods imlemented in [27] relied Date: March 31,

2 2 KRZYSZTOF KLOSIN significantly on the fact that the class number of Q(i) is one and could not be directly generalized to deal with other imaginary quadratic fields. In this aer we develo new tools - among them a new notion of an adelic Maass sace and a Rankin-Selberg tye formula - which work in sufficient generality. As a consequence we are in articular able to extend the results of [27] to all imaginary quadratic fields of rime discriminant D K, i.e., to all modular forms φ of any rime level D K, of arbitrary weight k 1 (with k divisible by the number of roots of unity contained in K), and nebentyus χ K being the quadratic character associated with the extension K/Q. Our result on the one hand rovides evidence for the Bloch- Kato conjecture for the motives ad 0 ρ φ ( 1) and ad 0 ρ φ (2) for a rather broad family of modular forms φ. On the other hand the congruence itself (between a Maass lift and a hermitian modular form orthogonal to the Maass sace) rovides a roof of the unitary analogue of Harder s conjecture (as formulated recently by Dummigan [12]) in the extreme (holomorhic) case. As alluded above, the first difficulty that one encounters in dealing with a general imaginary quadratic field is the lack of a roer notion of the Maass sace in this case. The definition introduced by Krieg [29] does not allow one to define the action of the Hecke oerators at non-rincial rimes. The more recent (very elegant) results due to Ikeda [24] while dealing with class number issues, are not quite sufficient for our uroses. So, in this aer we introduce a new adelic version of the Maass sace and carefully study its roerties, esecially its invariance under the action of the Hecke algebras. This rovides us with a correct analogue of the classical Maass lift to the sace of automorhic forms on U(2, 2)(A) for any imaginary quadratic field of rime discriminant. We then roceed to construct a congruence between the Maass lift and hermitian modular forms orthogonal to the Maass sace. The method of exhibiting elements in Selmer grous of automorhic forms via constructing congruences between automorhic forms on a higher-rank grou has been used by several authors. The original idea can be dated back to the influential aer of Ribet [33] on the converse to Herbrand s Theorem, where for a certain family of Dirichlet characters χ elements in the χ-eigensace of the class grou of a cyclotomic field are constructed by first exhibiting a congruence between a certain Eisenstein series (associated with χ) and a cus form on GL 2. Higher-rank analogues of this method have been alied to rovide evidence for (one inequality in) the Bloch-Kato conjecture for several motives by Bellaïche and Chenevier, Brown, Berger as well as Agarwal and the author ([2], [8], [4], [27], [1]). An extension of these ideas was also used to rove results towards the Main Conjecture of Iwasawa Theory by Mazur and Wiles, Urban ([30], [44]) and very recently by Skinner and Urban [41]. The general idea is the following. Given an automorhic form φ on an algebraic grou M (with an associated Galois reresentation ρ φ ) one lifts φ to an automorhic form on G in which M can be realized as a subgrou (in our case M = Res K/Q (GL 2/K ) is the Levi subgrou of a Siegel arabolic of G = U(2, 2)). The Galois reresentation attached to the lift is reducible and has irreducible comonents related to ρ φ. Assuming divisibility (by a uniformizer ϖ in some extension of Q l ) of a certain L-value associated with φ one shows (this is usually the technically difficult art) that the lift is congruent (mod ϖ) to an automorhic form π on G whose Galois reresentation ρ π is irreducible. Because of the congruence,

3 THE MAASS SPACE AND THE BLOCH-KATO CONJECTURE 3 the mod ϖ reduction of ρ π must be reducible, but (because ρ π was irreducible) it can be chosen to reresent a non-slit extension of its irreducible comonents, thus giving rise to a non-zero element in some Selmer grou (related to ρ φ ). Let φ S k 1 (D K, χ K ) be a newform. In our case the lifting rocedure is the Maass lift, which roduces an automorhic form f φ,χ on U(2, 2)(A) (which deends on a certain character χ of the class grou of K) whose associated automorhic reresentation is CAP in the sense of Piatetski-Shairo [32]. Even though the desired congruence is between Hecke eigenvalues of the lift f φ,χ and those of π, we first construct a congruence between Fourier coefficients of these forms and only then deduce the Hecke eigenvalue congruence. The former congruence is achieved by first defining a certain hermitian modular form Ξ (essentially a roduct a Siegel Eisenstein series and a hermitian theta series) and writing it as: Ξ = Ξ, f φ,χ f φ,χ, f φ,χ f φ,χ + g, where g is a hermitian modular form orthogonal to f φ,χ. The form Ξ has nice arithmetic roerties (in articular its Fourier coefficients are ϖ-adically integral) and we show that the inner roducts can be exressed by certain L-values. In articular the inner roduct in the denominator is related to L alg (Symm 2 φ, k). Choosing Ξ so that the secial L-values contributing to the inner roduct in the numerator make it a ϖ-adic unit and assuming the ϖ-adic valuation of L alg (Symm 2 φ, k) is ositive we get a congruence between f φ,χ and a scalar multile g of g. To ensure that g itself is not a Maass lift we construct a certain Hecke oerator T h that kills the Maass art of g. Let us now briefly elaborate on the technical difficulties that one encounters in the current aer as oosed to the case of K = Q(i) which was studied in [27]. First of all, as mentioned above, the Maass sace and the Maass lift in the case of the field Q(i) are well-understood thanks to the work of Kojima, Gritsenko and Krieg ([28], [15], [16], [29]) and in [27] we simly invoke the relevant definitions and roerties of these objects. In the current aer we introduce a notion of an adelic Maass sace for a general imaginary quadratic field and rove that it is a Hecke-stable subsace of the sace of hermitian modular forms. This result does not use the assumtion that D K is rime. One of the difficulties in extending the classical notion of Kojima and Krieg is the fact that when the class number of K is greater than one the classical Maass sace (which was defined by Krieg for all imaginary quadratic fields) is not stable under the action of the local Hecke algebras at non-rincial rimes of K. This is one of the reasons why we chose to formulate the theory in an adelic language, even though it would in rincile be ossible to extend the classical definitions of Krieg and work with several coies of the hermitian uer half-sace. However, we think that the action of the Hecke oerators as well as the role layed by a central character are most transarent in the adelic setting. When D K is rime we are able to relate our lift to the results of Krieg and Gritsenko and derive exlicit formulas for the descent of the Hecke oerators. We also rove an L-function identity relating the standard L-function of a Maass lift to the L-function of the base change to K of the modular form φ. All this is the content of section 5. A yet another notion of the Maass lift has in the meantime been introduced by Ikeda [24] using a different aroach. This notion

4 4 KRZYSZTOF KLOSIN agrees with ours in the case of a trivial central character, but not all the formulas necessary for our arithmetic alications are resent in [24]. On the other hand a lot of attention in [27] was devoted to comuting the Petersson norm of a Maass lift ([27], section 4). Here, however, we use a formula due to Sugano (cf. [24]) to tackle the roblem. To derive the congruence one also needs to be able to exress the inner roduct Ξ, f φ,χ by certain L-values related to φ. This calculation (drawing heavily on the work of Shimura) becomes somewhat involved in the case of class number larger than one. The relevant comutations are carried out in section 7. Since it does not add much to the comutational comlexity we rove all the results for the grou U(n, n) for a general n > 1, obtaining this way a general Rankin-Selberg tye formula that might be of indeendent interest (Theorem 7.8). In that section we also rove the integrality of the Fourier coefficients of a certain hermitian theta series involved in the definition of Ξ. Finally in the current aer our construction of the Hecke oerator T h is somewhat cleaner, because we work more with comleted Hecke algebras. This is carried out in section 6. In section 8 we collect all the results to arrive at the desired congruence, first between the Fourier coefficients of f φ,χ and a hermitian modular form orthogonal to the Maass sace (Theorem 8.10) and then between the Hecke eigenvalues of f φ,χ and those of some hermitian Hecke eigenform f also orthogonal to the Maass sace (Corollary 8.16). The latter can only be achieved modulo the first ower of ϖ even if L alg (Symm 2 φ, k) is divisible by a higher ower of ϖ. This is not a shortcoming of our method but a consequence of the fact that there may be more than one f congruent to f φ,χ and the L-value (conjecturally) controls contributions from all such f. In fact this is recisely what we rove by studying congruences between f φ,χ and all the ossible eigenforms f orthogonal to the Maass sace and as a result give a lower bound (in terms of L alg (Symm 2 φ, k)) on the index of an analogue of the classical Eisenstein ideal (which we in our case call the Maass ideal) in the aroriate local Hecke algebra (section 8.3). Finally, we demonstrate how our results imly the holomorhic case of the analogue of Harder s conjecture for the grou U(2, 2) (section 8.4). As exlained above the congruence can then be used to deduce the existence of certain non-zero elements in the Selmer grous Hf 1(K, ad0 ρ f ( 1)) and Hf 1(K, ad0 ρ f (2)) and hence get a result towards the Bloch-Kato conjecture for these motives. For this we use a theorem of Urban [44]. The relevant results are stated in section 9. We note that the above Selmer grous are over K, while the conjecture relates L alg (Symm 2 φ, k) to the order of the corresonding Selmer grou over Q. In the current aer in section 9 we establish (using some results of Bellaïche and Chenevier [3]) a criterion under which the extension we construct using our congruence lies in the art of Hf 1(K, ad0 ρ φ ( 1)) fixed under the action of the comlex conjugation and hence gives rise to elements in H 1 (Q, ad 0 ρ φ ( 1)), as redicted by the conjecture. This result alies also to the case K = Q(i). The author would like to thank Chris Skinner who initially suggested this roblem to him. Over the course of work on this aer the author benefited greatly from conversations and corresondence with many eole and would like to exress his articular gratitude to Günther Harder, Tamotsu Ikeda and Jacques Tilouine. He is also grateful to the Deartment of Mathematics of the Université Paris 13, where art of this work was carried out, for a friendly and stimulating atmoshere during the author s stay there in 2009 and 2010 and similarly to the

5 THE MAASS SPACE AND THE BLOCH-KATO CONJECTURE 5 Max-Planck-Institut in Bonn, where the author sent the Summer of Finally, we would like to thank Neil Dummigan for sending us his rerint on the analogue of Harder s conjecture for the grou U(2, 2). 2. Notation and terminology In this section we introduce some basic concets and establish notation which will be used throughout this aer unless exlicitly indicated otherwise Number fields and Hecke characters. Throughout this aer l will always denote a fixed odd rime. We write i for 1. Let K be a fixed imaginary quadratic extension of Q, and let O K be the ring of integers of K. We will write Cl K for the class grou of K and h K for # Cl K. For α K, denote by α the image of α under the non-trivial automorhism of K. Set Nα := N(α) := αα, and for an ideal n of O K, set Nn := #(O K /n). As remarked below we will always view K as a subfield of C. For α C, α will denote the comlex conjugate of α and we set α := αα. Let L be a number field with ring of integers O L. For a lace v of L, denote by L v the comletion of L at v and by O L,v the valuation ring of L v. If is a lace of Q, we set L := Q Q L and O L, := Z Z O L. We also allow =. Set Ẑ = lim Z/nZ = Z and similarly ÔK = v O K,v. For a finite, let n val denote the -adic valuation on Q. For notational convenience we also define val ( ) :=. If α Q, then α Q := val(α) denotes the -adic norm of α. For =, Q = R = is the usual absolute value on Q = R. In this aer we fix once and for all an algebraic closure Q of the rationals and algebraic closures Q of Q, as well as comatible embeddings Q Q C for all finite laces of Q. We extend val to a function from Q into Q. Let L be a number field. We write G L for Gal(Q/L). If is a rime of L, we also write D G L for the decomosition grou of and I D for the inertia grou of. The chosen embeddings allow us to identify D with Gal(L /L ). We will always write Frob D /I to denote the arithmetic Frobenius. For a local field E (which for us will always be a finite extension of Q for some rime ) and a choice of a uniformizer ϖ E, we will write val ϖ : E Z for the ϖ-adic valuation on E. For a number field L let A L denote the ring of adeles of L and ut A := A Q. Write A L, and A L,f for the infinite art and the finite art of A L resectively. For α = (α ) A set α A := α Q. By a Hecke character of A L (or of L, for short) we mean a continuous homomorhism ψ : L \ A L C. The trivial Hecke character will be denoted by 1. The character ψ factors into a roduct of local characters ψ = v ψ v, where v runs over all laces of L. If n is the ideal of the ring of integers O L of L such that ψ v (x v ) = 1 if v is a finite lace of L, x v O L,v and x 1 no L,v no ideal m strictly containing n has the above roerty, then n will be called the conductor of ψ. If m is an ideal of O L, then we set ψ m := ψ v, where the roduct runs over all the finite laces v of L such that v m. For a Hecke character ψ of A L, denote by ψ the associated ideal character. Let ψ be a Hecke character of A K. We will sometimes think of ψ as a character of

6 6 KRZYSZTOF KLOSIN (Res K/Q GL 1/K )(A), where Res K/Q stands for the Weil restriction of scalars. We have a factorization ψ = ψ into local characters ψ : ( ) Res K/Q GL 1/K (Q ) C. For M Z, we set ψ M :=, M ψ. If ψ is a Hecke character of A K, we set ψ Q = ψ A The unitary grou. For any affine grou scheme X over Z and any Z-algebra A, we denote by x x the automorhism of (Res OK /Z X OK )(A) induced by the non-trivial automorhism of K/Q. Note that (Res OK /Z X OK )(A) can be identified with a subgrou of GL n (A O K ) for some n. In what follows we always secify such an identification. Then for x (Res OK /Z X OK )(A) we write x t for the transose of x, and set x := x t and ˆx := (x t ) 1. Moreover, we write diag(a 1, a 2,..., a n ) for the n n-matrix with a 1, a 2,... a n on the diagonal and all the off-diagonal entries equal to zero. We will denote by G a the additive grou and by G m the multilicative grou. To the imaginary quadratic extension K/Q one associates the unitary similitude grou scheme over Z: G n := GU(n, n) = {A Res OK /Z GL 2n AJĀt = µ(a)j}, [ ] I where J = n, with I n denoting the n n identity matrix and µ(a) G m. I n We will also make use of the grous and U n = U(n, n) = {A GU(n, n) µ(a) = 1}, SU(n, n) = {A U n det A = 1}. [ ] x For x Res OK /Z(GL n ), we write x for U ˆx n. Since the case n = 2 will be of articular interest to us we set G = G 2, U = U 2. Note that if is inert or ramified in K, then K /Q is a degree two extension of local fields and a a induces the non-trivial automorhism in Gal(K /Q ). If slits in K, denote by ι,1, ι,2 the two distinct embeddings of K into Q. Then the ma a b (ι,1 (a)b, ι,2 (a)b), induces a Q -algebra isomorhism K = Q Q, and a a corresonds on the right-hand side to the automorhism defined by (a, b) (b, a). We denote the isomorhism Q Q K by ι. For a matrix g = (g ij ) with entries in Q Q we also set ι (g) = (ι (g ij )). For a slit rime the ma ι 1 identifies U n (Q ) with U n, = {(g 1, g 2 ) GL 2n (Q ) GL 2n (Q ) g 1 Jg t 2 = J}. Note that the ma (g 1, g 2 ) g 1 gives a (non-canonical) isomorhism U n (Q ) = GL 2n (Q ). Similarly, one has G n (Q ) = GL 2n (Q ) G m (Q ). In U = U 2 we choose a maximal torus a T = b â a, b Res K/Q G m/k ˆb,

7 THE MAASS SPACE AND THE BLOCH-KATO CONJECTURE 7 and a Borel subgrou B = T U B with uniotent radical 1 α β γ U B = 1 γ ᾱφ φ 1 α, β, γ Res K/Q G a/k, φ G a, β + γᾱ G a. ᾱ 1 Let a T Q = b a 1 a 1 1 b 1 a, b G m denote the maximal Q-slit torus contained in T. Let R(U) be the set of roots of T Q, and denote by e j, j = 1, 2, the root defined by a 1 e j : a 2 a j. The choice of B determines a subset R + (U) R(U) of ositive roots. We have a 1 2 R + (U) = {e 1 + e 2, e 1 e 2, 2e 1, 2e 2 }. We fix a set (U) R + (U) of simle roots (U) := {e 1 e 2, 2e 2 }. If θ (U), denote the arabolic subgrou corresonding to θ by P θ. We have P (U) = U and P = B. The other two ossible subsets of (U) corresond to maximal Q-arabolics of U: the Siegel arabolic P := P {e1 e 2} = M P U P with Levi subgrou {[ } A M P = A Res Â] K/Q GL 2/K, and (abelian) uniotent radical 1 b 1 b 2 U P = 1 b 2 b 4 1 b 1, b 4 G a, b 2 Res K/Q G a/k 1 the Klingen arabolic Q := P {2e2} = M Q U Q with Levi subgrou x [ ] M Q = a b a b ˆx x Res K/Q G m/k, U(1, 1), c d c d and (non-abelian) uniotent radical 1 α β γ U Q = 1 γ 1 α, β, γ Res K/Q G a/k, β + γᾱ G a ᾱ 1

8 8 KRZYSZTOF KLOSIN Similarly in U n we denote by T n the diagonal torus and by P n the Siegel arabolic (with Levi isomorhic to Res K/Q GL n/k ). For an associative ring R with identity and an R-module N we write M n (N) for the R-module of n n-matrices with entries in N. Let x = [ C A D B ] M 2n(N) with A, B, C, D M n (N). Define a x = A, b x = B, c x = C, d x = D. For M Q, N Z such that MN Z we will denote by D n (M, N) the grou U n (R) K 0,n,(M, N) U n (A), where (2.1) K 0,n, (M, N) = {x U n (Q ) a x, d x M n (O K, ), b x M n (M 1 O K, ), c x M n (MNO K, ) }. If M = 1, denote D n (M, N) simly by D n (N) and K 0,n, (M, N) by K 0,n, (N). For any finite, the grou K 0,n, := K 0,n, (1) = U n (Z ) is a maximal (oen) comact subgrou of U n (Q ). Note that if N, then K 0,n, = K 0,n, (N). We write K 0,n,f (N) := K 0,n,(N) and K 0,n,f := K 0,f (1). Note that K 0,n,f is a maximal (oen) comact subgrou of U n (A f ). Set {[ ] } K 0,n, + A B := U B A n (R) A, B GL n (C), AA + BB = I n, AB = BA. Then K + 0,n, is a maximal comact subgrou of U n(r). We will denote by K 0,n, the subgrou of G n (R) generated by K + 0,n, and J. Then K 0,n, is a maximal comact subgrou of G n (R). Let We have U(m) := {A GL m (C) AA = I m }. K + 0,n, = U n(r) U(2n) U(n) U(n), where the last isomorhism is given by [ ] A B (A + ib, A ib) U(n) U(n). B A Finally, set K 0,n (N) := K + 0,n, K 0,n,f(N) and K 0,n := K 0,n (1). The last grou is a maximal comact subgrou of U n (A). Similarly, we define K 1,n (N) = K + 0,n, K 1,n,f(N), where K 1,n,f (N) = K 1,n,(N), K 1,n, (N) = {x K 0,n, (N) a x I n M n (NO K, )}. Let M Q, N Z be such that MN Z. We define the following congruence subgrous of U n (Q): (2.2) Γ h 0,n(M, N) := U n (Q) D n (M, N), Γ h 1,n(M, N) := {α Γ h 0,n(M, N) a α 1 M n (NO K )},, Γ h n(m, N) := {α Γ h 1,n(M, N) b α M n (M 1 NO K )} and set Γ h 0,n(N) := Γ h 0,n(1, N), Γ h 1,n(N) := Γ h 1,n(1, N) and Γ h n(n) := Γ h n(1, N). When n = 2 we dro it from notation. Note that the grous Γ h 0(N), Γ h 1(N) and Γ h (N) are U n -analogues of the standard congruence subgrous Γ 0 (N), Γ 1 (N) and Γ(N) of SL 2 (Z). In general the suerscrit h will indicate that an object is in some way related to the grou U n. The letter h stands for hermitian, as this is the standard name of modular forms on U n.

9 THE MAASS SPACE AND THE BLOCH-KATO CONJECTURE Modular forms. In this aer we will make use of the theory of modular forms on congruence subgrous of two different grous: SL 2 (Z) and U(Z). We will use both the classical and the adelic formulation of the theories. In the adelic framework one usually seaks of automorhic forms rather than modular forms and in this case SL 2 is usually relaced with GL 2. For more details see e.g. [14], chater 3. In the classical setting the modular forms on congruence subgrous of SL 2 (Z) will be referred to as ellitic modular forms, and those on congruence subgrous of Γ Z as hermitian modular forms. Since the theory of ellitic modular forms is well-known we will only summarize the main facts below. Section 3 will be devoted to hermitian modular forms Ellitic modular forms. The theory of ellitic modular forms is well-known, so we omit most of the definitions and refer the reader to standard sources, e.g. [31]. Let H := {z C Im (z) > 0} denote the comlex uer half-lane. In the case of ellitic modular forms we will denote by Γ 0 (N) the subgrou of SL 2 (Z) consisting of matrices whose lowerleft entries are divisible by N, and by Γ 1 (N) the subgrou of Γ 0 (N) consisting of matrices whose uer left entries are congruent to 1 modulo N. Let Γ SL 2 (Z) be a congruence subgrou. Set M m (Γ) (res. S m (Γ)) to denote the C-sace of ellitic modular forms (res. cus forms) of weight m and level Γ. We also denote by M m (N, ψ) (res. S m (N, ψ)) the sace of ellitic modular forms (res. cus forms) of weight m, level N and character ψ. For f, g M m (Γ) with either f or g a cus form, and Γ Γ a finite index subgrou, we define the Petersson inner roduct f, g Γ := f(z)g(z)(im z) m 2 dx dy, and set f, g := Γ \H 1 [SL 2 (Z) : Γ ] f, g Γ, where SL 2 (Z) := SL 2 (Z)/ I 2 and Γ is the image of Γ in SL 2 (Z). The value f, g is indeendent of Γ. Every ellitic modular form f M m (N, ψ) ossesses a Fourier exansion f(z) = n=0 a(n)qn, where throughout this aer in such series q will denote e(z) := e 2πiz. For γ = [ ] a b c d GL + 2 (R), set j(γ, z) = cz + d. Let D = D K be a rime. In this aer we will be articularly interested in the sace S m (D K, χ K ), where χ K is the quadratic character of (Z/DZ) associated with the extension K = Q( D). Regarded as a function Z {1, 1}, it assigns the value 1 to all rime numbers such that () slits in K and the value 1 to all rime numbers such that () is inert in K. Note that since the character χ K is rimitive, the sace S m (D K, χ K ) has a basis consisting of rimitive normalized eigenforms. We will denote this (unique) basis by N. For f = n=1 a(n)qn N, set f ρ := n=1 a(n)qn N. Fact 2.1. ([31], section 4.6) One has a() = χ K ()a() for any rational rime D K. This imlies that a() = a() if () slits in K and a() = a() if () is inert in K.

10 10 KRZYSZTOF KLOSIN For f N and E a finite extension of Q l containing the eigenvalues of T n, n = 1, 2,... we will denote by ρ f : G Q GL 2 (E) the Galois reresentation attached to f by Deligne (cf. e.g., [10], section 3.1). We will write ρ f for the reduction of ρ f modulo a uniformizer of E with resect to some lattice Λ in E 2. In general ρ f deends on the lattice Λ, however the isomorhism class of its semisimlification ρ ss f is indeendent of Λ. Thus, if ρ f is irreducible (which we will assume), it is well-defined. 3. Hermitian modular forms 3.1. Classical theory. For n > 1, set i n := ii n and define H n := {Z M n (C) i n (Z Z ) > 0}. We call H n the hermitian uer half-lane of degree n. The grou G + n (R) = {x G n (R) µ(x) > 0} acts transitively on H n via gz := (a g Z + b g )(c g Z + d g ) 1. Definition 3.1. We say that a subgrou Γ G + n (R) is a congruence subgrou if Γ is commensurable with U n (Z), and there exists N Z >0 such that Γ Γ h (N) := {g U n (Z) g I 2n (mod N)}. Note that every congruence subgrou Γ must be contained in U n (Q), because commensurability with U n (Z) the fact that Γ G + n (R) force Γ G n (Q) and µ(γ) to be a finite subgrou of R + hence to be trivial. For g G + n (R) and Z H n set j(g, Z) := det(c γ Z + d γ ), and for a ositive integer k, a non-negative integer ν and a function F : H n C define F k,ν g(z) := det(g) ν j(g, Z) k F (gz). When ν = 0, we will usually dro it from notation and simly write F k g(z). Definition 3.2. Let Γ G + n (R) be a congruence subgrou. We say that a function F : H n C is a hermitian semi-modular form of weight (k, ν) and level Γ if F k,ν γ = F for every γ Γ. If in addition F is holomorhic, we call it a hermitian modular form of weight (k, ν) and level Γ. The sace of hermitian semi-modular (res. modular) forms of weight (k, ν) and level Γ will be denoted by Mn,k,ν sh (Γ) (res. Mn,k,ν h (Γ)). We also set M n,k,ν h = M n,k,ν h (U n(z)). If n = 2 or ν = 0 we dro them from notation. Define a ositive integer J (K) in the following way: { 1 J (K) = 2 #O K #O K is even #O K #O. K is odd Note that J (K) = 1 when D K > 12. Remark 3.3. Suose Γ U n (Z). It is a Theorem of Hel Braun ([7], Theorem I on. 143) that det U n (Z) = {u 2 u O K }. This in articular imlies that (det U n (Z)) ν = {1} if J (K) ν. In such case, we have Mn,k,ν sh sh (Γ) = Mn,k (Γ).

11 THE MAASS SPACE AND THE BLOCH-KATO CONJECTURE 11 If Γ = Γ h 0,n(N) for some N Z, then we say that F is of level N. Forms of level 1 will sometimes be referred to as forms of full level. One can also define hermitian semi-modular forms with a character. Let Γ = Γ h 0,n(N) and let ψ : A K C be a Hecke character such that for all finite, ψ (a) = 1 for every a O K, with a 1 NO K,. We say that F is of level N and character ψ if F m γ = ψ N (det a γ )F for every γ Γ h 0,n(N). Denote by Mn,k sh (N, ψ) (res. M n,k h (N, ψ)) the C-sace of hermitian semi-modular (res. modular) forms of weight k, level N and character ψ. If n = 2 we dro it from notation. Write Z H n as Z = X + i n Y, where X = Re (Z) and Y = Im (Z). Let M n = G n 2 a denote the additive grou of n n matrices. A hermitian semi-modular form of level Γ h n(m, N) ossesses a Fourier exansion F (Z) = c F (τ, Y )e(tr τx), τ S n(m)(z) where S n (M)(Z) = {x S n (Z) tr xl(m) Z} with S n = {h Res OK /Z M n/ok h = h} and L(M) = S n (Z) M n (MO K ). As usually when n = 2, we dro it from notation. As we will be articularly interested in the case when M = 1, we set S := S(1) = {[ ] t1 t 2 M t 2 t 2 (K) t 1, t 3 Z, t O K If F is holomorhic the deendence of c(h, Y ) on Y is exlicit: c F (h, Y ) = e(tr (i n hy ))c F (h), where c F (h) deends only on h. Then one can write F (Z) = c F (h)e(tr (hz)). h S n(q) For F Mn,k h (Γ) and α G+ n (R) one has F k α Mn,k h (α 1 Γα) and there is an exansion F k α = c α (τ)e(tr τz). τ S n(q) We call F a cus form if for all α G + n (R), c α (τ) = 0 for every τ such that det τ = 0. Denote by Sn,k h (Γ) (res. Sh n,k (N, ψ)) the subsace of cus forms inside Mn,k h (Γ) (res. M n,k h sh sh (N, ψ)). If ψ = 1, set Mn,k (N) := Mn,k (N, 1) and Sh n,k (N) := (N, 1). If n = 2 we dro it from notation. S h n,k Theorem 3.4 (q-exansion rincile, [22], section 8.4). Let l be a rational rime and N a ositive integer with l N. Suose all Fourier coefficients of F Mn,k h (N, ψ) lie inside the valuation ring O of a finite extension E of Q l. If γ U n (Z), then all Fourier coefficients of F m γ also lie in O. If F and F are two hermitian modular forms of weight k, level Γ and character ψ, and either F or F is a cus form, we define for any finite index subgrou Γ of Γ, the Petersson inner roduct F, F Γ := F (Z)F (Z)(det Y ) m 4 dxdy, Γ \H n }.

12 12 KRZYSZTOF KLOSIN where X = Re Z and Y = Im Z, and F, F = [U n (Z) : Γ ] 1 F, F Γ, where U n (Z) := U n (Z)/ i 2n and Γ is the image of Γ in U n (Z). The value F, F is indeendent of Γ Adelic theory. Notation 3.5. We adot the following notation. If H is an algebraic grou over Q, and g H(A), we will write g H(R) for the infinity comonent of g and g f for the finite comonent of g, i.e., g = (g, g f ). Definition 3.6. Let K be an oen comact subgrou of G n (A f ). Write Z n for the center of G n. Let M k,ν (K) denote the C-sace consisting of functions f : G n(a) C satisfying the following conditions: f(γg) = f(g) for all γ G n (Q), g G n (A), f(gκ) = f(g) for all κ K, g G n (A), f(gu) = (det u) ν j(u, i n ) k f(g) for all g G n (A), u K = K 0,n, (see (10.7.4) in [38]), f(ag) = a 2nν nk f(g) for all g G n (A) and all a C = Z n (R) G n (R). Let ψ : A K C be a Hecke character of conductor dividing N. Set M k(n, ψ) := {f M k(k 1,n (N)) f(γg(κ, κ f )) = ψ N(det(a κf )) 1 j(κ, i n ) k f(g)}. Remark 3.7. Note that the center Z n (R) of G n (R) acts via the infinite art of a Hecke character of infinity tye ( 2nν nk, 0). In articular this action is trivial if ν = k/2. It is well-known (see e.g., [9], Theorem 3.3.1) that for any finite subset B of GL n (A K,f ) of cardinality h K with the roerty that the canonical rojection c K : A K Cl K restricted to det B is a bijection, the following decomosition holds (3.1) GL n (A K ) = b B GL n (K) GL n (C)b GL n (ÔK). We will call any such B a base. We always assume that a base comes with a fixed ordering, so in articular if we consider a tule (f b ) b B indexed by elements of B, and aly a non-trivial ermutation σ to the elements f b, we do not consider the tules (f b ) b B and (f σ(b) ) b B to be the same. Proosition 3.8. Let K be a comact subgrou of G n (A f ) such that det K Ô K. There exists a finite subset C U n (A f ) such that the following decomosition holds (3.2) G n (A) = c C G n (Q)G + n (R)cK. Moreover, each element of C can be taken to be of the form b for some b in a fixed base B. The same holds for U n in lace of G n. Proof. This is roved like Lemmas 5.11(4) and 8.14 of [38]. We will call any set C of cardinality h K for which the decomosition (3.2) holds a unitary base.

13 THE MAASS SPACE AND THE BLOCH-KATO CONJECTURE 13 Corollary 3.9. If (h K, 2n) = 1 a base B can be chosen so that for all b B the matrices b and b are scalar matrices and bb = b b = I n. Proof. It follows from the Tchebotarev Density Theorem, that elements of Cl K can be reresented by rime ideals. Since all the inert ideals are rincial, Cl K can be reresented by rime ideals lying over slit rimes of the form O K =. Let Σ be a reresenting set consisting of such ideals. As (2n, h K ) = 1, the set Σ 2n consisting of elements of Σ raised to the ower 2n is also a reresenting set for Cl K. Moreover, as is a rincial ideal, = 1 as elements of Cl K, hence Σ := { n n } Σ also reresents all the elements Cl K. Elements of Σ can be written adelically as α n, with α = (1, 1,..., 1,, 1, 1,... ) A K,f, where aears on the -th lace and 1 aears at the -th lace. Set b = α I n. Then we can take B = {b } n n Σ and we have b = α I 2n. It is also clear that bb = b b = I n. Proosition Suose that K G n (A f ) is a comact subgrou such that det K Ô K. Let N k,ν (K) denote the C-sace of functions f : U n(a) C satisfying the conditions of Definition 3.6, but with g U n (A). Then the ma f f Un(A) gives an isomorhism M k,ν (K) = N k,ν (K). Proof. One has the following short exact sequence of grou schemes over Z: We first show that the induced ma 1 U n G n µ Gm 1. (3.3) f f Un(A) : M k,ν(k) N k,ν(k) is injective. Indeed, let f M k,ν (K). Using Proosition 3.8 we write any g G n (A) as g = γg R c with γ G n (Q), g R G + n (R), c U n (A f ). Then f(g) = f(g R c). Let x = diag((µ(g R )) 1/2, (µ(g R )) 1/2,... ). Then µ(x) = µ(g R ) and x Z n (R). Set y = x 1 g R. Then µ(y) = 1. Hence by Definition 3.6 f(g) = f(xyc) = µ(g R ) nν nk/2 f(yc) = f(yc), so f is comletely determined by what it does on U n (A). It remains to show the surjectivity of (3.3). To do this we need to show that every f N k,ν (K) has an extension to G n(a). Fix a unitary base C. Let f N k,ν (K). Let g = γg R b κ with γ G n (Q), g R G + n (R), c C and κ K. Write g R = µ(gr ) y. Then y U n (R). Set f(g) := µ(g R ) nν nk/2 f(yc). We need to show that this extension of f is well-defined. Let g = γ g R cκ be a different decomosition of g with γ G n (Q), g R G+ n (R), κ K. Then g = (γg R, γcκ) = (γ g R, γ cκ ), where the first comonent is the -comonent of g and the second is the finite comonent of g. Thus g R = (γ ) 1 γg R and c = (γ ) 1 γcκκ. The latter equality imlies that µ((γ ) 1 γ) Ẑ Q = {±1} which combined with the first equality and the fact that g R, g R G+ n (R) imlies that µ(g R g 1 R ) = µ((γ ) 1 γ) = 1, i.e., in articular µ(g R ) = µ(g R ) and (γ ) 1 γ U n (Q). Write g R = µ(g R )y with y U n (R). We have (3.4) µ(g R) nν nk/2 f(y c) = µ(g R) nν nk/2 f((µ(g R) 1/2 (γ ) 1 γg R, c) = µ(g R ) nν nk/2 f((µ(g R ) 1/2 (γ ) 1 γg R, (γ ) 1 γcκκ ) = µ(g R ) nν nk/2 f(yc).

14 14 KRZYSZTOF KLOSIN In view of Proosition 3.10 in what follows we will often not distinguish between automorhic forms defined on G n (A) and those on U n (A). Every f M n,k,ν (K) ossesses a Fourier exansion, i.e., for every q GL n (A K ), and every h S(Q) there exists a comlex number c f (h, q) such that one has ([ ] [ ]) In σ q (3.5) f = c I n ˆq f (h, q)e A (tr hσ) h S n(q) for every σ S n (A). Here e A is defined in the following way. Let a = (a v ) A, where v runs over all the laces of Q. If v =, set e v (a v ) = e 2πiav. If v =, a finite rime, set e v (a v ) = e 2πiy, where y is a rational number such that a v y Z. Then we set e A (a) = v e v(a v ). Suose 2 h K. For g U n (A), write g = γg 0 c U n (Q)U n (R)U n (A f ) with c U n (A f ) and g 0 such that det g 0 = re iϕ satisfies 0 ϕ < 2π/J (K). Note that such a g 0 exists and det g ν 0 is indeendent of the choice of g 0. Proosition Let f M n,k,ν (K). Let g = (g, 1) U n (R)U n (A f ). Set Z := g i n. Let C be a unitary base. For c C, set f c (Z) = (det g ) ν j(g, i n ) k f(g c) and write Γ c for U n (Q) (G + n (R) ckc 1 ). The ma f (f c ) c C defines a C-linear isomorhism Φ C : M n,k,ν (K) c C M sh n,k,ν (Γ c). Proof. This follows from [38], section 10. If h K is odd, B is a base and C = { b } b B, we write Γ b instead of Γ b and f b instead of f b for b B, and Φ B instead of Φ C. Definition Let C be a unitary base. A function f M n,k,ν (K) whose image under the isomorhism Φ lands in c C M n,k,ν h (Γ c) will be called an adelic hermitian modular form of weight (k, ν) and level K. The sace of hermitian modular forms of weight (k, ν) will be denoted by M n,k,ν (K). Moreover, we set M n,k,ν := M n,k,ν (U n (Ẑ)). When ν = 0 or n = 2 we dro them from notation. We clearly have (3.6) M n,k,ν (K) = Mn,k,ν(Γ h c ). c C Let χ : Cl K C be a character and choose a base B consisting of scalar matrices b such that bb = I n. Such a base always exists when (h K, 2n) = 1 by Corollary 3.9. Write Z n for the center of G n. Let z = γz 0 b κ with γ Z n (Q), z 0 Z n (R), b B and κ (K Z n (A f )) be an element of the center with z 0 = ζi 2n. If f M n,k,ν (K), then Set f(zg) = f(z 0 b g) = ζ 2nν nk f( b g). M χ n,k,ν (K) = {f M n,k,ν(k) f( b g) = χ(b)f(g)}, where we consider b as an element of Cl K under the identification B = Cl K given by b c K (det b) with c K : A K Cl K. Then (3.7) M n,k,ν (K) = M χ n,k,ν (K). χ Hom(Cl K,C )

15 THE MAASS SPACE AND THE BLOCH-KATO CONJECTURE 15 Proosition Assume J (K) ν and (2n, h K ) = 1. Let B be a base as in Corollary 3.9. Let K G n (A f ) be an oen comact subgrou. Then Γ := U n (Q) (G + n (R) b K 1 b ) is indeendent of b. Assume Γ U n (Z). Let β : A K C be an unramified (everywhere) Hecke character such that β(a ) = a ν for all a C. We will denote the set of such characters Char(n, ν) (note that #Char(n, ν) = h K ). Then Mn,k,ν h (Γ) = M n,k h (Γ) and one has the following commutative diagram in which all the mas are isomorhisms (3.8) M h n,k (K) Ψ β M h n,k,ν (K) Φ 0 c C M n,k h (Γ) ι β Φ ν c C M n,k h (Γ) where ι β (f c ) = β(det c)f c, for g = γg 0 cκ U n (Q)U n (R)cK one has Ψ β (f)(g) = β(det g)f(g), and for h U n (R) Φ 0 (f) c (hi n ) = j(h, i n ) k f(hc) and Φ ν (f) c (hi n ) = (det h) ν j(h, i n ) k f(hc). The ma Ψ β is Hecke-equivariant; more recisely for T = KaK with a U n (A) one has Ψ β (T f)(x) = β(det a)(t Ψ β (f))(x) (for the definition of the Hecke action see section 4). Proof. This is straightforward using the results of this section (cf. (3.6) and Remark 3.3). 4. Hecke oerators 4.1. Hermitian Hecke oerators. We study Hecke oerators acting on the sace M k,ν of hermitian modular forms on G(A) = G 2 (A). We also set U = U 2. Let be a rational rime write K for G(Z ). Let H be the C-Hecke algebra generated by the double cosets K gk, g G(Q ) with the usual law of multilication (cf. [38], section 11), and H + H be the subalgebra generated by K gk with g U(Q ). If K gk H, there exists a finite set A g G(Q ) such that K gk = α A g K α. For f M k,ν, g G(Q ), h G(A), set ([K gk ]f)(h) = α A g f(hα 1 ). It is clear that [K gk ]f M k,ν. Remark 4.1. Let K 0, := K U(Z ). Every element of H can be written as K gk with g a diagonal matrix. For κ K write m κ = diag(1, 1, µ(κ), µ(κ)). Then h κ = κm 1 κ K 0,. Since g is diagonal, m κ commutes with g, hence we get From this it follows that K 0, gk 0, = K gk = K gk 0,. α A g K 0, α = K gk =, α A g K g The case of a slit rime. Let be a rime which slits in K. Write () =. Recall that G(Q ) = GL 4 (Q ) G m (Q ). An element g of G(Q ) can be written as g = (g 1, g 2 ) GL 4 (Q ) GL 4 (Q ) with g 2 = µ(g)j(g t 1) 1 J. Set T := K (diag(1,,, ), diag(1, 1,, 1))K, T := K (diag(1, 1,, ), diag(1, 1,, ))K,

16 16 KRZYSZTOF KLOSIN := K (I 4, I 4 )K. It is easy to see that the C-algebra H is generated by the oerators T, T, T,, and their inverses. Proosition 4.2. We have the following decomositions ([ 1 a b c ] T = K, (4.1) (4.2) T = a,b,c Z/Z d,e Z/Z f Z/Z b,c,d,e Z/Z a,c,f Z/Z e,f Z/Z a,b Z/Z d Z/Z ([ ] 1 d e K, [ 1 b 1 c a 1 [ 1 d 1 e 1 ]) ]) ([ ] [ ]) f 1 K 1 f, 1 1 K ([ 1 K ([ 1 K ([ f 1 K ([ f 1 K ([ 1 K ([ 1 ], [ ]). b d ] [ 1 b c ]) 1 c e 1 d e, b c a ] e 1, a 1 d ] 1 K ([ 1 1 ], ] [ 1 a c ]), 1 f [ 1 e 1 f ]) ] [ 1 a b ]), 1, [ 1 d 1 [ ]) 1 1 ]). Proof. This follows easily from the corresonding decomositions for the grou GL 4 (Q ) The case of an inert rime. Let be a rime which is inert in K. Set T := K diag(1, 1,, )K, T 1, := K diag(1,, 2, )K, := K I 4 K. The oerators T, T 1,, and their inverses generate the C-algebra H. Proosition 4.3. We have the following decomositions [ 1 b c ] 1 c d T = K [ ] 1 e K 1 (4.3) b,d Z/Z c O K /O K a O K /O K b Z/Z [ 1 a K b a 1 e Z/Z ] K [ 1 1 ].

17 THE MAASS SPACE AND THE BLOCH-KATO CONJECTURE 17 (4.4) T 1, = a,c O K /O K b Z/Z a O K /O K K b,d Z/Z bd 0 (mod ) [ 1 a b+ac c ] c K [ a 2 K [ 2 a a 1 b d c O K /O K d Z/ 2 Z ] K [ 2 1 ] K [ ] b (Z/Z) c (O K /O K ) c ] 1 c d 2 [ ] b c K c c 2 b 1 Proof. See the roof of Lemma 5.3 in [27] and references cited there Action of the Hecke oerators on the Fourier coefficients. Let S = S 2 be as in section 3. Write S := S(Z ) for {h S(Q ) tr (S(Z )h) Z }. For a matrix h S(A) such that h S for every rime, set and ɛ (h) = max{n Z ɛ(h) = ɛ(h). 1 n h S } Note that ɛ (h) 0 for every and ɛ(h) = ɛ(h f ). For f M k,ν, q GL 2 (A K ) and h S(Q) we write c f (h, q) for the (h, q)- Fourier coefficient of f as in (3.5) The case of a slit rime. Let be a rime which slits in K. Let be a rime of K lying over and denote by its conjugate. As before we simultaneously identify G(Q ) with a subgrou of GL 4 (Q ) GL 4 (Q ) (the first factor corresonds to and the second one to ) and with a subgrou of G(A). Set T,1 := 1 T, T,2 := 1 T. Note that the oerators T,1, T,2 and their inverses generate H +. Define the following elements of GL 2 (Q ) GL 2 (Q ) which we regard as elements of GL 2 (A K ), ([ ) a α a =, I 1] 2, a = 0, 1,..., 1, ([ ] ) 1 α =, I 2, ([ ] ) (4.5) β =, I 2, ([ ) 1 a γ a =, I ] 2, a = 0, 1,..., 1 ([ ] ) γ =, I 1 2.

18 18 KRZYSZTOF KLOSIN We will write π A K for the adele whose th comonent is and whose all other comonents are 1. Write π = γ π, b κ with γ K, π, = γ 1 C, b A K,f, κ Ô K, so that val (b b ) = 0 (this is always ossible). Proosition 4.4. One has the following formulas 2 a=0 c f (h, qα a ) + a=0 c f (h, q ˆα a ) f = T,1 f; c f (h, q) = 4 c f (h, qβ ) + c f (h, q ˆβ ) + a=0 b=0 c f (h, qγ aˆγ b ) f = T,2 f; γ 2k 4ν c f (h, qb 1 ) f = f. Proof. This is an easy calculation using Proosition The case of an inert rime. Let be a rime which is inert in K. Set T,0 := 1 T 1,. Define the following elements of GL 2 (K ) GL 2 (A K ): [ a α a = a O 1] K /O K, [ ] 1 α =, (4.6) [ ] 1 a 1 β a = 1 a O K /O K, [ ] 1 β =. 1 Write P 1 (O K /O K ) for the disjoint union of O K /O K and. Let h S(Q) and q GL 2 (A K,f ) and assume that q hq S. Since D K, this imlies that qhq M 2 (O K, ), where q denotes the -th comonent of q. Set val (det(q hq)) = 0; s(h, q) := ( 1) val (det(q hq)) > 0, ɛ (q hq) = 0; 2 ( 1) ɛ (q hq) > 0. Proosition 4.5. Assume that q hq S. One has the following formulas: s(h, q)c f (h, q) + 4 a P 1 (O K /O K ) c f (h, qα a ) + a P 1 (O K /O K ) c f (h, qβ a ) f = T,0 f; c f (h, q) = 2k+4 c f (h, q) + c f ( 1 h, q) + k+1 a P 1 (O K /O K ) c f (h, β a q) f = T f; 4ν 2k c f (h, q) f = f. If q hq S, c f (h, q) = 0 in all of the above cases. Finally let us also note that in the inert case the algebra H + oerators T,0 and U := 1 T 2 and their inverses. is generated by the 5. Maass sace Let S = S 2, S be as in section 3 and S, ɛ, ɛ as in section 4.

19 THE MAASS SPACE AND THE BLOCH-KATO CONJECTURE Definition and basic roerties. Definition 5.1. Let B be a base. We say that f M k, k/2 is a B-Maass form if there exist functions c b,f : Z 0 C, b B, such that for every q GL 2 (A K ) and every h S(Q) the Fourier coefficient c f (h, q) satisfies (5.1) c f (h, q) = det q k e 2πtr (q hq ) det γ b,q k ( d k 1 c b,f D K d 2 det h val(det q f q f ) ), d Z + d ɛ(q f hq f ) where q f = γ b,q bκ q GL 2 (K)bK for a unique b B. Here K = GL 2 (ÔK) is a maximal comact subgrou of GL 2 (A K,f ). Also, here and in what follows we will often treat the K-oints as embedded diagonally in the A K,f -oints (i.e., instead of writing q f = γ b,q q bκ q with q = γ 1 b,q GL 2(C) we will simly write q f = γ b,q bκ q as above.) Remark 5.2. Note that by [38], Proosition 18.3(2), c f (h, q) 0 only if (q hq) S, so ɛ (q f hq f) 0. Also, note that Definition 5.1 is indeendent of the decomosition q f = γ b,q bκ q GL 2 (K)bK. Indeed, if q f = γ b,q bκ q GL 2 (K)bK is another decomosition of q f, then so det(γ b,q )k = det γ k b,q. det γ b,q det γ 1 b,q = det(κ q(κ q) 1 ) Ô K K = O K, Definition 5.3. The C-subsace of M k, k/2 consisting of B-Maass forms will be called the B-Maass sace. Definition 5.4. Let B be a base. We will say that q GL 2 (A K ) belongs to a class b B if there exist γ GL 2 (K), q GL 2 (C) and κ K such that q = γbq κ. It is clear that the class of q deends only on q f. Lemma 5.5. Suose r GL 2 (A K ) and q GL 2 (A K ) belong to the same class and r f = γq f κ GL 2 (K)q f GL 2 (ÔK). Then k (5.2) c f (h, r) = det r det q e 2πtr (r hr q γ hγq ) det γ k c f (γ hγ, q). Proof. It follows from the roof of art (4) of Proosition 18.3 of [38], that (5.3) c f (h, r) = det r k e 2πtr (r hr ) c rf (h), where f rf (Z) = h S c rf (h)e 2πitr hz. As is easy to see (cf. for examle the Proof of Lemma 10.8 in [38]), f rf = [ ] f qf γ 1 k. Hence γ (5.4) c rf (h) = det γ k c qf (γ hγ). The Lemma follows from combining (5.3) with (5.4).

20 20 KRZYSZTOF KLOSIN Proosition 5.6. Choose a base B and let f M k, k/2. If there exist functions c b,f : Z 0 C, b B, such that for every b B and every h S(Q), the Fourier coefficient c f (h, b) satisfies condition (5.1) with c b,f in lace of c b,f, then f is a B-Maass form and one has c b,f = c b,f for every b B. Proof. Fix B and f M k, k/2. Suose there exist c b,f such that (5.1) is satisfied for all airs (h, b). Let q = γbxκ = (γx, γbκ) GL 2 (C) GL 2 (A K,f ), where γ GL 2 (K), x GL 2 (C) and κ K. Then by Lemma 5.5, c f (h, q) = det q k e 2πtr (q hq γ hγ) det γ k c f (γ hγ, b). Since condition (5.1) is satisfied for (h, b), we know that ( c f (h, b) = e 2πtr h d k 1 c b,f D K d 2 det h val(det b b) ). Thus c f (γ hγ, b) = e 2πtr (γ hγ) d Z + d ɛ(b hb) d k 1 c b,f ( D K d 2 det(γ hγ) val(det b b) ). So, d Z + d ɛ(b γ hγb) (5.5) c f (h, q) = det q k e 2πtr (q hq ) det γ k ( d k 1 c b,f D K d 2 det h det(γ γ) val(det b b) ). d Z + d ɛ(b γ hγb) The claim now follows since ɛ(b γ hγb) = ɛ(qf hq f) and det(γ γ) Q +, so det(γ γ) = val(det γ γ). Proosition 5.7. If B and B are two bases, then the B-Maass sace and the B - Maass sace coincide, i.e., the notion of a Maass form is indeendent of the choice of the base. Proof. Let B and B be two bases. Write q f = γ b,q bκ B = γ b,qb κ B with b B, b B, γ b,q, γ b,q GL 2 (K) and κ B, κ B K. Suose f is a B-Maass form, i.e., there exist functions c b,f for b B, such that for every q and h, c f (h, q) = t det γ b,q k d k 1 c b,f (s), d Z + d ɛ(q f hq f ) where t = det q k e 2πtr (q hq ) and s = D K d 2 det h val(det q f q f ). Our goal is to show that there exist functions c b,f for b B, such that for every q and h, (5.6) c f (h, q) = t det γ b,q k d k 1 c b,f (s). We have d Z + d ɛ(q f hq f ) (5.7) det γ b,q = det γ b,q det(b b 1 ) det(κ 1 B κ B ).

21 THE MAASS SPACE AND THE BLOCH-KATO CONJECTURE 21 Since det(b b 1 ) corresonds to a rincial fractional ideal, say (α b,b ), under the ma ((α b,b ) ) val((α b,b )), using [9], Theorem 3.3.1, we can write det(b b 1 ) = α b,b κ b,b A K,f with κ b,b Ô K. Then it follows from (5.7) that β := κ b,b det(κ 1 B κ B ) Ô K K = O K. Hence β k = 1. Thus det γ b,q k = det γ b,q k α b,b k. Note that α b,b k is well defined and only deends on b and b (i.e., it is indeendent of q and h). Set c b,f (n) = α b,b k c b,f (n). Then it is clear that c b,f satisfies (5.6). Definition 5.8. From now on we will refer to B-Maass forms simly as Maass forms. Similarly we will talk about the Maass sace instead of B-Maass saces. This is justified by Proosition 5.7. The Maass sace will be denoted by M M k, k/2. We now recall the definition of Maass sace introduced in [29]. We will refer to it as the U(Z)-Maass sace. Assume J (K) k 2, so that by Proosition 3.13 the sace Mk h = M k, k/2 h = M k, k/2 h (U(Z)). We say that F (Z) = h S c F (h)e 2πitr (hz) Mk h is a U(Z)-Maass form if there exists a function α F : Z 0 C such that for every h S, one has (5.8) c F (h) = d k 1 α F (D K d 2 det h). The subsace of M h k d Z + d ɛ(h) consisting of U(Z)-Maass forms will be denoted by M h,m k. Proosition 5.9. If 2 h K, then the Maass sace M M k, k/2 is isomorhic (as a C-linear sace) to #B coies of the U(Z)-Maass sace M h,m k. Proof. Since the Maass sace is indeendent of the choice of a base B by Proosition 5.7, we may choose B as in Corollary 3.9 and #B = #C = h K, with C as in Proosition 3.8. The ma Φ B : M k, k/2 b B M k h is an isomorhism. Let f M M k, k/2 and set (f b) b B = Φ B (f). Set α fb := c b,f. Then using (5.3), and the fact that the matrices b commute with h and b b = I 2, we see that condition (5.1) for c f (h, b) translates into condition (5.8) for c fb (h). Hence Φ B (M M k, k/2 ) b B M h,m k. On the other hand if (f b ) b B b B M h,m k, set c b,f := α fb. Then conditions (5.8) for c fb (h) translate into conditions (5.1) for c f (h, b). By Proosition 5.6 this imlies that f is a Maass form Invariance under Hecke action. It was roved in [29] that the U(Z)-Maass sace is invariant under the action of a certain Hecke oerator T associated with a rime which is inert in F. On the other hand Gritsenko in [16] roved the invariance of the U(Z)-Maass sace under all the Hecke oerators when K = Q(i). In this section we show that the Maass sace M M k, k/2 is in fact invariant under all the local Hecke algebras (for rimes D K ) without imosing restrictions on the class number. Theorem Let D K be a rational rime. The Maass sace is invariant under the action of H +, i.e., if f M M k, k/2, and g U(Q ), then [K gk ]f M M k, k/2.