Automorphic forms of higher order

In this paper a theory of Hecke operators for higher order modular forms is established. The definition of cusp forms and attached L-functions is extended beyond the realm of parabolic invariants. The role of representation theoretic methods is clarified and, motivated by higher order forms, new convolution products of L-functions are introduced.


Introduction
Higher-order modular forms have in recent years arisen in various contexts and they have been studied as analytic functions [8,10]. In particular, their spaces [1,[3][4][5] have been investigated. Parallel to that, L-functions were attached to higher-order forms [6,7,9] and some of their basic aspects studied. In [2], a cohomology theory for higher invariants is developed, and an Eichler-Shimura isomorphism for higher-order forms is established.
The current paper serves the following purposes: • to establish a theory of Hecke operators on higher-order forms; • to extend the definition of higher forms beyond parabolic invariants; • to clarify the role of representation theoretic methods in the theory; and • to introduce new convolution products of L-functions of higher forms. The first item fills a long-standing gap in the theory of higher-order forms by constructing a natural Hecke action. This is surprising, as there is no adelic counterpart of higher-order forms. The Hecke operators form bounded self-adjoint operators on direct limits of spaces of higher-order forms. It is an on-going long-term project of the authors to better understand the spectral decompositions of Hecke operators. For the second item, one gets that in the case of higher-order forms, the crucial Fourier expansion is replaced by a 'Fourier-Taylor expansion', which is introduced in this paper. For the third item, representation theoretic methods, it turns out that an intervention of Lie groups, as Dieudonné terms it, is possible in the theory and, in fact, higher-order forms can be incorporated into the same representational context as the classical automorphic forms. One would thus expect a distinction from classical forms in the intervention of adeles. Indeed, it turns out that there is no intervention of adeles, as there are no higher forms on the adelic level. The last item in the list, the convolution product, is inspired by the second insofar as the L-functions of higher-order forms are special cases of the convolution products. We show analyticity and the functional equation in greater generality.

Higher invariants
Let R be a commutative ring with unit and let Γ be a group. Let R[Γ] be the group algebra and I Γ ⊂ R[Γ] the augmentation ideal. Note that I Γ is a free R-module with basis (γ − 1) γ∈Γ . For an R[Γ]-module V , the usual invariants are the elements of the R-module H 0 (Γ, V ) = V Γ = {v ∈ V : I Γ v = 0}. We define the set of higher invariants to be H 0 q (Γ, V ) = {v ∈ V : I q+1 Γ v = 0}, for q = 0, 1, 2, . . ., where I k Γ is the kth power of the ideal I Γ .
For v ∈ V one has v ∈ H 0 q+1 (Γ, V ) if and only if (γ − 1)v ∈ H 0 q (Γ, V ) for every γ ∈ Γ. Note that if Γ is perfect, that is, if [Γ, Γ] = Γ, then I q+1 Γ = I Γ for every q 0 and so, in that case, there are no higher invariants except the usual invariants. This is due to the observation which shows that if Γ is perfect, then I 2 Γ = I Γ and hence inductively I q+1

Hecke operators
We now come to Hecke operators. For this, let (G, Γ) be a Hecke pair; that is, G is a group and Γ is a subgroup such that for every g ∈ G the set ΓgΓ/Γ is finite.
(ii) Let G be a topological group and let Γ be a compact open subgroup. Then (G, Γ) is a Hecke pair, since the compact set ΓgΓ can be covered by finitely many open sets of the form xΓ, x ∈ G.
The Hecke algebra H = H R (G, Γ) is the R-module of all functions f : Γ\G/Γ → R of finite support with the convolution product In particular, one has where ΓgΓ = · ∪ n j=1 g j Γ.
As v is Γ-invariant, this expression does not depend on the choice of the representatives g j .
For q 1 and v ∈ H 0 q (Γ, V ), however, the expression n j=1 g j v will in general depend on the choice of the representatives (g j ). Any other set of representatives is of the form (g j γ j ) for some γ j ∈ Γ. Note that .
The group Γ permutes the finite set ΓgΓ/Γ by left multiplication. Let Γ(g) ⊂ Γ be the subgroup of all elements that act trivially on ΓgΓ/Γ. Then Γ(g) is a finite-index normal subgroup of Γ. Note that Proof. (a) We have to show that the sum is annihilated by for any given σ 0 , . . . , σ q ∈ Γ(g). By the definition of Γ(g), it follows that for every j and every k there exist η k j ∈ Γ such that σ k g j = g j η k j . Hence The proof of (b) is similar.
The lemma implies that we get a well-defined Hecke operator . We need to extend this construction to finite-index subgroups Σ ⊂ Γ. First note that (G, Σ) is a Hecke pair again. We abbreviatē . As every finite-index subgroup contains a finite-index normal subgroup, it suffices to assume that Σ is normal in Γ. We define Σ(g) to be the kernel of the homomorphism Σ → Per(ΓgΓ/Σ). Note that Γ is present in the definition, although not in the notation. Then Σ(g) = γ∈Γ γ(Σ ∩ gΣg −1 )γ −1 , and Σ(g) is normal of finite index in Γ. We define as follows. Write ΓgΓ as a disjoint union of Σ-cosets · ∪ j h j Σ and set The same reasoning as before shows the well-definedness of T ΓgΓ . The factor 1/[Γ : Σ] will make the Hecke operator compatible with change of groups as follows. Assume that Σ ⊂ Σ is another finite-index normal subgroup. Then Σ (g) ⊂ Σ(g) as well and the inclusions show that there is a natural restriction homomorphism res Σ Σ :H q (Σ, V ) →H q (Σ , V ). As the intersection of two finite-index subgroups is a finite-index subgroup, these spaces form a direct system indexed by the set of all finite-index normal subgroups Σ of Γ. Lemma 1.4. For any two finite-index normal subgroups Σ ⊂ Σ of Γ and g ∈ G, the following diagram commutes.H
where ΓgΓ = · ∪ i k i Σ. Suppose that Σ = · ∪ j g j Σ . Then, by Lemma 1.3(b), the sum (1) equals In applications, it will be necessary to consider subsystems like the system of congruence subgroups defined as follows. A subgroup Σ of Γ is called a congruence subgroup if it contains a subgroup of the form Γ(g 1 )(g 2 ) . . . (g n ) for some g 1 , . . . , g n ∈ G.

Lemma 1.5. The intersection of two congruence subgroups is a congruence subgroup.
Proof. We claim that We spell out the argument in the case m = n = 1 and leave the obvious iteration to the reader. Recall that Σ(g) is defined as the kernel of the map Σ → Per(ΓgΓ/Σ). Hence a given γ ∈ Γ lies in Γ(g) if and only if γ acts trivially on ΓgΓ/Γ. It lies in Γ(g)(h) if and only if it also acts trivially on ΓhΓ/Γ(g). But then it acts trivially on ΓhΓ/Γ, and hence γ lies in Γ(g) ∩ Γ(h).
where the limit is taken over all normal congruence subgroups Σ of Γ. Further, let L all q (Γ, V ) denote the same direct limit, but now over all finite-index normal subgroups of Γ. The lemma above shows that one gets a well-defined operator and likewise for L all q (Γ, V ). For the rest of the section, we consider the case L q (Γ, V ) alone, but everything will also apply to L all q (Γ, V ).
Proposition 1.6. The map 1 ΓgΓ → T ΓgΓ extends uniquely to a representation of the Hecke algebra H(G, Γ) on L q (Γ, V ).
Proof. Uniqueness is clear as the 1 ΓgΓ span the Hecke algebra. We only have to show that the ensuing linear map is a representation. For this, we write it in a different manner. Let f ∈ H(G, Γ) and let v ∈H q (Σ, V ). Define the support of f is contained in Γg 1 Γ ∪ . . . ∪ Γg n Γ. This is an action of H which extends the above map.
commutes. Therefore the group G acts on the limit L q (Γ, V ). It is sometimes easier to understand L q (Γ, V ) as a G-module rather than as a Hecke module.

Unitary Hecke modules
Suppose now that for every congruence subgroup Σ the spaceH q (Σ, V ) is a Hilbert space in such a way that is a unitary map. The first condition gives L q (Γ, V ) the structure of a pre-Hilbert space. The second implies that G acts on this pre-Hilbert space by unitary maps. If this is the case for every q 1, we call V a unitary Hecke module. Theorem 1.7. Let V be a unitary Hecke module. For each g ∈ G the operator T ΓgΓ is a bounded operator on the pre-Hilbert space L q (Γ, V ). The operator norm satisfies Proof. For v ∈H q (Σ, V ) with ||v|| = 1, one has To compute the adjoint, let v, w ∈H q (Σ, V ). Then as the map xΣ → γ 1 xΣ is a bijection on G/Σ. Further, as Σ is normal in Γ, the group Γ acts on G/Σ via xΣ → xγΣ. Hence we get
Finally, assume the existence of G * . Note that Σ is a lattice in G * as well and that as the set ΣgΣ/Σ is one orbit under the left translation action of Σ and the stabilizer of the point gΣ/Σ is Σ ∩ gΣg −1 . Accordingly Let μ be the left Haar measure on G * ; then μ(gΣg −1 \G * ) = μ(Σ\G * ) and so This implies the claim.
Corollary 1.8. Let (G, Γ) be a Hecke pair such that there exists a locally compact group G * as in the theorem; then the pair (G, Γ) is unimodular.
Example 1.9. For the sake of completeness we give an example of a Hecke pair (G, Γ) that is not unimodular. Let p be a prime and let G be the semidirect product Q p Q × p . Therefore G is the topological space Q p × Q × p with the multiplication (x, y)(x , y ) = (x + yx , yy ). Let Γ be the compact open subgroup Z p Z × p and let g = (0, p). Then |ΓgΓ/Γ| = p, whereas |Γ\ΓgΓ| = 1.
Remark. Note that if V is a unitary Hecke module, then the representation of G on the pre-Hilbert space L q (Γ, V ) is unitary.

Lowering the order
There is a canonical injective linear map given as follows: first note that there is a canonical isomorphism Hom(Γ, V )). Using this, we can define l q as This indeed is well defined as for γ, τ ∈ Σ one has (γτ − 1) ≡ (γ − 1) + (τ − 1) mod I 2 Σ . We call l q the order-lowering homomorphism. It can be used to establish a unitary structure as is shown in the next section.
In the following sections we will give examples for spaces V to which the Hecke calculus applies. For each of these spaces the following problems arise.
• Determine the spectral decomposition of T ΓgΓ on the Hilbert completion of L q (Γ, V ). If the Hecke algebra is commutative, one can give a simultaneous spectral decomposition. Are the Hecke operators compact?
• Determine the difference that it makes for the spectrum of the Hecke algebra whether one starts with congruence subgroups alone or all finite-index normal subgroups Σ. This problem should be related to the Selberg conjecture.
• Determine the action of the Hecke algebra in terms of the Fourier-Taylor expansions (defined below) and on the ensuing L-functions.
Instead of the Hecke action, it is sometimes more useful to consider the G-action on L q (Γ, V ). Note that the order-lowering homomorphism can be iterated tō Then H Γ is a G-module and, taking limits, one gets an injection of G-modules, Sometimes this map will be surjective on the Hilbert space completions. Thus the spectral problem decomposes into • determining the spectral decomposition of (the Hilbert space completion of) the module H Γ , and • determining the decomposition of tensor products. These aims will be pursued in future work.

Modular forms
is independent of z ∈ H. Let S k (Σ) be the subspace of all cusp forms f for Σ, which are characterized by the fact that for every cusp c of Γ one has Note that every cusp form f has rapid decay at the cusps, which means that for every cusp c one has f | k σ Γ c (x + iy) = O(e −αy ) for some α > 0.

Theorem 2.1. There is a canonical choice of inner products that makes V a unitary Hecke module.
Proof. As a first step we have to construct a canonical inner product on M k (Σ). For this, note that for f ∈ M k (Σ) and g ∈ S k (Σ) one has the Petersson inner product where μ is the G R -invariant measure dx dy y 2 . We extend it to an inner product on M k (Σ) as follows. The Petersson inner product defines an orthogonal projection P : M k (Σ) → S k (Σ). Let C be the set of cusps of Γ. For f, g ∈ M k (Σ) let where the sum runs over all Σ-equivalence classes of cusps c. The form f, g = f, g cusp + P (f ), P (g) Pet is a positive-definite inner product such that for Σ ⊂ Σ the inclusion M k (Σ) → M k (Σ ) is an isometry and for g ∈ G Q the map g : M k (Σ) → M k (gΣg −1 ) is unitary. This settles the case q = 1. Suppose that the case q has been taken care of; then consider the order-lowering homomorphism l q+1 :H q+1 (Σ, V ) → Hom(Σ, C) ⊗H q (Σ, V ). The Eichler-Shimura map gives a canonical isomorphism The right-hand side is equipped with a canonical inner product by the above, such that for Σ ⊂ Σ one has a commutative diagram as follows.
The Petersson inner product thus establishes a canonical inner product on Hom(Σ, C) such that the restriction map Hom(Σ, C) → Hom(Σ , C) is isometric and for g ∈ G Q the map g : Hom(Σ, C) → Hom(gΣg −1 , C) is unitary. Now equipH q+1 (Σ, M k ) with the inner product from the above injection into Hom(Σ, C) ⊗H q (Σ, V ) to make M k a unitary Hecke module in a canonical way.
The inner product ·, · onH q (Σ, M k ) can inductively be written explicitly as follows. We . Assume that the inner product ·, · has been defined onM q−1 k /M q−2 k , and let {f i } d i=1 be an orthonormal basis and suppose that, for every σ ∈ Σ, Then the definition of inner product onH q (Σ, M k ) can be written as It should be noted thatÖ. Imamoglu and C. O'Sullivan have given an alternative definition of an inner product on a subspace ofM 1 k /M 0 k and (conjecturally) forM q k /M q−1 k (q 2) in their forthcoming paper 'Higher-order cusp forms: inner products'. Their construction relies on delicate analytic manipulations, but the inner product it gives can be proved to have essentially the same value as our inner product.

Cusp forms
In this section, let Γ be a congruence subgroup of G Z .
By linearity, we extend the definition f | k σ to elements σ of the group ring R[Γ]. Let k 0 be even and let S k (Γ) be the space of cusp forms of weight k.
We now define cusp forms of order q, which are the cuspidal analogues of the elements of M q k discussed at the end of Subsection 2.1. First letS k,0 (Γ) = S k,0 (Γ) = S k (Γ); so classical cusp forms are of order 0. Next suppose thatS k,q (Γ) and S k,q (Γ) are already defined and let S k,q+1 (Γ) be the space of all functions f with: as y → ∞ for some α > 0 ('rapid decay at the cusps'). Further, let S k,q+1 (Γ) be the set of all f ∈S k,q+1 (Γ) with f | k (γ − 1) = 0 for every parabolic element γ of Γ. Note that for f ∈ S k,q+1 (Γ) and γ ∈ Γ one has f | k (γ − 1) ∈ S k,q (Γ).
CompareS k,1 with P S k,2 of [6], where a classification and a converse-theorem type of result for such functions is proved.
Note that the space V of all holomorphic functions on H that satisfy the above growth condition at each cusp serves as a G-module where, for instance, G = SL 2 (Q) and Γ = SL 2 (Z). In this way one obtains a theory of Hecke operators for the spaceS k,q (Γ)/S k,q−1 (Γ). Also, in view of the discussion at the end of Subsection 2.1, we have an inner product on the same space that makes it a unitary Hecke module.

Fourier-Taylor expansion
Set T = ( 1 1 0 1 ) and let V 0 be the space of all holomorphic functions f on the upper half plane with f | 0 T = f , that is, periodic with period 1. Inductively, let V q+1 be the space of all holomorphic functions on H such that f | 0 (T − 1) ∈ V q . We also set V −1 = {0}. Note that S k,q (Γ) ⊂ V q if Γ contains the translation z → z + 1.

Proposition 2.2. Every f ∈ V q has a Fourier-Taylor expansion
f (z) = n∈Z e 2πinz (a n,0 + a n,1 z + · · · + a n,q z q ) for uniquely determined coefficients a k,j ∈ C. For every j and every y > 0 the sequence (a n,j e −2πny ) n∈N is rapidly decreasing. The map T : Proof. We prove the proposition by induction on q. For q = 0, every f ∈ V 0 is periodic, and therefore has a Fourier expansion which, as f is holomorphic, is of the form f (z) = n a n e 2πinz . Next assume that the claim is proven for q and let f ∈ V q+1 . By induction, f (z + 1) − f (z) = n e 2πinz (a n,0 + · · · + a n,q z q ).
Thus h(z) = f (z) − g(z) ∈ V 0 . As g and h possess Fourier-Taylor expansions, so does f . By induction, the coefficients a n,k are uniquely determined, so are those of h(z), which implies that the coefficients of f are uniquely determined.

Intervention of Lie groups
Let G R = AN K R be the Iwasawa decomposition, and let k : G R → K R be the corresponding projection. Then For a given function f on the upper half plane, define the function ψ f on G R by The next lemma shows that, via the identification ψ, the action | k on functions on H becomes the action by left translation.
Proof. The proof consists of a computation based on the identity where z = gi ∈ H.

Lemma 2.4. Suppose that Γ is torsion-free. Then η q is surjective.
Proof. If the group Γ has genus g and s cusps, then there are 2g hyperbolic elements γ i and s parabolic elements γ 2g+i generating Γ and satisfying the relation Because of this relation, every element in Hom(Γ, C) ⊗q ⊗ F 0 is uniquely determined by its values at (γ i1 , . . . , γ iq ) for i ∈ {1, . . . , N} q , where N = max(2g, 2g + s − 1). Therefore, to establish surjectivity, it suffices to show that for every choice of functions f i1,...,iq ∈ F 0 there is an F ∈F q such that The space L 2 (Γ\G) = L 2 (Γ\G)/N 0 is a Hilbert space. Fix a Hilbert space structure on Hom(Γ, C) and equip the space L 2 q (Γ\G), which is mapped bijectively onto Hom(Γ, C) ⊗q ⊗ L 2 (Γ\G), with the induced Hilbert space structure.

No intervention of adeles
Let A = A fin × R be the adele ring over Q. Let K Γ be a compact open subgroup of G A fin ; then Γ = K Γ ∩ G Q is a congruence subgroup, and the natural map which maps Γx to G Q xK Γ , is a G R -equivariant, continuous bijection. This gives a natural isomorphism In other words, automorphic forms on Γ\G can be lifted to G Q \G A . This is what Dieudonné calls the 'intervention of adeles'. We ask for higher forms in the adelic setting. Note first that there are no higher G Q -invariants, as the group G Q is perfect. Also the K Γ -action does not yield higher-order forms, at least not in the space L 2 (G Q \G A ), as the group K Γ is compact and acts through a continuous representation on the Hilbert space L 2 (G Q \G A ); see Proposition 1.1.

Higher-order cusp forms
Let Γ be a lattice in G such that ∞ is a cusp of Γ of width 1. Let f ∈S k,q (Γ). Then f has the Fourier-Taylor expansion e 2πinz (a n,0 + a n,1 z + . . . + a n,q z q ).
For ν = 0, . . . , q define Let S = 0 −1 1 0 . After replacing Γ with a conjugate if necessary, we may assume that 0 is a cusp of Γ, too. Then the group S −1 ΓS has ∞ for a cusp. Let w = w Γ > 0 be its width. Let S w be the matrix S diag( √ w, √ w −1 ). Then ∞ is a cusp of width 1 of the group Thenf ∈S k,q (Γ). Note that S 2 w = 1, and so wΓ = w Γ andΓ = Γ as well aŝ f = f .
As an example, let N ∈ N and consider the group Γ = Γ 0 (N ) consisting of all a b c d ∈ SL 2 (Z) such that c ≡ 0 mod N . In this case one has w = N andΓ = Γ.
For 0 ν q − 1 the L-function L ν (f, s) extends to an entire function.

Convolution of L-functions
In [1], it is shown that if f (z) = ∞ n=1 a n e 2πinz and g(z) = ∞ n=1 b n e 2πinz are cusp forms of weight k ∈ Z 2 and 2, respectively, then the nth Fourier coefficient of the second-order form Following [6], one can define the L-function of F (z) by means of the Dirichlet series This function of s admits meromorphic continuation and satisfies a functional equation (see [6]). Again we assume that 0 and ∞ are cusps of Γ, the width of ∞ being 1. We define w andΓ as in Theorem 3.1. Let k, l be even integers greater than or equal to 0 and let f ∈ S k (Γ), g ∈ S l (Γ). Denote their respective Fourier coefficients by a n and b n . For complex numbers s and t with large-enough real parts we set For Re(s) large enough, we observe that the Mellin transform ( We shall analytically continue (L f #L g )(s, t) by a repeated application of the Riemann trick. We decompose the integrals into sums of the form w . Then Λ f,g (s, t) will be the sum of four terms, The last summand defines a holomorphic function on C 2 , as f and g are rapidly decreasing at ∞.
Recall that we have f i 1 wy = (i √ wy) kf (y) and likewise for g and l.
The substitution x → 1/wx in the outer integral shows that A equals (ix)g i 1 wx + y y t−1 x k−s−1 dy dx.
As f and g are rapidly decreasing at ∞, this integral converges absolutely for Re(t) > 0 and defines a holomorphic function in (s, t) in that region. The inner integral can be integrated by parts to get Hence A extends to a meromorphic function on Re(t) > −1 with a simple pole at t = 0. Iteration of that argument shows that A is entire except for simple poles at t = 0, −1, −2, . . .. The other summands can be treated similarly. Since the Γ-function has poles at t = 0, −1, −2, . . . , we have proved the following theorem.
Theorem 3.2. The function L f #L g extends to an entire function on C 2 .
If t = p ∈ N we set (L f # p L g )(s) = (L f #L g )(s, p). Then (L f # p L g )(s) extends to an entire function for every p ∈ N. Next we fix p = l − 1, where l is the weight of g. Then we can obtain a functional equation for a related Dirichlet series, which however involves 'lower-order' L-functions L f , L g .
Specifically we consider The relation with values of L f # l−1 L g is given by the following result. Proof.
From the binomial expansion of (y − x) l−2 we deduce that this equals This gives the result.
The analytic continuation of Λ f,g can be deduced from this proposition together with Theorem 3.2 and the analytic continuation of Γ(s)L f (s). The functional equation is given by the following proposition. Proof. The change of variables x → 1/wx implies that Λ f,g (s) equals The same change of variables on y gives which is the desired statement.