PProducts of Vector Valued Eisenstein Series
Martin Westerholt-Raum Abstract:
We prove that products of at most two vector valued Eisenstein series that originate in level 1 spanall spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to aresult that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main featureof the proof are vector valued Hecke operators. We recover several classical constructions from them, includingclassical Hecke operators, Atkin-Lehner involutions, and oldforms. As a corollary to our main theorem, weobtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagierin the context their previously mentioned result. vector valued Hecke operators (cid:4) period relations (cid:4) cusp expansions of modular formsMSC Primary: 11F11 (cid:4)
MSC Secondary: 11F67
Introduction 11 Preliminaries 6 I N a paper from 1951, published in 1952, Rankin [Ran52] derived an expression in terms of periodsfor scalar products 〈 E l E k − l , f 〉 for cuspidal Hecke eigenforms f of weight k . This served Kohnenand Zagier [KZ84] in connecting modular forms with period polynomials. As an immediate conse-quence of their work one concludes that products of at most two Eisenstein series span all spaces ofmodular forms for SL ( (cid:90) ). In fact, it is possible to describe linear relations of the E l E k − l by periodpolynomials [Sko93]. The author thanks the Max Planck Institute for Mathematics for their hospitality. The paper was partially written, while hewas supported by the ETH Zurich Postdoctoral Fellowship Program and by the Marie Curie Actions for People COFUNDProgram. – – a r X i v : . [ m a t h . N T ] M a r roducts of Vector Valued Eisenstein Series — Introduction M. Westerholt-Raum Let us write M k for the space of level 1, weight k modular forms, and E k for the subspace of Eisen-stein series. The aforementioned consequence of Kohnen’s and Zagier’s result reads as follows:M k = E k + span ≤ l ≤ k − E l · E k − l . (0.1)This was considered by Imamo˘glu and Kohnen [IK05] in the case of Γ (2), and also generalized to Γ ( p ) ⊂ SL ( (cid:90) ) by Kohnen and Martin [KM08]. A similar statement, based on products of at most two“toric” modular forms, was found by Borisov and Gunnells in [BG03].Note that we can view (0.1) as a statement about products of Eisenstein series focusing on the weightaspect. We study the level aspect of the same problem: Can cusp forms of any level be expressedas products of at most two Eisenstein series, varying the level but fixing the weight? The affirmativeanswer, which we provide, is most naturally phrased in terms of vector valued modular forms. For acomplex representation ρ of SL ( (cid:90) ) and k ∈ (cid:90) write M k ( ρ ) for the space of all vector valued modularforms of weight k and type ρ . By definition, f ∈ M k ( ρ ) satisfies f (cid:161) a τ + bc τ + d (cid:162) = ρ (cid:161)(cid:161) a bc d (cid:162)(cid:162) ( c τ + d ) k f ( τ )for all (cid:161) a bc d (cid:162) ∈ SL ( (cid:90) ). A corresponding subspace E k ( ρ ) of Eisenstein series can be defined in a naturalway, which is explained in Section 3. For every N , we will define in Section 2 a vector valued Heckeoperator T N . It yields a map T N : E k ( ) → E k ( ρ T N ), also described in Section 2, where is the trivialrepresentation of SL ( (cid:90) ) and ρ T N in the easiest case is the permutation representation of SL ( (cid:90) ) oncosets Γ ( N )\SL ( (cid:90) ). In genearl, ρ T N = T N defined in Section 2.1. For the time being, it suffices toknow that the components of T N E k ( ) can be expressed in terms of E k (cid:161) a τ + bd (cid:162) , where (cid:161) a b d (cid:162) ∈ GL ( (cid:81) ). Asa further ingredient to formulate our Main Theorem, note that by composition, homomorphisms φ : ρ → σ of representations give rise to maps between spaces of modular forms φ : M k ( ρ ) → M k ( σ ). Theorem I.
Let ρ be a complex representation of SL ( (cid:90) ) whose kernel contains a congruence subgroup.For even integers k ≥ and l such that l , k − l ≥ , we have M k ( ρ ) = E k ( ρ ) + span < N , N (cid:48) ∈ (cid:90) φ : ρ TN ⊗ ρ TN (cid:48) → ρ φ (cid:179) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:180) . (0.2) The sum runs over homomorphisms φ from ρ T N ⊗ ρ T N (cid:48) to ρ for positive integers N and N (cid:48) . Remark II. (1) The range of N and N (cid:48) can be bounded by means of Hecke theory. For example, con-sider the case that the representation ρ corresponds to the subgroup Γ ( M ) ⊆ SL ( (cid:90) ) for some positiveM ∈ (cid:90) . That is consider the case that ρ = Ind SL ( (cid:90) ) Γ ( M ) using notation of Section 1.3. The Hecke alge-bra, viewed as a subalgebra of the endomorphism algebra of M k ( Γ ( N )) , is finitely generated, say by(classical) Hecke operators T N with N ≤ N . Then it suffices to let N and N (cid:48) in (0.2) run through integersbetween and N . Precise estimates of how many Hecke operators are required to obtain the all of M k ( ρ ) are not yet available, but planned for a sequel on computational aspects.(2) By introducing holomorphic projections of products of almost holomorphic Eisenstein series, wecould lower the bound on k to k ≥ and the bounds on l and k − l to l , k − l ≥ . This also will bediscussed in more detail in a sequel on computational aspects. Derivatives of Eisenstein series, whichare directly related to almost holomorphic modular forms, were already introduced into the subject atthe end of Section 2 in [KZ84]. – – roducts of Vector Valued Eisenstein Series — Introduction M. Westerholt-Raum (3) It is currently not clear whether the extra space of Eisenstein series in (0.2) is needed. This canbe directly related to a purely representation theoretic question. Using notation from Section 3, the keyquestion is whether for arbitrary representations ρ with congruence subgroup kernel we haveV ( ρ )(1) T = span < N , N (cid:48) ∈ (cid:90) φ : ρ TN ⊗ ρ TN (cid:48) → ρ φ (cid:179) T N V ( ) ⊗ T N (cid:48) V ( ) (cid:180) . Hecke operators.
For simplicity, assume that f is a level 1 modular form of weight k . Classical Heckeoperators can be written as a sum N k − (cid:80) m ∈ ∆ N f | k m , where ∆ N is the set of all (cid:161) a b d (cid:162) with ad = N and 0 ≤ b < d . Instead of summing over ∆ N , we introduce vector valued modular forms, separatingcontributions of each m ∈ ∆ N . The vector valued modular form T N f takes values in the (cid:67) vector spacewith basis ∆ N . Its m -th component for m ∈ ∆ N is defined as f | k m . We show in Section 2.2 that thisis a vector valued modular form. Vector valued Hecke operators can be applied to any vector valuedmodular form. We will defined them in two steps. Given a representation ρ , we find another one T M ρ .Modular forms of type ρ , we show, are mapped to modular forms of type T M ρ . Compatibility with products.
A crucial property of vector valued Hecke operators, compared to clas-sical ones, is that T N ( f ⊗ g ) can be recovered from ( T N f ) ⊗ ( T N g ), while f g | T N (cid:54)= ( f | T N )( g | T N ) ingeneral for scalar valued modular forms—cf. [Duk99] for a related and amusing topic. Explicitly, thesums (cid:80) m , m (cid:48) (cid:161) f | k m (cid:162)(cid:161) g | l m (cid:48) (cid:162) and (cid:80) m f g | k + l m are not equal in any formal basis. Exceptional equali-ties can occur, but this is forced by dimensions. On the other hand, for a single m ∈ ∆ N , we have (cid:161) f | k m (cid:162)(cid:161) g | l m (cid:162) = f g | k + l m . The tensor product of T N f and T N g has components (cid:161) f | k m (cid:162)(cid:161) g | l m (cid:48) (cid:162) for m , m (cid:48) ∈ ∆ N . Picking the components with m = m (cid:48) we obtain T N ( f g ). Relation to classical Atkin-Lehner-Li theory.
Atkin-Lehner-Li theory is governed by classical Heckeoperators and two additional families of operators. The Atkin Lehner involutions W M for M | N , in thesimplest case M = N , map a modular form f to a suitable scalar multiple of τ − k f ( − N τ ). The oldformoperator, tentatively denoted by sc M (notation is alluding to reSCaling) for a positive integer M , maps f to f ( M τ ). Both constructions can be phrased in terms of elements of ∆ M . We will find a suitable m ∈ ∆ M such that f | k m = W M ( f ), and another m ∈ ∆ M such that f | k m = sc M ( f ). In other words,classical Atkin-Lehner-Li theory is subsumed by vector valued Hecke operators.We reformulate this observation: For simplicity let us fix a congruence subgroup Γ = Γ ( N ) ⊆ SL ( (cid:90) ).There is a representation ρ Γ so that M k ( Γ ), the space of modular forms of weight k for the group Γ , iscanonically isomorphic to M k ( ρ Γ ). This isomorphism, here and later, will be denoted by Ind : M k ( Γ ) → M k ( ρ Γ ). The Hecke operators and the Atkin-Lehner involutions are maps from M k ( Γ ) to itself. Theoldform constructions yield maps from M k ( Γ (cid:48) ) to M k ( Γ ) where Γ ⊂ Γ (cid:48) ⊆ SL ( (cid:90) ). In Proposition 2.12and Remark 2.13 of Section 2.3, we give explicit maps π Hecke : T M ρ Γ −→ ρ Γ , π AL : T M ρ Γ −→ ρ Γ , π old : T M ρ Γ −→ ρ Γ such thatInd ◦ ( | k T M ) = π Hecke ◦ T M ◦ Ind, Ind ◦ W M = π AL ◦ T M ◦ Ind, and Ind ◦ α M = π old ◦ T M ◦ Ind.The maps π Hecke , π AL , and π old intertwine the induction map and the three families of operators. Relation to other vector valued Hecke operators.
Weil representations are not in the scope of thepresent paper. Nonetheless, some words seem due: Vector valued modular forms for Weil repre-sentations and one certain kind of Hecke operators for them, denoted by ↑ AH and ↓ AH , were consid-ered in [Sch11]. The same operators are foundational to the newform theory that Bruinier develops – – roducts of Vector Valued Eisenstein Series — Introduction M. Westerholt-Raum in [Bru12] for Weil representation attached to cyclic discriminant modules. A vector valued Hecke op-erator V l was derived in [Rau12] by means of Jacobi forms. All these operators should relate to theHecke operators in the present paper. The operator ↑ AH is analogous to the intertwining property de-scribed in Lemma 2.14. The operator ↓ AH should be its adjoint. Finally, V l seems to be the same as T l ,after applying the theta decomposition and a suitable change of basis. We suggest to consider theseclaims in the context of a newform theory for vector valued modular forms for Weil representations—very much in the spirit of [Bru12].Hecke operators also previously appeared in work by Bruinier and Stein [BS10]. Note that in thatpaper half-integral weight modular forms occur, while we treat integral weight modular forms. Onecan extend our construction to modular forms for representations of the metaplectic double coverof SL ( (cid:90) ), but the details remain to be worked out. The question in [BS10] can be rephrased as whetherfor half-integral weights and Weil representations it is possible to find an analogue to the map π Hecke above. When put in this way, it is a purely representation theoretic question. It was establishedin [BS10] that there is such a map when working with projective representations, but in general thereis a non-trivial obstruction to lifting it to an actual representation.
Relation to automorphic representations.
We sketch a relation between vector valued modular formsand automorphic representation theory. Our description of their links, condensed to half a page, isdoomed to be imprecise, since we now invoke the machinery of adelic automorphic representationtheory. Nevertheless, it yields one fruitful way to think about vector valued Hecke operators. We workover (cid:81) so that at every finite place v corresponds to a prime in the classical sense.We start by relating vector valued modular forms and automorphic representations. Let us fix anewform f , and consider the associated adelic automorphic representation (cid:36) f = (cid:78) (cid:48) v (cid:36) f , v . At eachfinite place v , we obtain a vector w f , v ∈ (cid:36) f , v that corresponds to f . There is a subgroup K ⊆ GL ( (cid:90) p )such that the space of K fixed vectors in (cid:36) f , v is 1-dimensional, spanned by w f , v .Conjugating any such K by elements γ of GL ( (cid:90) p ), we obtain other vectors γ w f , v ∈ (cid:36) f , v that arefixed by γ K γ − . Since K \GL ( (cid:90) p ) is a finite set, one can assemble all γ w f , v for γ ∈ K \GL ( (cid:90) p ) to asingle vector of finite length. From this perspective, vector valued modular forms are a way to view all K with dim (cid:36) Kf , v = (cid:36) f , v is spherical, that is K = GL ( (cid:90) p ). The vector valued Heckeoperator applied to Ind( f ) for a newform f exposes further parts of (cid:36) f , v . It is standard in context ofstrong approximation to view the previously defined set ∆ N for a p -power N as a subset of representa-tives of GL ( (cid:90) p )\GL ( (cid:81) p ). Define K N as the intersection of all γ K γ − for γ ∈ ∆ N . Then T N Ind( f ), at theplace v , is the same as the previously discussed vector of K N fixed vectors of (cid:36) f , v . The dimension of (cid:36) K N f , v increases as N growth, and this is the reason why T N Ind( f ) has increasingly many components.From this perspective, it becomes immediately clear, why classical Atkin-Lehner-Li theory must beencoded in vector valued Hecke operators. Period relations.
A proof of (0.1) based on [KZ84] relies on an explicit formulation of the EichlerShimura isomorphism in terms of period polynomials. In particular, linear relations between the E l E k − l can be rephrased in terms of L -values. As we will explain later, the proof of our main theoremrests on completely different observations. It is possible to give a vanishing criterion for cusp forms asa corollary to our main theorem. However, the resulting convolution L -series, given in Corollary 4.5,can not be simplified in all cases. The reason is that the expansion of newforms at cusps not mappedto ∞ by Atkin-Lehner involutions are not necessarily newforms. In other words, given a level N new-form f and a cusp c for which there is no M | N with W M c = ∞ , then there is no a priori reason why theexpansion of f at c is even a Hecke eigenform; let alone a newform. – – roducts of Vector Valued Eisenstein Series — Introduction M. Westerholt-Raum We refer the reader to the explicit vector valued Eichler Shimura theorem proved by Pa¸sol andPopa [PP13] in analogy with the level 1 case. It would be interesting to relate our Main Theorem toa statement about periods and compare it to Corollary 4.5. For the time being, we present the latter ascuriosity, for which we neither have an application nor a satisfying explanation.
Computing modular forms and their cusp expansions.
Our Main Theorem allows us to span spacesof cusp forms by products of at most two Eisenstein series. This provides an alternative to algorithmsbased on modular symbols [Cre97; Man72]. In a sequel to this paper, we will report on how perfor-mance of these two methods compare.Fix a modular form f for a Dirichlet character χ mod N . In the context of, for example, Borcherdsproducts [Bor98] it is an interesting question how Fourier expansions of f at cusps of Γ ( N )\ (cid:72) can becomputed. If N is square free then modular symbols provide us with a satisfactory theory, which isemployed, e.g., in Sage [Ste+14]. Indeed, Atkin Lehner operators act on modular symbols, and in thiscase permute cusps transitively.The case of non square free level N is more complicated. Atkin Lehner involutions fail to act transi-tively on cusps, and hence modular symbols are insufficient to obtain all cusp expansions. In [Rau12],we found an algorithm to compute them, which however does not perform very well in practice. Itsruntime is subexponential, but not polynomial with respect to the level. Combining our Main The-orem with formulas for cusp expansions of Eisenstein series obtained in [Wei77], we obtain anotherpromising approach to computing cusp expansions. It has the charm of relying only on multiplica-tion of power series and computation of tensor products. The former is straightforward, and the lattercan be quickly reduced to linear algebra over cyclotomic fields, which for example is implementedefficiently in the computer algebra system Magma. Runtime.
Let us sketch how to compare the runtime of the best known algorithms for computingFourier expansions of modular forms [EC11] and an algorithm which builds on the present work.Writing M • ( χ ) for the M • -module of modular forms for a Dirichlet character χ . To compute Fourierexpansions for all elements of M • ( χ ) it suffices to determine those of a minimal set of generators. Thedimension formula for modular forms implies that their weight is less than 12. We therefore con-sider computations of modular forms for a fixed weight k . We bound the time required to compute allFourier coefficients c ( n ) for n < n with n → ∞ of all modular forms of level N < N with N → ∞ .This is justified at least with an eye to combinatorial applications—see [And74] and work following theideas described there.Let us first assume that we have uniquely identified a modular form. The algorithm by Couveignesand Edixhoven [EC11] essentially requires a piece of the Jacobian that corresponds to an irreducibleGalois representation. Given such a representation, it computes the n -th Fourier coefficient of theattached newform in TIME(log( n )). We are suprressing issues with mod l Galois representations forthe benefit of [EC11]. Repeating this for all n < n yields TIME( n log( n )). An algorithm based onthe present paper would require for the computation of Fourier coefficients of a modular form an ex-pression in terms of Eisenstein series and vector valued Hecke operators. Given such a representationthe remaining computations are to evaluate Fourier coefficients of level 1 Eisenstein series, and tomultiply Fourier expansions of length n . Both yield TIME( n log( n )). Summing up, in the aspect ofcomputing all Fourier coefficients of a given modular form up to a given precision the two algorithmsperform asymptotically equally fast.We have to discuss how to obtain a Galois representation, and how to evaluate (0.2). Galois represen-tations can be computed using modular symbols [Ste07]. Computing modular symbols for a Dirichletcharacter modulo N relies on computing the reduced echelon normal form of a size O ( kN ) matrix – – roducts of Vector Valued Eisenstein Series — 1 Preliminaries M. Westerholt-Raum with entries in a cyclotomic field of order N (and degree ϕ ( N )). In the present consideration, we ne-glect utilization of degeneracy maps, whose impact on runtime to the author’s knowledge has notbeen effectively estimated. Let ω ∈ (cid:82) be the runtime exponent for computing reduced echelon normalforms. Then computing modular symbols for one fixed character yields TIME( k ω N ω ). Summing overall characters modulo N < N yields runtime TIME( k ω N ω + ).We pass to (0.2). Using Sturm bounds and inspecting the proof of the main theorem it seems plau-sible to assume that it suffices to decompose T N ⊗ T (cid:112) N to find an expression for level N modularforms for all divisors of N —a precise analysis would be involved and is planned for the aforemen-tioned sequel on practical aspects. Using the MeatAxe algorithm (cf. Chapter 7.4 of [HEO05]), we canreduce computations to the echelon form of a matrix of size O ( N ) with coefficients in an order N cyclotomic field. This yields a runtime estimate TIME( k + N ω /4 + ), where the additional term k stemsfrom the contribution of the weight to Sturm bounds.The FLINT: Fast Library for Number Theory uses the Strassen algorithm, yields 2.8 < ω <
3, whiletheoretical estimates show that ω < ω /4. One should Keep in mind that the decomposition of T N ⊗ T (cid:112) N can probably be sped up using for example the well-known classification of representations ofSL ( (cid:81) p ). Method of proof.
We discuss the method of proof, comparing it to [KZ84]. In the classical setting,one fixes a cusp form f that is orthogonal to all E l E k − l . Relating this to vanishing of periods oneconcludes that the period polynomial of an attached cusp form f (cid:48) , which in general is different from f ,vanishes. This, in particular, makes use of an explicit Eichler Shimura correspondence. In our case, weachieve to reduce ourselves to the case of newforms f , such that the necessary vanishing statement isstraightforward.Reduction to newforms for Γ ( N ) ⊆ SL ( (cid:90) ) is performed by vector valued Hecke operators T N , in-troduced in Section 2. We show that span φ φ (cid:161) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:162) in (0.2) yields a Hecke module inthe classical sense. If an irreducible Hecke module which appears in M k (cid:161) Γ ( N ) (cid:162) , the space of modularforms for Γ ( N ), is missed by our construction, then we fix a suitable modular form in there. Morespecifically, we may choose a newform f , and are hence reduced to this case.In the course of the proof, we exploit the symmetry of (0.2) to focus on the case l ≤ k /2. This is nec-essary to conclude that the L -series L ( f , k − l ) that appears in the proof is nonzero. The case l = k /2is special, since L ( f , k /2) may indeed vanish. Note that this difficulty has already appeared in [BG01],and after the first preprint version of the present paper in [DN15]. Vector valued Hecke operators al-lows us to circumvent it. We employ them to show that L ( f × χ , k /2) vanishes for certain Dirichletcharacters χ and then make use of Waldspurger’s nonvanishing results. Acknowledgment.
The author is grateful to one of the referees for extraordinarily extensive and veryhelpful comments, improving readability of this paper. §1.1 Classical modular forms.
As usual, write (cid:72) for the Poincaré upper half plane { τ ∈ (cid:67) : Im τ > ( (cid:82) ) by Möbius transformations. Writing γ = (cid:161) a bc d (cid:162) for an element of SL ( (cid:82) ), wehave γτ = a τ + bc τ + d . This action gives rise to a family | k of actions on holomorphic functions f on (cid:72) , whichis indexed by k ∈ (cid:90) . We set ( f | k γ )( τ ) = ( c τ + d ) − k f ( γτ ). For a character χ of Γ ⊆ SL ( (cid:90) ) and γ ∈ Γ , weset ( f | k , χ γ )( τ ) = ( c τ + d ) − k χ ( γ − ) f ( γτ ). – – roducts of Vector Valued Eisenstein Series — 1 Preliminaries M. Westerholt-Raum Definition 1.1.
Let Γ ⊆ SL ( (cid:90) ) be a finite index subgroup and χ a character of Γ . We call a holomorphicfunction f : (cid:72) → (cid:67) a modular form of weight k for Γ and χ if(i) for all γ ∈ Γ , we have f | k , χ γ = f , and(ii) for every γ ∈ SL ( (cid:90) ) , (cid:161) f | k γ (cid:162) ( τ ) is bounded as τ → i ∞ .The space of such functions is denoted by M k ( Γ , χ ) = M k ( χ ) and M k ( Γ ) , if χ is trivial. The correspondingsubspaces of cusp forms are denoted by S k ( Γ , χ ) = S k ( χ ) and S k ( Γ ) . Denoting the entries of γ ∈ SL ( (cid:90) ) by a ( γ ), b ( γ ), c ( γ ), and d ( γ ), we define subgroups Γ ( N ) = (cid:169) γ ∈ SL ( (cid:90) ) : c ( γ ) ≡ N ) (cid:170) , Γ ( N ) = (cid:169) γ ∈ SL ( (cid:90) ) : c ( γ ) ≡ a ( γ ) ≡ d ( γ ) ≡ N ) (cid:170) , Γ ( N , N ) = (cid:169) γ ∈ SL ( (cid:90) ) : c ( γ ) ≡ N ), a ( γ ) ≡ d ( γ ) ≡ N ) (cid:170) , and Γ ( N ) = (cid:169) γ ∈ SL ( (cid:90) ) : b ( γ ) ≡ c ( γ ) ≡ a ( γ ) ≡ d ( γ ) ≡ N ) (cid:170) .Dirichlet characters mod N give rise to characters of Γ ( N ) by means of χ ( γ ) = χ ( d ( γ )). The standardparabolic subgroup of SL ( (cid:90) ) is defined as Γ ∞ = (cid:169) γ ∈ SL ( (cid:90) ) : c ( γ ) = (cid:170) . §1.2 Vector valued modular forms. A (complex) representation ρ of Γ ⊂ SL ( (cid:90) ) is a group homomor-phism from Γ to GL (cid:161) V ( ρ ) (cid:162) for a complex vector space V ( ρ ), which is called the representation spaceof ρ . We focus on finite dimensional representations, in which case the dual of ρ , denoted by ρ ∨ ,can be defined by ρ ∨ ( γ ) = t ρ ( γ ) − . Its representation space V ( ρ ∨ ) is the space of linear function-als V ( ρ ) ∨ on V ( ρ ). The trivial representation of any group is denoted by , and the group will beclear from the context. A Γ -representation ρ is called irreducible if the only Γ invariant subspaces of V ( ρ ) are V ( ρ ) and {0}. Given a representation ρ and an arbitrary irreducible one ρ (cid:48) , we call ρ ( ρ (cid:48) ) : = V ( ρ (cid:48) ) ⊗ Hom( ρ (cid:48) , ρ ) ⊆ V ( ρ ) the ρ (cid:48) -isotypical component of ρ . It is the maximal subrepresentation of ρ for which Hom( ρ (cid:48)(cid:48) , ρ ) (cid:54)= {0} implies that an irreducible ρ (cid:48)(cid:48) is isomorphic to ρ (cid:48) . In particular, if ρ (cid:48) is nota subrepresentation of ρ , then ρ ( ρ (cid:48) ) = {0}. The isotrivial component of a representation ρ is definedas ρ ( ). Recall that in our setting all unitary representations are completely reducible as a direct sum,and to every inclusion ρ (cid:44) → σ there is a corresponding projection σ (cid:16) ρ and vice versa.For any k ∈ (cid:90) and any finite dimensional representation ρ of SL ( (cid:90) ), we set (cid:161) f | k , ρ γ (cid:162) ( τ ) = ( c τ + d ) − k ρ ( γ − ) f ( γτ ). Definition 1.2.
Fix a finite dimensional complex representation ρ of SL ( (cid:90) ) . A holomorphic functionf : (cid:72) → V ( ρ ) , we say, is a modular form of weight k and type ρ , if(i) for all γ ∈ SL ( (cid:90) ) , we have f | k , ρ γ = f , and(ii) for every v ∈ V ( ρ ) ∨ , (cid:161) v ◦ f (cid:162) ( τ ) is bounded as τ → i ∞ .The space of such functions is denoted by M k ( ρ ) . A cusp forms is a modular form f ∈ M k ( ρ ) satisfying (cid:161) v ◦ f (cid:162) ( τ ) → as τ → i ∞ for all v ∈ V ( ρ ) ∨ . The space of cusp forms is denoted by S k ( ρ ) .Remark 1.3. (1) We have defined vector valued modular forms for all finite dimensional, complexrepresentations of SL ( (cid:90) ). Our main interest, however, lies in modular forms for congruence sub-groups. This is why we will, from here on, restrict ourselves to representations with finite index ker-nel. Since such ρ : SL ( (cid:90) ) → GL (cid:161) V ( ρ ) (cid:162) factor through a finite quotient SL ( (cid:90) )/ ker ρ , they are unita-rizable. In particular, we can and will assume that ρ is unitary with respect to a scalar product, say, v , w → 〈 v , w 〉 = 〈 v , w 〉 ρ on V ( ρ ). – – roducts of Vector Valued Eisenstein Series — 1 Preliminaries M. Westerholt-Raum (2) Representations of SL ( (cid:90) ) whose kernel is a congruence subgroup, can be factored as a ten-sor product (cid:78) p ρ p of representations whose level is a p -power. Such representations were studiedin [Sil70]. In particular, a complete classification is available.Given v ∈ V ( ρ ), we write 〈 f , v 〉 for the function τ (cid:55)→ 〈 f ( τ ), v 〉 , which is a modular form for ker ρ . Fortwo functions f : (cid:72) → V ( ρ ) and g : (cid:72) → V ( σ ), we write f ⊗ g : (cid:72) → V ( ρ ⊗ σ ) for ( f ⊗ g )( τ ) = f ( τ ) ⊗ g ( τ ).Morphisms of representations φ : σ → ρ give rise to maps φ : M k ( σ ) → M k ( ρ ) on modular forms, in ananalogous way. Proposition 1.4. If ρ = ρ ⊕ ρ , then there is a canonical isomorphism of M k ( ρ ) and M k ( ρ ) ⊕ M k ( ρ ) .Proof. Let ι : ρ (cid:44) → ρ and ι : ρ (cid:44) → ρ be the inclusions associated with the given decomposition of ρ .For f ∈ M k ( ρ ) and f ∈ M k ( ρ ), we have ( ι ◦ f ) + ( ι ◦ f ) ∈ M k ( ρ ) as is easily verified. The inverse isgiven by the corresponding projections π : ρ (cid:16) ρ and π : ρ (cid:16) ρ .We proceed to the definition of Peterssson scalar products. Let ρ and σ be two representations ofSL ( (cid:90) ) and fix an embedding ι : (cid:44) → ρ ⊗ σ , where σ is the complex conjugate of σ . Since we are in theunitary setting, there is an identification of σ = σ ∨ be means of the scalar product on V ( σ ). In casethat ρ = σ , there is a canonical choice given by ι (1) = (cid:80) v v ⊗ v , where v runs through an orthonormalbasis of V ( ρ ). In general, we have Hom( , ρ ⊗ σ ∨ ) ∼= Hom( σ , ρ ). For f ∈ S k ( ρ ) and g ∈ S k ( σ ), we define 〈 f , g 〉 ι = (cid:90) SL ( (cid:90) )\ (cid:72) 〈 f ⊗ g , ι (1) 〉 dx dyy − k . (1.1)As usually, we can extend the Petersson scalar product to the case f ∈ M k ( ρ ) by applying Borcherdsregularization, explained at the beginning of Section 6 of [Bor98]. Cusp forms are, as seen when un-folding, orthogonal to Eisenstein series, defined in Section 3. Details on the unfolding of regularizedintegrals can be found in [Bru02], on page 47ff. §1.3 Induced representations. Recall that we focus on finite dimensional representations whosekernel has finite index in SL ( (cid:90) ). The induced representation Ind Γ (cid:48) Γ ρ attached to Γ ⊆ Γ (cid:48) ⊆ SL ( (cid:90) ) and arepresentation ρ of Γ can thus be defined by V (cid:161) Ind Γ (cid:48) Γ ρ (cid:162) : = V ( ρ ) ⊗ (cid:67) [ B ] and Ind ρ ( γ (cid:48) ) (cid:161) v ⊗ e β (cid:162) = (cid:161) ρ (cid:161) I − β ( γ (cid:48)− ) (cid:162) v (cid:162) e βγ (cid:48)− (1.2)for a fixed system B of representatives of Γ \ Γ (cid:48) containing the identity element, and the following cocy-cle I . For β ∈ B and γ (cid:48) ∈ Γ (cid:48) , we set βγ (cid:48) = I β ( γ (cid:48) ) βγ (cid:48) where βγ (cid:48) ∈ B and I β ( γ (cid:48) ) ∈ Γ . This indeed defines acocycle, that is, we have I β ( γ (cid:48) γ (cid:48) ) = I β ( γ (cid:48) ) I βγ (cid:48) ( γ (cid:48) ). To ease notation, we extend it to all of Γ (cid:48) by setting I β = I β . We write e β , β ∈ B for the canonical orthonormal basis of (cid:67) [ B ].We define a map Ind on modular forms. For any k ∈ (cid:90) and any Dirichlet character χ mod N , we setInd : M k ( Γ ( N ), χ ) −→ M k (cid:161) Ind Γ ( N ) χ (cid:162) , f (cid:55)−→ (cid:88) γ ∈ Γ ( N )\SL ( (cid:90) ) (cid:161) f | k γ (cid:162) e γ , (1.3)where γ runs trough a fixed system of representatives of Γ ( N )\SL ( (cid:90) ). The group and character of afunction will always be fixed separately, so that we throughout write Ind( f ) without referring to thegroup or character attached to f . Proposition 1.5.
The map
Ind in (1.3) is an isomorphism. – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum Proof.
It is clear that Ind is injective. Its inverse is given by f (cid:55)→ 〈 f , e I 〉 for f ∈ M k (Ind Γ ( N ) χ ), where I is the 2 × Proposition 1.6.
For classical modular forms f ∈ M k ( χ ) and g ∈ S k ( χ ) with χ a Dirichlet charactermodulo N and the associated vector valued forms Ind f , Ind g ∈ M k (Ind Γ ( N ) χ ) , we have 〈 f , g 〉 = 〈 Ind f , Ind g 〉 ι , with ι (1) = ( (cid:90) ) : Γ ( N )] (cid:88) γ ∈ Γ ( N )\SL ( (cid:90) ) e γ ⊗ e γ ,where [SL ( (cid:90) ) : Γ ( N )] is the index of Γ ( N ) in SL ( (cid:90) ) .Proof. Employing the definition of 〈 Ind f , Ind g 〉 ι , we obtain (cid:90) SL ( (cid:90) )\ (cid:72) (cid:68) Ind f ⊗ Ind g , (cid:80) γ e γ ⊗ e γ [SL ( (cid:90) ) : Γ ( N )] (cid:69) dxdyy − k = I − N (cid:90) SL ( (cid:90) )\ (cid:72) (cid:88) γ (cid:161) f | k γ (cid:162)(cid:161) g | k γ (cid:162) dxdyy − k = I − N (cid:90) Γ ( N )\ (cid:72) f g dxdyy − k ,where I N = [SL ( (cid:90) ) : Γ ( N )]. The definition of vector valued Hecke operators involves two steps. In Section 2.1, we study Heckeoperators on representations, and Section 2.2 contains a treatment of Hecke operators on modularforms. We show in Section 2.3 that classical Hecke operators for modular forms attached to Γ ( N ) anda Dirichlet character χ can be reconstructed from the new definition. §2.1 Hecke operators on representations. For a positive integer M , let ∆ M = (cid:169) (cid:161) a b d (cid:162) : ad = M , 0 ≤ b < d (cid:170) be a set of upper triangular matrices of determinant M that are inequivalent with respect to theaction of SL ( (cid:90) ) from the left. Given any 2 × m with integer coefficients and determinant M ,there is γ (cid:48) ∈ SL ( (cid:90) ) such that γ (cid:48) m = m for some matrix m ∈ ∆ M . We will use this overline-notationthroughout.We have a right action of SL ( (cid:90) ) on ∆ M defined by ( m , γ ) (cid:55)→ m γ with γ (cid:48) m γ = m γ for some γ (cid:48) ∈ SL ( (cid:90) ).It is readily verified that the associated map I m ( γ ) = γ (cid:48) is a cocyle, satisfying I m ( γ γ ) = I m ( γ ) I m γ ( γ ).Note that the subscript m makes it impossible to confuse I m and I γ defined in Section 1.3.The cocycle property of I m ( γ ) guaranties that, given a representation ρ , then T M ρ which is definedby V ( T M ρ ) : = V ( ρ ) ⊗ (cid:67) (cid:163) ∆ M (cid:164) and ( T M ρ )( γ ) ( v ⊗ e m ) : = ρ (cid:161) I − m ( γ − ) (cid:162) ( v ) ⊗ e m γ − (2.1)is a also representation. Indeed, we have( T M ρ )( γ ) (cid:161) ( T M ρ )( γ )( v ⊗ e m ) (cid:162) = ( T M ρ )( γ ) (cid:161) ρ (cid:161) I − m ( γ − ) (cid:162) v ⊗ e m γ − (cid:162) = ρ (cid:161) I − m γ − ( γ − ) (cid:162) ρ (cid:161) I − m ( γ − ) (cid:162) v ⊗ e m γ − γ − = ρ (cid:161) I m ( γ − ) I m γ − ( γ − ) (cid:162) − v ⊗ e m ( γ γ ) − = ρ (cid:161) I − m (cid:161) ( γ γ ) − (cid:162)(cid:162) v ⊗ e m ( γ γ ) − = ( T M ρ )( γ γ )( v ⊗ e m ) . – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum Definition 2.1.
Let M be a positive integer. Then the M -th vector valued Hecke operator on representa-tions is the assigment T M : ρ (cid:55)−→ T M ρ ,where T M ρ is the representation defined in (2.1) .Remark 2.2. For the purpose of reference, we note that ( T M ρ )( γ − ) and ( T M ρ ) ∨ ( γ ) act by( T M ρ )( γ − ) ( v ⊗ e m ) : = ρ ( I − m ( γ ))( v ) ⊗ e m γ , ( T M ρ ) ∨ ( γ ) ( v ⊗ e m ) : = ρ ∨ ( I m ( γ ))( v ) ⊗ e m γ .The next two technical lemmas will be used frequently without further mentioning them. Lemma 2.3. If ρ has finite index kernel, then T M ρ has finite kernel, too. If ker ρ is a congruence sub-group, then so is ker ( T M ρ ) .Proof. A necessary condition for γ ∈ ker T M ρ is m γ − = m for all m ∈ ∆ M . For our purpose, it sufficesto observe that the principal congruence subgroup Γ ( M ) acts trivially on ∆ M . Given γ ∈ Γ ( M ), we have γ ∈ ker T M ρ if I − m ( γ − ) = m − γ m ∈ ker( ρ ) for all m . In other words, we haveker (cid:161) T M ρ (cid:162) ⊆ Γ ( M ) ∩ (cid:92) m ∈ ∆ M m (ker ρ ) m − .If Γ ( N ) ⊆ ker ρ for some N , we readily verify that Γ ( M N ) ⊆ ker (cid:161) T M ρ (cid:162) , completing the proof. Lemma 2.4.
Assume that ρ is unitary with respect to the scalar product 〈· , · 〉 ρ on V ( ρ ) . Then T M ρ isunitary with respect to 〈 v ⊗ e m v , w ⊗ e m w 〉 = (cid:40) 〈 v , w 〉 ρ , if m v = m w ; , otherwise.Proof. This follows directly, since SL ( (cid:90) ) acts on the e m by permutations.The following two homomorphisms involving vector valued Hecke operators on representationsare central to our further considerations. Their analogues for attached spaces of modular forms isstated in Theorem 2.8. Note that to ease discussion from now on we will identify a representation withits representation space. In the next statement and throughout the paper, inclusions are denoted byarrows (cid:44) −→ and surjections are denoted by arrows (cid:16) . Proposition 2.5.
For every positive integer M , for every two finite dimensional SL ( (cid:90) ) -representations ρ and σ , and for every homomorphism φ : ρ → σ between them, the following maps are homomorphismsof SL ( (cid:90) ) -representations:T M φ : T M ρ −→ T M σ , v ⊗ e m (cid:55)−→ φ ( v ) ⊗ e m ; (2.2) (cid:44) −→ T M , c (cid:55)−→ c (cid:88) m ∈ ∆ M e m ; (2.3)( T M ρ ) ⊗ ( T M σ ) − (cid:16) T M ( ρ ⊗ σ ) , ( v ⊗ e m ) ⊗ ( w ⊗ e m (cid:48) ) (cid:55)−→ (cid:40) ( v ⊗ w ) ⊗ e m , if m = m (cid:48) ; , otherwise. (2.4) If ψ : σ → (cid:36) is a further homomorphism of SL ( (cid:90) ) -representations, then T M ( ψ ◦ φ ) = ( T M ψ ) ◦ ( T M φ ) . – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum Proof.
The equality( T M ρ )( γ ) ( T M φ )( v ⊗ e m ) = σ (cid:161) I − m ( γ − ) (cid:162)(cid:161) φ ( v ) ⊗ e m γ − (cid:162) = φ (cid:161) ρ (cid:161) I − m ( γ − ) (cid:162) v (cid:162) ⊗ e m γ − ,which yields ( T M φ ) ( T M ρ )( γ )( v ⊗ e m ), establishes that T M φ is a homomorphism. Compatibility withcomposition of homomorphisms is readily verified.To show that (2.3) defines a homomorphism, it suffices to observe that SL ( (cid:90) ) acts on ∆ M by per-mutations. In particular, the action on (cid:80) m e m is trivial.For the same reason, m γ − = m (cid:48) γ − implies that m = m (cid:48) . In particular, for m (cid:54)= m (cid:48) we have (cid:161) T M ρ ⊗ T M σ (cid:162) ( γ ) (cid:161) ( v ⊗ e m ) ⊗ ( w ⊗ e m (cid:48) ) (cid:162) = (cid:161) I − m ( γ − ) v ⊗ e m γ − (cid:162) ⊗ (cid:161) I − m (cid:48) ( γ − ) w ⊗ e m (cid:48) γ − (cid:162) (cid:55)−→ (cid:161) T M ρ ⊗ T M σ (cid:162) ( γ ) (cid:161) ( v ⊗ e m ) ⊗ ( w ⊗ e m ) (cid:162) = (cid:161) ρ ( I − m ( γ − )) v ⊗ e m γ − (cid:162) ⊗ (cid:161) σ ( I − m ( γ − )) w ⊗ e m γ − (cid:162) (cid:55)−→ (cid:161) ρ ( I − m ( γ − )) v ⊗ σ ( I − m ( γ − )) w (cid:162) ⊗ e m γ − = (cid:161) T M ( ρ ⊗ σ ) (cid:162) ( γ ) (cid:161) ( v ⊗ w ) ⊗ e m (cid:162) . §2.2 Hecke operators on modular forms. For m = (cid:161) a b d (cid:162) ∈ GL ( (cid:82) ) and f : (cid:72) → (cid:67) , we define (cid:161) f | k m (cid:162) ( τ ) = (cid:161) ad (cid:162) k f (cid:161) a τ + bd (cid:162) . Definition 2.6.
Fix a representation ρ of SL ( (cid:90) ) with finite index kernel and a positive integer M . Givenf ∈ M k ( ρ ) , we define T M f by (cid:161) T M f (cid:162) ( τ ) = (cid:88) m ∈ ∆ M (cid:161) f | k m (cid:162) ⊗ e m . (2.5) Proposition 2.7.
If f ∈ M k ( ρ ) , then T M f ∈ M k ( T M ρ ) .Proof. We have to check that ( T M f ) | k γ = ( T M ρ )( γ ) f for all γ ∈ SL ( (cid:90) ). Our proof is a direct computation using the definitions given in (2.1) and (2.6). (cid:161) T M f (cid:162) | k , T M ρ γ = (cid:88) m ∈ ∆ M (cid:161)(cid:161) f | k m (cid:162) ⊗ e m (cid:162) | k , T M ρ γ = (cid:88) m ∈ ∆ M ρ ( I − m ( γ )) (cid:161) f | k m γ (cid:162) ⊗ e m γ = (cid:88) m ∈ ∆ M ρ ( I − m ( γ )) (cid:161) f | k I m ( γ ) m γ (cid:162) ⊗ e m γ = (cid:88) m ∈ ∆ M (cid:161) f | k m γ (cid:162) ⊗ e m γ = T M f .In the third equality, we have used the fact that m γ = I m ( γ ) m γ , by definition of the cocycle I . Thefourth equation follows from f ∈ M k ( ρ ). The last equality follows when replacing m γ by m (cid:48) ∈ ∆ M . Theorem 2.8.
Fix a positive integer M and two representations ρ and σ of SL ( (cid:90) ) . Write π M , ρ , σ for thehomomorphism of representations defined in (2.4) . For any weight k ∈ (cid:90) , it gives rise to a linear map ofmodular forms by composition: M k (cid:161) T M ρ ⊗ T M σ (cid:162) −→ M k (cid:161) T M ( ρ ⊗ σ ) (cid:162) , f (cid:55)−→ π M , ρ , σ ◦ f . (2.6) This map intertwines the tensor product of modular forms and the vector valued Hecke operator. Thatis, we have π M , ρ , σ ◦ (cid:161) ( T M f ) ⊗ ( T M g ) (cid:162) = T M ( f ⊗ g ) (2.7) for any two modular forms f ∈ M k ( ρ ) and g ∈ M l ( σ ) . – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum Remark 2.9.
Compare Theorem 2.8 to the fact that ( f | k T M )( g | l T M ) (cid:54)= f g | k + l T M for general f ∈ M k ( Γ )and g ∈ M l ( Γ ) and finite index subgroups Γ ⊆ SL ( (cid:90) ), where | T M is the classical Hecke operator. Proof of Theorem 2.8.
The first part follows immediately from the fact that π M , ρ , σ is a homomorphism.The second part is quickly verified: π M , ρ , σ (cid:179) (cid:88) m , m (cid:48) ∈ ∆ M (cid:161) f | k m ⊗ e m (cid:162) ⊗ (cid:161) g | l m (cid:48) ⊗ e m (cid:48) (cid:162)(cid:180) = (cid:88) m ∈ ∆ M (cid:161) f | k m (cid:162) ⊗ (cid:161) g | l m (cid:162) ⊗ e m . §2.2.1 Hecke operators from induced representations. We have decided to define Hecke operatorsin a way that resembles the classical construction most. Instead however, we could equivalently define T M ρ as a suitable SL ( (cid:90) )-invariant subspace of an induced representation. We discuss this idea in abit more detail.Given a representation ρ of SL ( (cid:90) ) let (cid:98) ρ = Ind GL + ( (cid:81) )SL ( (cid:90) ) ρ ,where GL ( (cid:81) ) + ⊂ GL ( (cid:81) ) is the subgroup of matrices with positive determinant. Since GL ( (cid:81) ) + isdiscrete, a definition of the induction analogous with the one in Section 1.3 works in this case, ignoringslight topological complications. Fixing a set of representatives of SL ( (cid:90) )\GL ( (cid:81) ) + , set γδ = ˆ I γ ( δ ) γδ for γ and γδ one of these representative and ˆ I γ ( δ ) ∈ SL ( (cid:90) ). Clearly, ˆ I γ ( δ ) is a cocyle.For simplicity we assume that the chosen set of representatives of SL ( (cid:90) )\GL ( (cid:81) ) + comprises all ∆ M .We claim that ι Ind, M : T M ρ (cid:44) −→ (cid:98) ρ , v ⊗ e m (cid:55)−→ v ⊗ e m (2.8)is a well-defined inclusion of SL ( (cid:90) )-representations. In particular, (cid:76) M T M ρ (cid:44) → (cid:98) ρ . Note that there isan associated projection π Ind, M , that maps e γ with γ ∈ SL ( (cid:90) )\GL ( (cid:81) ) + to either e γ if γ ∈ ∆ M or to zero,otherwise.It is clear that (2.8) is well-defined, because of our assumption that ∆ M is part of the representativesof SL ( (cid:90) )\GL ( (cid:81) ) + . Equivariance with respect to SL ( (cid:90) ) is almost part of the definition of the cocycles.Indeed, for m ∈ ∆ M and δ ∈ SL ( (cid:90) ), we have ˆ I m ( δ ) m δ = m δ = I m ( δ ) m δ .Next, we compare the Hecke operator on modular forms and the induction of modular forms. Given f ∈ M k ( ρ ), we set (cid:100) Ind( f ) = (cid:80) γ (cid:161) f | k γ (cid:162) ⊗ e γ , where the sum runs over SL ( (cid:90) )\GL ( (cid:81) ) + . We claim that π Ind, M intertwines T M and (cid:100) Ind. That is, we have π Ind, M (cid:161)(cid:100) Ind( f ) (cid:162) = T M f for all f ∈ M k ( ρ ). The proof is straightforward and again makes use of the fact that the cocycles I m ( δ )and ˆ I m ( δ ) are essentially the same. §2.2.2 Adjunction. In (1.1), we have already introduced a scalar product on all spaces of vector valuedmodular forms that we consider. In the context of vector valued Hecke operator, we are naturally ledto ask for a corresponding adjunction formula. To describe it, let m = (cid:161) d − b − c a (cid:162) be the adjoint of a 2 × m . Set for a representation ρ and a positive integer M ι adj : ρ (cid:44) −→ T M T M ρ , v (cid:55)−→ (cid:88) m ∈ ∆ M v ⊗ e m ⊗ e m . (2.9) – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum A corresponding projection is defined by π adj : T M T M ρ − (cid:16) ρ , (cid:88) m , m (cid:48) ∈ ∆ M v m , m (cid:48) ⊗ e m ⊗ e m (cid:48) (cid:55)−→ (cid:88) m ∈ ∆ M v m , m . (2.10) Proposition 2.10.
For every representation ρ and every positive integer M the maps ι adj and π adj areinclusions and projections of representations, respectively.Given modular forms f ∈ M k ( ρ ) and g ∈ M k ( T M ρ ) , we have 〈 T M f , g 〉 = 〈 f , π adj T M g 〉 = 〈 ι adj f , T M g 〉 . (2.11) Proof.
It is clear that π adj is adjoint to ι adj so that is suffices to show that the latter is a homomorphismof representations. Given v ∈ V ( ρ ) and δ ∈ SL ( (cid:90) ), we have to check that( T M T M ρ )( δ − ) (cid:161) ι adj ( v ) (cid:162) = ι adj (cid:161) ρ ( δ − )( v ) (cid:162) .The right hand side side equals (cid:88) m ∈ ∆ M ρ ( δ − )( v ) ⊗ e m ⊗ e m ,while the left hand side is( T M T M ρ )( δ − ) (cid:88) m ∈ ∆ M v ⊗ e m ⊗ e m = (cid:88) m ∈ ∆ M (cid:161) T M ρ (cid:162)(cid:161) I − m ( δ ) (cid:162) (cid:161) v ⊗ e m (cid:162) ⊗ e m δ = (cid:88) m ∈ ∆ M ρ (cid:161) I − m ( I m ( δ )) (cid:162) ( v ) ⊗ e mI m ( δ ) ⊗ e m δ .To establish the first part of the proposition, we therefore have to show that m δ = mI m ( δ ) and I m ( I m ( δ ) = δ .We show the second equality first. Observe that we have mI m ( δ ) m δ = mm δ = M δ implying that mI m ( δ ) = δ M (cid:161) m δ (cid:162) − ∈ ∆ M . (2.12)In other words, δ satisfies the defining property of I m ( I m ( δ )). We deduce the first equality by evaluat-ing the product mI m ( δ ) m δ = I − m ( I m ( δ )) mI m ( δ ) I − m ( δ ) m δ = I − m ( I m ( δ )) mm δ = I − m ( I m ( δ )) M δ = M ,where the last equality follows from rewriting (2.12). This finishes our proof that ι adj is an inclusion ofrepresentations.It remains to argue that (2.11) is true, which says that 〈 T M f , g 〉 = 〈 f , π adj T M g 〉 = 〈 ι adj f , T M g 〉 .Using that π adj and ι adj are adjoint, it suffices to consider equality of 〈 T M f , g 〉 and 〈 ι adj f , T M g 〉 . Inorder to keep computations as explicit as possible let v i ∈ V ( ρ ) and v ∨ i be an orthonormal basis and itsdual. Let f i and g i , m be the corresponding components of f and g . The left hand side of (2.11) equals (cid:90) SL ( (cid:90) )\ (cid:72) (cid:88) i , m (cid:161) f i | k m (cid:162)(cid:161) g i , m (cid:162) dxdyy − k . – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum The right hand side is (cid:90) SL ( (cid:90) )\ (cid:72) (cid:88) i , m , m (cid:48) f i δ ( m , m (cid:48) ) (cid:161) g i , m | k m (cid:48) (cid:162)(cid:162) dxdyy − k = (cid:90) SL ( (cid:90) )\ (cid:72) (cid:88) i , m , m (cid:48) f i (cid:161) g i , m | k m (cid:162) dxdyy − k ,where δ denotes the Kronecker delta function. The remainder of the proof is the same as in the classi-cal setting. We exploit invariance of the measure, applying m / (cid:112) M ∈ SL ( (cid:82) ). This yields (cid:90) SL ( (cid:90) )\ (cid:72) (cid:88) i , m , m (cid:48) (cid:161) f i | k m / (cid:112) M (cid:162)(cid:161) g i , m | k m m / (cid:112) M (cid:162)(cid:162) dxdyy − k = (cid:90) SL ( (cid:90) )\ (cid:72) (cid:88) i , m , m (cid:48) (cid:161) f i | k m / (cid:112) M (cid:162)(cid:161) g i , m | k (cid:112) M I (cid:162)(cid:162) dxdyy − k = (cid:90) SL ( (cid:90) )\ (cid:72) (cid:88) i , m , m (cid:48) (cid:161) f i | k m (cid:162)(cid:161) g i , m (cid:162) dxdyy − k ,which is the left hand side of (2.11), as stated above. §2.3 Connections with known constructions. In this section, we illustrate how to obtain classicalconstructions on scalar valued modular forms in terms of the operators T N . For 0 < N ∈ (cid:90) , we set ρ N = Ind Γ ( N ) , and for a Dirichlet character χ mod N , we set ρ χ = Ind Γ ( N ) χ . In particular, if χ is thetrivial character mod N , then ρ χ = ρ N .As a first step, we identify T M with a sum of ρ N ’s for suitable N . Lemma 2.11.
For any positive integer M , we haveT M ∼= (cid:77) a | M Ind Γ ( M / a ) = (cid:77) a | M ρ M / a . (2.13) Proof.
By definition, the representation T M is given by γ e m = e m γ − , m ∈ ∆ M . (2.14)To decompose the permutation representation T M into irreducible components, it suffices to deter-mine the orbits of SL ( (cid:90) ) acting on SL ( (cid:90) ) ∆ M from the right. This is achieved by classical Hecke theory,which says that SL ( (cid:90) ) ∆ M SL ( (cid:90) ) = (cid:91) a ; a | M SL ( (cid:90) ) (cid:161) M / a a (cid:162) SL ( (cid:90) )is a disjoint union of orbits. It thus suffices to determine the stabilizer of SL ( (cid:90) ) (cid:161) M / a a (cid:162) for each a .Given a positive integer N = M / a , the stabilizer of SL ( (cid:90) )diag( N , 1) equals Γ ( N ), where diag( N , 1)is the matrix (cid:161) N
00 1 (cid:162) . Indeed, a direct computation shows that the right action of Γ ( N ) preserves it. Onthe other hand, it is known by Hecke theory that the SL ( (cid:90) )-double coset generated by diag( N , 1) hasthe same cardinality as the projective line over (cid:90) / N (cid:90) . This shows that the index of the stabilizer inSL ( (cid:90) ) equals the index of Γ ( N ), completing our argument.The remaining section builds up on the following purely representation theoretic computation.We set m M = diag( M , 1), m (cid:48) M = diag(1, M ), and m M , b = (cid:161) M b M (cid:162) . Given M and an exact divisor N of M , we write γ M , N ∈ SL ( (cid:90) ) for some fixed matrix that satisfies γ M , N ≡ (cid:161) − (cid:162) (mod M ) and γ M , N ≡ (cid:161) (cid:162) (mod M / N ). To simplify notation, we assume that γ M , N is one of the fixed representatives of Γ ( N )\SL ( (cid:90) ). Further, set e M ( x ) = exp(2 π i x / M ) for x ∈ (cid:67) , and let G ( (cid:178) , e M ( b )) = (cid:80) a (mod M ) (cid:178) ( a ) e M ( ab )be the Gauss sum attached to a Dirichlet character (cid:178) modulo M . For χ a Dirichlet character mod N ,we define a (monoid) character on the set of matrices m = (cid:161) a bc d (cid:162) with N | c by χ ( m ) = χ ( d ). – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum Proposition 2.12.
Let χ be a Dirichlet character mod N . For a positive integer M , the maps ι Hecke : ρ χ −→ T M ρ χ , e γ (cid:55)−→ M k − (cid:88) m ∈ ∆ M χ ( m ) χ (cid:161) I I ( I m ( γ )) (cid:162) e I m ( γ ) ⊗ e m γ , (2.15) ι AL : ρ χ (cid:48) −→ T M ρ χ , e γ (cid:55)−→ χ (cid:161) I I ( γ M , N I m M ( γ )) (cid:162) e γ M , N I mM ( γ ) ⊗ e m M γ , (2.16) ι old : ρ χ (cid:48) −→ T M ρ χ , e γ (cid:55)−→ M − k χ (cid:161) I I ( I m M ( γ )) (cid:162) e I mM ( γ ) ⊗ e m M γ , (2.17) ι Γ ( N ) : ρ Γ ( N ) −→ T N ρ Γ ( N , N ) , e γ (cid:55)−→ M k e I m (cid:48) N ( γ ) ⊗ e m (cid:48) N γ , (2.18) ι twist : ρ χ (cid:48) −→ T M ρ χ , e γ (cid:55)−→ M − (cid:88) b (mod M ) G ( (cid:178) , e M ( − b )) χ (cid:161) I I ( I m M , b ( γ )) (cid:162) e I mM , b ( γ ) ⊗ e m M , b γ , (2.19) ι id : ρ χ −→ T M ρ χ , e γ (cid:55)−→ e γ ⊗ e diag( M , M ) (2.20) are inclusions under the following circumstances:(i) The map (2.15) is an inclusion if M is coprime to N .(ii) The map (2.16) is an inclusion if M | N and M is coprime to N / M , and χ (cid:48) is the Dirichlet charactermod N that equals χ mod N / M and that equals χ mod M .(iii) The map (2.17) is an inclusion for χ (cid:48) the mod M N Dirichlet character defined by χ .(iv) The map (2.18) is always an inclusion.(v) The map (2.19) is an inclusion if (cid:178) is a Dirichlet character mod M , and χ (cid:48) the mod N M Dirichletcharacter defined by χ(cid:178) .(vi) The map (2.20) is always an inclusion.Remark 2.13. Using the scalar product on induced representations and their images under vector-valued Hecke operators (cf. Lemma 2.4), we obtain from each inclusion defined in the previous propo-sition a corresponding projection. More specifically, we define a projection π associated to an inclu-sion by requiring that 〈 π ( v ), w 〉 = 〈 v , ι ( w ) 〉 for all vectors v and w in the domain of π and ι , respectively.We denote the resulting projections by π Hecke : T M ρ χ −→ ρ χ , π AL : T M ρ χ −→ ρ χ , π old : T M ρ χ −→ ρ χ (cid:48) , π Γ ( N ) : T N ρ Γ ( N , N ) −→ ρ Γ ( N ) , π twist : T M ρ χ −→ ρ χ (cid:48) , π id : T M ρ χ −→ ρ χ . (2.21) Proof of Proposition 2.12.
There are at least two possible proofs of Proposition 2.12. We will prove thefirst two cases by direct computation to illustrate various cocycle relations that enter the calculations.Then we give a more general, representation theoretic argument. Given γ ∈ Γ ( N )\SL ( (cid:90) ) and δ ∈ SL ( (cid:90) ), we will check by computing the left and right hand side separately that ι (cid:161) ρ ( δ − ) e γ (cid:162) = ρ (cid:48) ( δ − ) ι ( e γ ) ,where ρ is the repesentation on the domain of (2.15), (2.16), etc., ρ (cid:48) is the corresponding representationon the codomain, and ι is the inclusion of ρ into ρ (cid:48) . – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum We verify that (2.15) is an inclusion if M and N are coprime. Applying the definition of inducedrepresentations ρ χ and of ι Hecke , we find that M − k ι Hecke (cid:161) ρ χ ( δ − ) e γ (cid:162) = M − k ι Hecke (cid:179) χ (cid:161) I − γ ( δ ) (cid:162) e γδ (cid:180) = χ (cid:161) I − γ ( δ ) (cid:162) (cid:88) m ∈ ∆ M χ ( m ) χ (cid:161) I I ( I m ( γδ )) (cid:162) e I m ( γδ ) ⊗ e m γδ = (cid:88) m ∈ ∆ M χ ( m ) χ (cid:161) I γ ( δ ) I − I ( I m ( γδ )) (cid:162) e I m ( γδ ) ⊗ e m γδ = (cid:88) m (cid:48) ∈ ∆ M χ ( m (cid:48) ) χ (cid:161) I I ( I m (cid:48) I γ ( δ ) ( γδ )) (cid:162) e I m (cid:48) I γ ( δ ) ( γδ ) ⊗ e m (cid:48) I γ ( δ ) γδ .To obtain the last equality, we have replaced m with m (cid:48) I γ ( δ ).On the other hand, we have M − k ( T M ρ χ )( δ − ) (cid:161) ι Hecke ( e γ ) (cid:162) = ( T M ρ χ )( δ − ) (cid:179) (cid:88) m ∈ ∆ M χ ( m ) χ (cid:161) mI I ( I m ( γ )) (cid:162) e I m ( γ ) ⊗ e m γ (cid:180) = (cid:88) m ∈ ∆ M χ ( m ) χ (cid:161) mI I ( I m ( γ )) (cid:162) (cid:161) ρ χ (cid:161) I − m γ ( δ ) (cid:162) e I m ( γ ) (cid:162) ⊗ e m γδ = (cid:88) m ∈ ∆ M χ ( m ) χ (cid:161) mI I ( I m ( γ )) (cid:162) χ (cid:161) I I m ( γ ) ( I m γ ( δ )) (cid:162) e I m ( γ ) I m γ ( δ ) ⊗ e m γδ = (cid:88) m ∈ ∆ M χ ( m ) χ (cid:161) mI I ( I m ( γδ )) (cid:162) e I m ( γδ ) ⊗ e m γδ .To obtain the last equality we apply the cocycle relation to the subscript of the first tensor component,and the relation I I (cid:161) I m ( γ ) (cid:162) I I m ( γ ) (cid:161) I m γ ( δ ) (cid:162) = I I (cid:161) I m ( γ ) I m γ ( δ ) (cid:162) = I I (cid:161) I m ( γδ ) (cid:162) to the argument of the character.To finish the case of (2.15), we only need to show equality between the arguments of the character χ and the subscripts of the tensor components. That is, we have to establish that I mI γ ( δ ) ( γδ ) = I m ( γδ ) and mI γ ( δ ) γδ = m γδ .The second equality is a simple consequence of the defining formula of I γ ( δ ). The first equality followsfrom the second one and from I m ( γδ ) mI γ ( δ ) γδ = I m ( γδ ) m γδ = m γδ .To establish the case of (2.16), consider the following equality, baring in mind that γ = γ .( T M ρ χ )( δ − ) (cid:161) ι AL ( e γ ) (cid:162) = ( T M ρ χ )( δ − ) (cid:179) χ (cid:161) I I ( γ M , N I m M ( γ )) (cid:162) e γ M , N I mM ( γ ) ⊗ e m M γ (cid:180) = χ (cid:161) I I (cid:161) γ M , N I m M ( γ ) (cid:162)(cid:162) (cid:161) ρ χ (cid:161) I − m M γ ( δ ) (cid:162) e γ M , N I mM ( γ ) ⊗ e m M γδ = χ (cid:161) I I (cid:161) γ M , N I m M ( γ ) (cid:162)(cid:162) χ (cid:161) I γ M , N I mM ( γ ) (cid:161) I m M γ ( δ ) (cid:162)(cid:162) e γ M , N I mM ( γ ) I mM γ ( δ ) ⊗ e m M γδ = χ (cid:161) I I (cid:161) γ M , N I m M ( γδ ) (cid:162)(cid:162) e γ M , N I mM ( γδ ) ⊗ e m M γδ .The argument of the character simplifies, because I I (cid:161) γ M , N I m M ( γ ) (cid:162) I γ M , N I mM ( γ ) (cid:161) I m M γ ( δ ) (cid:162) = I I (cid:161) γ M , N I m M ( γ ) I m M γ ( δ ) (cid:162) = I I (cid:161) γ M , N I m M ( γδ ) (cid:162) – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum by applying the cocycle relation twice.Comparing that expression to ι AL (cid:161) ρ χ (cid:48) ( δ − ) e γ (cid:162) = ι AL (cid:161) χ (cid:48) (cid:161) I γ ( δ ) (cid:162) e γδ (cid:162) = χ (cid:48) (cid:161) I γ ( δ ) (cid:162) χ (cid:161) I I (cid:161) γ M , N I m M ( γδ ) (cid:162)(cid:162) e γ M , N I mM ( γδ ) ⊗ e m M γδ we need to show that χ (cid:179) I I (cid:161) γ M , N I m M ( γδ ) (cid:162)(cid:180) = χ (cid:48) (cid:161) I γ ( δ ) (cid:162) χ (cid:179) I I (cid:161) γ M , N I m M ( γδ ) (cid:162)(cid:180) and m M γδ = m M γδ . (2.22)Since γδ = I γ ( δ ) γδ with I γ ( δ ) ∈ Γ ( N ), we immediately obtain the second equality: Indeed, by what isargued in the proof of Proposition 2.12, Γ ( N ) stabilizes m M , since we have M | N .It is more involved to illustrate the first auxiliary equality. The cocylce equality and mI γ ( δ ) = m imply that I m M ( γδ ) = I m M I γ ( δ ) ( γδ ) = I m M (cid:161) I γ ( δ ) (cid:162) I m M ( γδ ) .We therefore have to analyze χ (cid:179) I I (cid:161) γ M , N I m M ( γδ ) (cid:162)(cid:180) = χ (cid:179) I I (cid:161) γ M , N I m M (cid:161) I γ ( δ ) (cid:162) I m M ( γδ ) (cid:162)(cid:180) .Since we are interested only in a value of χ , which is a Dirichlet character mod N , it suffices to con-sider its argument mod N . Additionally, because I I is a cocycle for the action of SL ( (cid:90) ) on the cosets Γ ( N )\SL ( (cid:90) ), its value mod N depends only on its argument mod N . Therefore, all the following com-putations can be performed modulo N . Since further M and N / M are coprime, it indeed suffices tocompute mod M and mod N / M separately.We first consider values mod N / M , which is easier since γ M , N ≡ I (mod N / M ). We have I I (cid:161) γ M , N I m M (cid:161) I γ ( δ ) (cid:162) I m M ( γδ ) (cid:162) ≡ I I (cid:161) I m M (cid:161) I γ ( δ ) (cid:162) I m M ( γδ ) (cid:162) ≡ I m M (cid:161) I γ ( δ ) (cid:162) I I (cid:161) I m M ( γδ ) (cid:162) .The last congruence deserves further justification. However, we first finish the computation mod-ulo N / M . Namely, we have to compare χ (cid:161) I m M (cid:161) I γ ( δ ) (cid:162)(cid:162) and χ (cid:161) I γ ( δ ) (cid:162) . We have already noticed that I γ ( δ )stabilizes m M . Therefore I m M (cid:161) I γ ( δ ) (cid:162) = m M I γ ( δ ) m − M . The diagonal entries of m M I γ ( δ ) m − M coincideswith the ones of I γ ( δ ). Since χ and χ (cid:48) coincide mod N / M , this finishes the mod N / M computationsfor (2.22).Let us now argue that the last equality in the above equation holds. From the previously statedexpression for I m M (cid:161) I γ ( δ ) (cid:162) we see that it lies in Γ ( N / M ). Among other things, this implies that I m M (cid:161) I γ ( δ ) (cid:162) I m M ( γδ ) = I m M ( γδ ).By definition of the cocyles, we have I I (cid:161) I m M (cid:161) I γ ( δ ) (cid:162) I m M ( γδ ) (cid:162) I m M ( γδ ) ≡ I I (cid:161) I m M (cid:161) I γ ( δ ) (cid:162) I m M ( γδ ) (cid:162) I − m M (cid:161) I γ ( δ ) (cid:162) I m M ( γδ ) ≡ I m M (cid:161) I γ ( δ ) (cid:162) I m M ( γδ )and I I (cid:161) I m M ( γδ ) (cid:162) I m M ( γδ ) ≡ I m M ( γδ ).Combining both equalities, we obtain I m M (cid:161) I γ ( δ ) (cid:162) I I (cid:161) I m M (cid:161) I γ ( δ ) (cid:162) I m M ( γδ ) (cid:162) ≡ I I (cid:161) I m M ( γδ ) (cid:162) , – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum which yields precisely the relation that we have employed above. This finishes our computationsmod N / M .Computations mod M depends on γ M , N ≡ (cid:161) −
11 0 (cid:162) , γ M , N I m M (cid:161) I γ ( δ ) (cid:162) ∈ Γ ( M ), and on χ (cid:48) being theinverse of χ mod M . The rest of the considerations is the same as before. We leave details to thereader, and finish discussion of (2.16).The remaining cases (2.17), (2.18), (2.19), and (2.20) follow a similar pattern, but in particular (2.19)would be very involved to establish. We therefore give an alternative proof, which is more conceptual,but allows for less insight into what transformations show up.Consider the case (2.19), and fix χ and χ (cid:48) as in the assumptions. For any k ∈ (cid:90) and any f ∈ M k ( χ ),we have, as in the proof of Proposition 2.19 thattwist (cid:178) ( f ) = M − (cid:88) b G ( (cid:178) , e M ( b )) f | k m M , b ∈ M k ( χ (cid:48) ).Comparing with the definition of induction of represenations in (1.2), the spacespan (cid:169) twist (cid:178) ( f ) | k γ : γ ∈ SL ( (cid:90) ) (cid:170) with left representation γ g = g | k γ − is isomorphic to Ind ρ χ (cid:48) , since f ∈ M k ( χ (cid:48) ). As a space of functions,it coincides with span (cid:169)(cid:161) M − (cid:88) b G ( (cid:178) , e M ( b )) f | k m M , b (cid:162) | k γ : γ ∈ SL ( (cid:90) ) (cid:170) .The latter is a subspace of span (cid:169) f | k γ m : γ ∈ SL ( (cid:90) ), m ∈ ∆ M (cid:170) ,which is isomorphic to T M ρ χ when considered as a space with left representation γ g = g | k γ − . Theembedding is given via the conjugate of (2.19), which proves the proposition. §2.3.1 The identity map. For later use, we have to recover f from T M f . Lemma 2.14.
Let χ be a Dirichlet character mod N . Fix a positive integer M . The inclusion ι id and thecorresponding projection π id intertwine the vector valued Hecke operator T M and the identity map withinduction from Γ ( N ) . For every f ∈ M k ( χ ) and v ∈ V ( ρ χ ) , we have 〈 Ind( f ), v 〉 = (cid:68) π id (cid:161) T M Ind( f ) (cid:162) , v (cid:69) = (cid:68) T M Ind( f ), ι id ( v ) (cid:69) .Proof. In light of Proposition 2.12 and the definition of π id in terms of ι id , it suffices to check that theleft and right hand side agree for v = e I . We have 〈 Ind( f ), e I 〉 = f and 〈 T M Ind( f ), ι id ( e I ) 〉 = 〈 T M Ind( f ), e I ⊗ e diag( M , M ) 〉 = f | k diag( M , M ) = f . – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum §2.3.2 Classical Hecke operators. One obviously wants to recover classical Hecke operators fromDefinition 2.6. Given f ∈ M k ( χ ), let f | T M be its image under the classical Hecke operator: (cid:161) f | k , χ T M (cid:162) ( τ ) = M k − (cid:88) d | M ≤ b < d d − k χ ( d ) f (cid:179) Md − τ + bd (cid:180) . Proposition 2.15.
Let χ be a Dirichlet character mod N . Fix a positive integer M that is coprime toN . The inclusion ι Hecke and the corresponding projection π Hecke intertwine the vector valued Heckeoperator and the classical Hecke operator with the induction map. For every f ∈ M k ( χ ) and v ∈ V ( ρ χ ) ,we have 〈 Ind (cid:161) f | k , χ T M (cid:162) , v 〉 = (cid:68) π Hecke (cid:161) T M Ind( f ) (cid:162) , v (cid:69) = (cid:68) T M Ind( f ), ι Hecke ( v ) (cid:69) .Proof. As in the proof of Lemma 2.14, we can focus on the case of v = e I . The left hand side thenequals f | k , χ T M , and the right hand side is 〈 T M Ind( f ), ι Hecke ( e I ) 〉 = (cid:68) (cid:88) m , γ (cid:161) f | k γ m (cid:162) e γ ⊗ e m , M k − (cid:88) m ∈ ∆ M χ (cid:161) m (cid:162) e I ⊗ e m (cid:69) = M k − (cid:88) m χ ( m ) f | k m = (cid:68) (cid:88) γ (cid:161) f | k , χ T M γ (cid:162) e γ , e I (cid:69) . §2.3.3 Atkin–Lehner involutions. Atkin–Lehner involutions for level N modular forms, which are de-fined under the assumptions of the next Proposition 2.16, map f ∈ M k ( N ) to W M ( f ) = f | k γ M , N m M . Proposition 2.16.
Let χ be a Dirichlet character mod N . Fix a positive integer M | N such that M andN / M are coprime. Let χ (cid:48) be the Dirichlet character mod N that equals χ mod N / M and that equals χ mod M . The inclusion ι AL and the corresponding projection π AL intertwine the Atkin Lehner operatorand Hecke operator with the induction map. For f ∈ M k ( Γ ( N ), χ ) and v ∈ V ( ρ χ ) , we have 〈 Ind Γ ( N ) W M ( f ), v 〉 = (cid:68) π AL (cid:161) T M Ind( f ) (cid:162) , v (cid:69) = (cid:68) T M Ind( f ), ι AL ( v ) (cid:69) .Proof. In analogy with the proof of Proposition 2.15 it suffices to note that for v = e I the right handside equals 〈 T M Ind( f ), χ (cid:48) ( γ M , N ) e γ M , N ⊗ e m M 〉 = χ (cid:48) (cid:161) I I ( γ M , N ) (cid:162) f | k γ M , N m M = f | k γ M , N m M . §2.3.4 Oldforms. Oldforms in the scalar valued setting are obtained as f ( M τ ) for a given modularform f . We recover this construction using vector valued Hecke operators. For convenience let (cid:161) sc M / M (cid:48) f (cid:162) ( τ ) = f (cid:161) M τ / M (cid:48) (cid:162) (2.23)be the map that rescales the argument of a function. Clearly, (cid:161) sc M f (cid:162) ( τ ) = M − k f | k (cid:161) M (cid:162) is the oldformconstruction. – – roducts of Vector Valued Eisenstein Series — 2 Vector Valued Hecke Operators M. Westerholt-Raum Proposition 2.17.
Let χ be a Dirichlet character mod N . Fix a positive integer M , and let χ M be themod M N Dirichlet character defined by χ . The inclusion ι old and the corresponding projection π old intertwine the oldform construction and the Hecke operator with the induction map. For f ∈ M k ( χ ) andv ∈ V ( ρ χ M ) , we have 〈 Ind (cid:161) sc M f (cid:162) , v 〉 = (cid:68) π old (cid:161) T M Ind( f ) (cid:162) , v (cid:69) = (cid:68) T M Ind( f ), ι old ( v ) (cid:69) .Proof. As in the other cases, we inspect the right hand side for v = e I . 〈 T M Ind( f ), M − k e I ⊗ e m M 〉 = M − k f | k e m M . §2.3.5 Rescaling for principal congruence subgroups. Like the oldform construction, it is very com-mon to consider f ( N τ ) for f a modular form for the principal congruence subgroup Γ ( N ). The re-sulting f ( N τ ) is a modular form for Γ ( N , N ). For technical reasons, we consider the reverse map. Proposition 2.18.
Let N be a positive integer. The inclusion ι Γ ( N ) and the projection π Γ ( N ) correspond-ing to it intertwine the map sc N from M k ( Γ ( N , N )) to M k ( Γ ( N )) . For f ∈ M k ( Γ ( N , N )) and v ∈ V ( ρ Γ ( N ) ) , we have 〈 Ind (cid:161) sc N f (cid:162) , v 〉 = (cid:68) π Γ ( N ) (cid:161) T N Ind( f ) (cid:162) , v (cid:69) = (cid:68) T N Ind( f ), ι Γ ( N ) ( v ) (cid:69) .Proof. We inspect the right hand side for v = e I . 〈 T N Ind( f ), N k e I ⊗ e m (cid:48) N 〉 = N k f | k e m (cid:48) N §2.3.6 Twists of modular forms. We show that also twists of modular forms by Dirichlet characterscan be recovered from vector valued Hecke operators. For a modular form f ∈ M k ( χ ) and a Dirich-let character (cid:178) mod M , there is a twist f (cid:178) = twist (cid:178) ( f ) ∈ M k ( χ (cid:48) ) with Fourier coefficients c ( f (cid:178) ; n ) = (cid:178) ( n ) c ( f ; n ), where χ (cid:48) is the mod N M Dirichlet character defined by χ(cid:178) . One can reconstruct f (cid:178) from T M f . Note that the modular form sc M twist (cid:178) f already appears as a component of T M Ind( f ), sinceit can be written by (2.24) below assc M f (cid:178) = sc M (cid:161) M − (cid:88) b (mod M ) G ( (cid:178) , e M ( − b )) f | k m M , b (cid:162) = M k − (cid:88) b (mod M ) G ( (cid:178) , e M ( − b )) f | k m M , b (cid:161) M (cid:162) = M k − (cid:88) b (mod M ) G ( (cid:178) , e M ( − b )) f | k (cid:161) b M (cid:162) . Proposition 2.19.
Let χ be a Dirichlet character mod N . Fix another Dirichlet character (cid:178) mod M , andlet χ (cid:48) be as above. The inclusion ι twist and the corresponding projection π twist intertwine twist (cid:178) and theHecke operator with the induction map. For f ∈ M k ( χ ) and v ∈ V ( ρ χ ) , we have 〈 Ind (cid:161) twist (cid:178) f (cid:162) , v 〉 = (cid:68) π twist (cid:161) T M Ind( f ) (cid:162) , v (cid:69) = (cid:68) T M Ind( f ), ι twist ( v ) (cid:69) .Proof. We have q n | k m M , b = e M ( nb ) q n . This allows us to write twists of a modular form f ∈ M k ( χ ) as f (cid:178) ( τ ) = (cid:88) n (cid:178) ( n ) c ( n ) q n = (cid:88) n (mod M ) (cid:178) ( n ) M − (cid:88) b (mod M ) e M ( − n b ) f | k m M , b = M − (cid:88) b (mod M ) G ( (cid:178) , e M ( − b )) f | k m M , b . (2.24)With this at hand, inspecting the right hand side of the proposition’s statement yields a proof. – – roducts of Vector Valued Eisenstein Series — 3 Vector Valued Eisenstein Series M. Westerholt-Raum Recall that we assume throughout that ρ has finite index kernel. Fix an even integer k > v ∈ V ( ρ ).We define the stabilizer Stab( v ) of v as { γ ∈ SL ( (cid:90) ) : ρ ( γ ) v = v (cid:170) . Write Γ ∞ ( v ) for the intersection of Γ ∞ and Stab( v ). Observe that Γ ∞ ( v ) has finite index in Γ ∞ , since ρ has finite index kernel. The series E k , v ( τ ) : = (cid:163) Γ ∞ : Γ ∞ ( v ) (cid:164) (cid:88) γ ∈ Γ ∞ ( v )\SL ( (cid:90) ) v | k , ρ γ (3.1)is well-defined, converges, and defines a modular form of weight k and type ρ . We define the Eisen-stein subspace of M k ( ρ ) as their span.E k ( ρ ) = span (cid:169) E k , v : v ∈ V ( ρ ) (cid:170) . (3.2)In the case of weight 2, the Hecke trick leads us to the definition E v ( τ ) : = (cid:163) Γ ∞ : Γ ∞ ( v ) (cid:164) lim s → (cid:88) γ ∈ Γ ∞ ( v )\SL ( (cid:90) ) y s v | ρ γ , (3.3)where y = Im τ > Lemma 3.1.
Assume that ρ has finite index kernel. The Eisenstein series E v is holomorphic if ρ doesnot contain the trivial representation. We have M ( ) = {0} for the trivial representation of SL ( (cid:90) ) .Proof. We use the ξ operator, first defined in [BF04], to prove the first part. It is defined by ξ ( E v ) : = i y ∂ τ E v ∈ M ( ρ ) .By the intertwining property of ξ , we find that ξ ( E v ) is a vector valued modular form of weight 0and type ρ . Assume that ρ does not contain . Then neither does ρ . Lemma 3.2 therefore impliesthat ξ ( E v ) vanishes. Since the kernel of ξ applied to C ∞ ( (cid:72) ) consists of holomorphic functions, thisproves that E v is holomorphic. The second part is a classical fact. Lemma 3.2.
Suppose that ρ is a representation with finite index kernel. If ρ does not contain the trivialrepresentation, then M ( ρ ) = {0} .Proof. Without loss of generality, we may assume that ρ is irreducible and not equal to the trivial rep-resentation. Suppose that there is 0 (cid:54)= f ∈ M ( ρ ). It is a classical fact that weight 0 modular forms forfinite index subgroups of SL ( (cid:90) ) are constant. In particular, we can view f as a nonzero vector in V ( ρ ),which is invariant under the action of SL ( (cid:90) ), contradicting irreducibility of ρ .Lemma 3.1 inspires the definitionE ( ρ ) = (cid:169) E v : v ∈ V ( ρ (cid:48) ) (cid:170) , where ρ = ρ ( ) ⊕ ρ (cid:48) .The matrix T = (cid:161) (cid:162) acts on the upper half space by translations τ (cid:55)→ τ +
1. In the context of vectorvalued modular forms, T -eigenspaces of a representation ρ encode information about possible expo-nents in the Fourier expansion. We describe vector valued Eisenstein series using this information.Write V ( ρ )(1) T : = (cid:169) v ∈ V ( ρ ) : ρ ( T ) v = v (cid:170) (3.4)for the isotrivial component of the restriction of ρ to the subgroup of SL ( (cid:90) ) generated by T . – – roducts of Vector Valued Eisenstein Series — 3 Vector Valued Eisenstein Series M. Westerholt-Raum Proposition 3.3.
We have, for k ≥ , E k ( ρ ) ∼= V ( ρ )(1) T and E ( ρ ) ∼= V ( ρ (cid:48) )(1) T ,where ρ = ρ ( ) ⊕ ρ (cid:48) as above.Proof. We have Γ ∞ ( v ) = Γ ∞ ( N ) : = Γ ∞ ∩ Γ ( N ) for some N . Summation over Γ ∞ ( N )\ Γ ∞ in the definitionof Eisenstein series corresponds to the projection onto V ( ρ )(1) T ⊆ V ( ρ ), where V ( ρ ) is decomposedwith respect to eigenvalues of T . §3.1 Hecke operators acting on Eisenstein series. Our proof of the Main Theorem requires the fol-lowing statement.
Proposition 3.4.
Let k ≥ be an even integer and ρ a representation of SL ( (cid:90) ) with finite index kernel.For every positive integer M , we have T M E k ( ρ ) ⊆ E k ( T M ρ ) .Proof. To reduce technicalities, we focus on the case k >
2, so that Eisenstein series can be definedwithout the Hecke trick. It suffices to consider Eisenstein series attached to v ∈ V ( ρ )(1) T . Fix such v ,set T M v = (cid:80) m ∈ ∆ M d ( m ) − k v ⊗ e m , and consider the Eisenstein series E k , T M v = M (cid:88) m ∈ ∆ M d ( m ) − k (cid:88) γ ∈ Γ ∞ ( M )\SL ( (cid:90) ) v ⊗ e m | k , T M ρ γ .Its m (cid:48) -th component is equal to M − k − (cid:88) m ∈ ∆ M γ ∈ Γ ∞ ( M )\SL ( (cid:90) ) m γ = γ (cid:48) m (cid:48) (cid:161) v | k m ⊗ e m (cid:162) | k , T M ρ γ = M − k − (cid:88) m ∈ ∆ M γ ∈ Γ ∞ ( M )\SL ( (cid:90) ) m γ = γ (cid:48) m (cid:48) (cid:161) ρ − ( γ (cid:48) ) v | k γ (cid:48) m (cid:48) (cid:162) ⊗ e m (cid:48) .For every γ (cid:48) ∈ Γ ∞ ( M )\SL ( (cid:90) ), we find unique m and γ such that m γ = γ (cid:48) m (cid:48) . Therefore, the abovesimplifies in the following way: M − k − (cid:88) γ (cid:48) ∈ Γ ∞ ( M )\SL ( (cid:90) ) (cid:161) v | k , ρ γ (cid:48) (cid:162) | k m (cid:48) ⊗ e m (cid:48) ,which equals the m (cid:48) -component of M − k T M E k , v . Since m (cid:48) ∈ ∆ M was arbitrary, we have shown that T M E k , v = M k E k , T M v ,which implies the proposition. §3.2 Two particular Eisenstein series. In the proof the main theorem, we need two particular Eisen-stein series which we introduce now. For w ∈ (cid:67) and two Dirichlet characters δ and (cid:178) , we set σ w , δ , (cid:178) ( n ) = (cid:88) < d | n δ ( d ) (cid:178) ( n / d ) d w . (3.5)The trivial Dirichlet character mod N will be denoted by N . If N =
1, then we instead write .To state the next two lemmas, recall from Section 2.3 our notations ρ N = Ind Γ ( N ) and ρ χ = Ind Γ ( N ) χ for a Dirichlet character χ mod N . – – roducts of Vector Valued Eisenstein Series — 3 Vector Valued Eisenstein Series M. Westerholt-Raum Lemma 3.5.
Given an even integer k ≥ and an even Dirichlet characters χ mod N > , there is anEisenstein series in E k ( ρ χ ) whose Γ ( N ) -th component (i.e. the component that corresponds to the trivialcoset in Γ ( N )/SL ( (cid:90) ) ) has Fourier expansion ∞ (cid:88) n = σ k − χ , ( n ) q n .Proof. Miyake [Miy89] in his Theorem 7.1.3 computes the Fourier expansions of an Eisenstein series(which he denotes by E k ( z ; χ , )), which is a modular form for the character χ by Lemma 7.1.1. Apply-ing the induction map (1.3) to Miyake’s construction yields exactly the searched for Eisenstein series. Lemma 3.6.
Fix an even integer k ≥ and an even Dirichlet character χ mod N . There is an adjointpair of inclusion and projection ι diag, χ : ρ N −→ ρ χ ⊗ ρ χ , e γ (cid:55)−→ e γ ⊗ e γ (3.6) π diag, χ : ρ χ ⊗ ρ χ −→ ρ N , e γ ⊗ e γ (cid:48) (cid:55)−→ (cid:40) e γ , if γ = γ (cid:48) ; , otherwise. (3.7) Let v = e Γ ( N ) ⊗ e Γ ( N ) ∈ V (cid:161) ρ χ ⊗ ρ χ (cid:162) . Then the Eisenstein series E k , χ , ∞ : = E k , v satisfiesE k , χ , ∞ ∈ ι diag, χ (cid:161) (cid:88) M | N T M E k ( ) (cid:162) .Proof. It is a straight forward computation to verify that ι diag, χ is an inclusion, and that it is adjoint to π diag, χ . To establish the remaining part of the lemma, is suffices to show that the Eisenstein series E k , v (cid:48) for v (cid:48) = e Γ ( N ) ∈ V ( ρ N ) lies in (cid:80) M | N T M E k ( ). Observe that E k , v (cid:48) corresponds, under the map (1.3), toan Eisenstein series that vanishes at every cusp but ∞ . Such a series is an oldform coming from level 1,by classical theory. Apply Proposition 2.17, which says that oldforms can be constructed using vectorvalued Hecke operators, to finish the proof. §3.3 A pairing of modular forms. Let ρ and ρ E be two representations of SL ( (cid:90) ) with finite indexkernel. For even l satisfying 2 ≤ l ≤ k −
2, we define F : E k − l ( ρ E ) ⊗ V ( ρ ⊗ ρ E ) ∨ −→ Hom (cid:161) S k ( ρ ), (cid:67) (cid:162) , E ⊗ v (cid:55)−→ (cid:161) f (cid:55)→ 〈 f , E ⊗ E l , v 〉 ι (cid:162) ,where the scalar product is taken with respect to the canonical inclusion ι of the trivial representationinto ρ ⊗ ρ E ⊗ (cid:161) ρ ⊗ ρ E (cid:162) ∨ = ρ ⊗ ρ E ⊗ (cid:161) ρ ⊗ ρ E (cid:162) ∨ . Proposition 3.7.
For k ≥ , ≤ l ≤ k − , and two SL ( (cid:90) ) -representations ρ and ρ E , let E ∈ E k − l ( ρ E ) ,f ∈ S k ( ρ ) , and v ∈ V ( ρ ⊗ ρ E ) ∨ (1) T . Under these assumptions we haveF ( E ⊗ v )( f ) = Γ ( k − π ) k − (cid:88) ≤ n ∈ (cid:81) n − k v (cid:161) c ( f ; n ) ⊗ c ( E ; n ) (cid:162) , (3.8) where v is the complex conjugate of v. – – roducts of Vector Valued Eisenstein Series — 4 Products of Eisenstein series M. Westerholt-Raum Proof.
For simplicity, we assume that l , k − l >
2, so that we need not apply the Hecke trick. The readerreadily verifies that all arguments remain valid after introducing auxiliary variables s and s that tendto 0 and factors y s , y s .Set σ = ( ρ ⊗ ρ E ) ∨ . We unfold the Petersson scalar product to obtain an explicit formula for 〈 f , E ⊗ E l , v 〉 ι : (cid:90) SL ( (cid:90) )\ (cid:72) (cid:68)(cid:161) f ⊗ ( E ⊗ E l , v ) (cid:162) ( τ ), ι (1) (cid:69) dx dyy − k = (cid:90) SL ( (cid:90) )\ (cid:72) (cid:68)(cid:179) f ⊗ (cid:161) E ⊗ (cid:88) γ v | l , σ γ (cid:162)(cid:180) ( τ ), ι (1) (cid:69) dx dyy − k = (cid:90) SL ( (cid:90) )\ (cid:72) (cid:88) γ (cid:161) v | l , σ γ (cid:162)(cid:161) f ⊗ E (cid:162) ( τ ) dx dyy − k = (cid:90) SL ( (cid:90) )\ (cid:72) (cid:88) γ (cid:161) v | l , σ γ (cid:162)(cid:161) f ⊗ E | k − l , ρ ⊗ ρ E γ (cid:162) ( τ ) dx dyy − k = (cid:90) SL ( (cid:90) )\ (cid:72) (cid:88) γ (cid:179) v (cid:161) f ⊗ E (cid:162) ( τ ) (cid:180)(cid:175)(cid:175)(cid:175) k γ dx dyy − k = (cid:90) Γ ∞ \ (cid:72) v (cid:179)(cid:161) f ⊗ E (cid:162) ( τ ) (cid:180) dx dyy − k .Since f is a cusp form, all the displayed integrals converge absolutely. In the first equality we haveinserted the definition of E l , v . The second equality holds by virtue of the definition of ι . More precisely,we have ι (1) = (cid:80) w w ⊗ w ∨ where w runs through an orthonormal basis of V ( ρ ⊗ ρ E ) and w ∨ denotesa corresponding dual basis element. For the third equality, we have employed modular invariance of f ⊗ E . The fourth equality follows from ( σ ( γ ) v )( σ ∨ ( γ ) w ) = v ( w ), which is true for all w ∈ V ( σ ) ∨ . Thefinal equality is based on SL ( (cid:90) )-invariance of the measure y − dxdy .As our next step, we carry out the integral with respect to x . Expanding f ⊗ E into a Fourier series,we obtain v (cid:161) ( f ⊗ E )( τ ) (cid:162) = (cid:88) n , n (cid:48) v (cid:161) c ( f ; n ) ⊗ c ( E ; n (cid:48) ) (cid:162) exp (cid:161) π i ( n τ − n (cid:48) τ ) (cid:162) .The integral with respect to x picks up terms with n = n (cid:48) . We may interchange integration and sum-mation, since the resulting right hand side converges absolutely. We therefore obtain 〈 f , E ⊗ E l , v 〉 = (cid:90) ∞ (cid:88) n v (cid:161) c ( f ; n ) ⊗ c ( E ; n ) (cid:162) exp( − π n y ) dyy − k = Γ ( k − π ) k − (cid:88) ≤ n ∈ (cid:81) n − k v (cid:161) c ( f ; n ) ⊗ c ( E ; n ) (cid:162) . §4.1 Rankin convolutions. The next proposition generalizes Rankin’s [Ran52] statement about L -se-ries of modular forms for SL ( (cid:90) ). Proposition 4.1.
Let f ∈ M k ( χ ) be a newform with Fourier coefficients c ( n ) . Then for Dirichlet charac-ters δ and (cid:178) , and two complex variables s , w with Re s > Re w + k + , we have ∞ (cid:88) n = σ w , δ , (cid:178) ( n ) c ( n ) n − s = L ( f × (cid:178) , s ) L ( f × δ , s − w ) L ( χδ(cid:178) , 2 s − w + − k ) .Proof. The proof is completely analogous to the one in [Ran52]. By assumptions, we have for m , n ∈ (cid:90) that c ( m ) c ( n ) = (cid:88) d | ( m , n ) χ ( d ) d k − c (cid:161) mnd (cid:162) . – – roducts of Vector Valued Eisenstein Series — 4 Products of Eisenstein series M. Westerholt-Raum Expanding L ( f × (cid:178) , s ) L ( f × δ , s − w ), then applying that relation, and finally simplifying, we obtain (cid:88) m , nd | ( m , n ) χ ( d ) (cid:178) ( m ) δ ( n ) c (cid:161) mnd (cid:162) d k − ( mn ) − s n w = (cid:88) d χδ(cid:178) ( d ) d k − + w − s (cid:88) m , n (cid:178) ( m ) δ ( n ) c ( mn )( mn ) − s n w .This equals the desired factorization L ( χδ(cid:178) , 2 s − w + − k ) (cid:88) m ; n | m (cid:178) ( mn ) δ ( n ) c ( m ) m − s n w = L ( χδ(cid:178) , 2 s − w + − k ) (cid:88) m σ w , δ , (cid:178) ( m ) c ( m ) m − s . §4.2 Proof of the Main Theorem. Throughout the subsection, we fix positive, even integers k and l such that l , k − l ≥
4. Set ρ T N = T N for any N . Note that these are the same representations ρ T N thatwe referred to in the introduction. For a finite dimensional complex representation ρ of SL ( (cid:90) ) wedefine the statement Span( ρ ) asSpan( ρ ) : M k ( ρ ) = E k ( ρ ) + span < N , N (cid:48) ∈ (cid:90) φ : ρ TN ⊗ ρ TN (cid:48) → ρ φ (cid:179) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:180) . (4.1)The Main Theorem says that Span( ρ ) is true for all ρ whose kernel is a congruence subgroup. It isfurther convenient to abbreviateEE k ( ρ ) : = E k ( ρ ) + span < N , N (cid:48) ∈ (cid:90) φ : ρ TN ⊗ ρ TN (cid:48) → ρ φ (cid:179) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:180) . (4.2)Since the weight is typically clear from the context, we will often suppress the subscript, writing EE( ρ ). Lemma 4.2.
For any homomorphism of representations ψ : ρ → ρ (cid:48) and for any positive integer M wehave ψ (cid:161) EE( ρ ) (cid:162) ⊆ EE( ρ (cid:48) ) and T M (cid:161) EE( ρ ) (cid:162) ⊆ EE (cid:161) T M ρ (cid:162) .Proof. To prove the first equality observe that ψ (cid:161) EE( ρ ) (cid:162) = ψ (cid:181) E k ( ρ ) + span < N , N (cid:48) ∈ (cid:90) φ : ρ TN ⊗ ρ TN (cid:48) → ρ φ (cid:179) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:180)(cid:182) = ψ (cid:161) E k ( ρ ) (cid:162) + span < N , N (cid:48) ∈ (cid:90) φ : ρ TN ⊗ ρ TN (cid:48) → ρ ψ ◦ φ (cid:179) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:180) .It is clear from the definition of Eisenstein series (3.1) that ψ (cid:161) E k ( ρ ) (cid:162) ⊆ E k ( ρ (cid:48) ). Further, compositionwith ψ yields a map ψ ∗ : Hom (cid:161) T N ⊗ T N (cid:48) , ρ (cid:162) −→ Hom (cid:161) T N ⊗ T N (cid:48) , ρ (cid:48) (cid:162) .We thus obtain the desired inclusion into EE( ρ (cid:48) ).We now address the case of Hecke operators. The inclusion of spaces of Eisenstein series T M E k ( ρ ) ⊆ E k ( T M ρ ) is stated in Proposition 3.4. We consider the space T M (cid:179) φ (cid:161) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:162)(cid:180) . – – roducts of Vector Valued Eisenstein Series — 4 Products of Eisenstein series M. Westerholt-Raum Recall that by Proposition 2.5, for each φ ∈ Hom( T N ⊗ T N (cid:48) , ρ ) there is a homomorphism T M φ : T M (cid:161) T N ⊗ T N (cid:48) (cid:162) −→ T M ρ that satisfies T M ◦ φ = ( T M φ ) ◦ T M . Therefore, we have T M (cid:179) φ (cid:161) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:162)(cid:180) = ( T M φ ) (cid:179) T M (cid:161) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:162)(cid:180) ⊆ ( T M φ ) (cid:179) π (cid:179) T MN E l ( ) ⊗ T MN (cid:48) E k − l ( ) (cid:162)(cid:180)(cid:180) ,where the last inclusion is a direct consequence of Theorem 2.8 and π : T MN ⊗ T MN (cid:48) − (cid:16) T M (cid:161) T N ⊗ T N (cid:48) (cid:162) is the projection displayed in (2.4). Inserting this into the definitions of EE( ρ ) and EE( T M ρ ), we provethe lemma. §4.2.1 Two lemmas about Span . We now establish two lemmas that will allows us to focus on the case ρ = Ind Γ ( N ) in the actual proof of the main theorem. Lemma 4.3.
Given representations ρ and ρ (cid:48) , we have Span( ρ ) ∧ Span( ρ (cid:48) ) ⇐⇒ Span( ρ ⊕ ρ (cid:48) ) .Proof. Proposition 1.4 says that M k ( ρ ⊕ ρ (cid:48) ) = M k ( ρ ) ⊕ M k ( ρ (cid:48) ). Proposition 3.3 implies the analogue forEisenstein series: We have E k ( ρ ⊕ ρ (cid:48) ) = E k ( ρ ) ⊕ E k ( ρ (cid:48) ).For simplicity defineH( N , N (cid:48) ) = Hom (cid:161) ρ T N ⊗ ρ T N (cid:48) , ρ (cid:162) and H (cid:48) ( N , N (cid:48) ) = Hom (cid:161) ρ T N ⊗ ρ T N (cid:48) , ρ (cid:48) (cid:162) .Now, assume that Span( ρ ) and Span( ρ (cid:48) ) both are true. ThenM k ( ρ ⊕ ρ (cid:48) ) = M k ( ρ ) ⊕ M k ( ρ (cid:48) ) = (cid:181) E k ( ρ ) + span φ ∈ H( N , N (cid:48) ) φ (cid:179) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:180)(cid:182) ⊕ (cid:181) E k ( ρ (cid:48) ) + span φ (cid:48) ∈ H (cid:48) ( N , N (cid:48) ) φ (cid:48) (cid:179) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:180)(cid:182) = E k ( ρ ⊕ ρ (cid:48) ) + span φ ⊕ φ (cid:48) ∈ H( N , N (cid:48) ) ⊕ H (cid:48) ( N , N (cid:48) ) ( φ ⊕ φ (cid:48) ) (cid:179) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:180) .Since Hom (cid:161) ρ N ⊗ ρ N (cid:48) , ρ ⊕ ρ (cid:48) (cid:162) = H( N , N (cid:48) ) ⊕ H (cid:48) ( N , N (cid:48) ), this shows that Span( ρ ⊕ ρ (cid:48) ) is true.Conversely, if Span( ρ ⊕ ρ (cid:48) ) is true, then apply the canonical projections π : ρ ⊕ ρ (cid:48) → ρ and π (cid:48) : ρ ⊕ ρ (cid:48) → ρ (cid:48) to both sides of the equalityM k ( ρ ⊕ ρ (cid:48) ) = E k ( ρ ⊕ ρ (cid:48) ) + span φ ⊕ φ (cid:48) ∈ H( N , N (cid:48) ) ⊕ H (cid:48) ( N , N (cid:48) ) ( φ ⊕ φ (cid:48) ) (cid:179) T N E l ( ) ⊗ T N (cid:48) E k − l ( ) (cid:180) to find that also Span( ρ ) and Span( ρ (cid:48) ) are true. Lemma 4.4.
Fixing a congruence subgroup Γ and a character χ of Γ , suppose that M k ( Γ , χ ) = (cid:88) i ψ i (cid:161) VM i (cid:162) , VM i ⊆ M k ( Γ i , χ i ) – – roducts of Vector Valued Eisenstein Series — 4 Products of Eisenstein series M. Westerholt-Raum for a finite number of congruence subgroups Γ i , characters χ i of Γ i , and maps ψ i : M k ( Γ i , χ i ) −→ M k ( Γ , χ ) .If for every i , there is a positive integer M i and a projection π i : T M i (cid:161) Ind Γ i χ i (cid:162) (cid:16) Ind Γ χ such that (cid:68) Ind ψ i ( f ), v (cid:69) = (cid:68) π (cid:161) T M i Ind( f ) (cid:162) , v (cid:69) for all v ∈ Ind Γ χ , then (cid:179) ∀ i : Ind VM i ⊆ EE (cid:161) Ind Γ i χ i (cid:162)(cid:180) =⇒ Span (cid:161)
Ind Γ χ (cid:162) .Proof. We apply the map Ind defined in (1.3) to the assumptionM k ( Γ , χ ) ⊆ (cid:88) i ψ i (cid:161) VM i (cid:162) , implying that M k (Ind Γ χ ) ⊆ (cid:88) i Ind (cid:179) ψ i (cid:161) VM i (cid:162)(cid:180) .On the right hand side we can employ the intertwining properties of the projections π i to obtainM k (Ind Γ χ ) ⊆ (cid:88) i π i ◦ T M (cid:161) Ind VM i (cid:162) .Let us now assume that Ind VM i ⊆ EE (cid:161) Ind Γ i χ i (cid:162) for all i , and let us deduce that this implies, as de-sired, that Span(Ind Γ χ ) is true. Continuing our previous chain of inclusions, we find thatM k (Ind Γ χ ) ⊆ (cid:88) i π i ◦ T M (cid:179) EE k (cid:161) Ind Γ i χ i (cid:162)(cid:180) ⊆ (cid:88) i π i (cid:179) EE k (cid:161) T M Ind Γ i χ i (cid:162)(cid:180) ⊆ (cid:88) i EE k (cid:179) π i T M (cid:161) Ind Γ i χ i (cid:162)(cid:180) .The last and next to last inclusions follow from Lemma 4.2, which says that the spaces EE are stable un-der the application of representation homorphisms and Hecke operators. It is part of the assumptionsthat π i T M i (cid:161) Ind Γ i χ i (cid:162) equals Ind Γ χ , so that we obtain thatM k (Ind Γ χ ) ⊆ (cid:88) i EE k (cid:161) Ind Γ χ (cid:162) = EE k (cid:161) Ind Γ χ (cid:162) . §4.2.2 Proof of the Main Theorem: Reduction to the case of ρ = ρ M . We now start to actually provethe Main Theorem. Using the tools developed in the previous two lemmas, we show that it suffices toestablish Span( ρ M ) for all M in order to settle all other cases.Recall the assumptions: ρ is a representation of SL ( (cid:90) ) whose kernel is a congruence subgroup.Further, k , l are even integers satisfying l , k − l ≥ ρ is finite dimensional and has finite index kernel, it is unitary. Therefore ρ = (cid:76) i ρ i fora finite number of irreducible ρ i . By Lemma 4.3, it thus suffices to treat irreducible ρ .(2) By the same argument, we can prove the theorem for ρ = Ind Γ ( M ) for all positive M insteadof irreducible ρ . Indeed, suppose that, given an irreducible ρ , we have Γ ( M ) ⊆ ker ρ for some M , then ρ ⊆ Ind Γ ( M ) . Lemma 4.3 shows that Span (cid:161) Ind Γ ( M ) (cid:162) implies Span( ρ ). – – roducts of Vector Valued Eisenstein Series — 4 Products of Eisenstein series M. Westerholt-Raum (3) Recall the rescaling operator sc M from (2.23). We haveM k ( Γ ( M )) ⊆ sc M (cid:161) M k ( Γ ( M , M )) (cid:162) .Proposition 2.18 asserts existence of an inclusion ι Γ ( M ) : ρ Γ ( M ) (cid:44) −→ T N ρ Γ ( M , M ) that intertwines sc M and the vector valued Hecke operator T M . Therefore Lemma 4.4 allows us torestrict ourselves to the case ρ = ρ Γ ( M ) = Ind Γ ( M ) for arbitrary M (we have replaced M by M in thisvery last sentence).(4) We have Ind Γ ( M ) = (cid:76) χ Ind Γ ( M ) χ , where χ runs through all Dirichlet characters mod M .Lemma 4.3 thus allows us to focus on the case ρ = ρ χ = Ind Γ ( M ) χ for arbitrary M and arbitrary χ .The fifth reduction step requires an inductive argument. Observe that k is even by the assumptions,and therefore M k ( χ ) ∼= M k (cid:161) Ind Γ ( M ) χ (cid:162) is trivial, if χ is odd. In other words, we can and will assume that χ is even. It therefore has a square root: χ = χ (cid:48) . Classical theory of modular forms asserts thatM k ( χ ) ⊆ twist χ (cid:48) (cid:161) M k ( N ) (cid:162) + M old k ( χ ) , M old k ( χ ) = (cid:88) M (cid:48) | MM (cid:48) (cid:54)= M (cid:88) χ M (cid:48) sc M / M (cid:48) (cid:161) M k ( χ M (cid:48) ) (cid:162) (4.3)where χ M (cid:48) runs through Dirichlet characters mod M (cid:48) . Indeed, given f ∈ M k ( χ ), we consider its twist f χ (cid:48) ∈ M k ( N · N ). The difference f − twist χ (cid:48) ( f χ (cid:48) ) has Fourier expansion supported on n with gcd( n , N ) (cid:54)=
1. In other words, it is an old form.We reduce ourselves to the proving that Span( ρ M ) holds for all M by employing Lemma 4.4 in con-junction with induction on M . For the time being assume that Span( ρ M ) is true for all M . We willestablish this in Sections 4.2.3, 4.2.4, and 4.2.5—without recursing to the statement Span( ρ χ ) for non-trivial χ .Fixing M , our induction hypothesis is that Span(Ind Γ ( M (cid:48) ) χ M (cid:48) ) holds for all M (cid:48) | M with M (cid:48) (cid:54)= M andall Dirichlet characters mod M (cid:48) . We then show that Span(Ind Γ ( M ) χ ) for any Dirichlet character χ mod M . If M = M >
1. Proposition 2.19 guarantees existence of ι twist inter-twining T M and the twisting twist χ (cid:48) . Proposition 2.17 asserts existence of ι old intertwining T M / M (cid:48) andthe old form construction sc M / M (cid:48) . Combine this with the inclusion in (4.3) to see that the assumptionsof Lemma 4.4 are satisfied. The statement Span(Ind Γ ( M ) χ ) is therefore implied by Span(Ind Γ ( M ) )and Span(Ind Γ ( M (cid:48) ) χ M (cid:48) ) for all M (cid:48) | M with M (cid:48) (cid:54)= M . The induction hypothesis implies the latter. Theformer is part of our standing assumption that Span( ρ M ) be true for all M . This finishes our argument. §4.2.3 Proof of the Main Theorem: Classical Atkin-Lehner-Li Theory. Summarizing, we have shownso far that is suffices to establish Span( ρ M ) for all positive M . More precisely, we have to show thatM k ( ρ M ) = EE k ( ρ M ), where EE k ( ρ ) was defined in (4.2) for any ρ . In this Section, we will neverthelesstreat ρ χ for arbitrary Dirichlet characters χ , since no additional complications arise from this level ofgenerality.We can identify M k ( ρ χ ) with the space of classical modular forms M k ( χ ) via the inverse to Ind givenin (1.3): Ind − : M k ( ρ χ ) −→ M k ( χ ), f (cid:55)−→ 〈 f , e Γ ( M ) 〉 . – – roducts of Vector Valued Eisenstein Series — 4 Products of Eisenstein series M. Westerholt-Raum Set EE (cid:48) k ( ρ χ ) = Ind − EE k ( ρ χ ) ⊆ M k ( χ ). As for EE we suppress the subscript of EE (cid:48) except for few cases.From classical Atkin-Lehner-Li theory we find thatM k ( χ ) = (cid:77) M (cid:48) | M sc M / M (cid:48) M new k ( Γ ( M (cid:48) ), χ ) ,where M new k ( Γ ( M (cid:48) ), χ ) is the space of level M (cid:48) newforms for χ if χ extends as a character of the group Γ ( M ) to Γ ( M (cid:48) ), and M new k ( Γ ( M (cid:48) ), χ ) = {0} if it does not. Proposition 2.17 and Lemma 4.4 show that itsuffices to establish that M new k ( χ ) ⊆ EE (cid:48) ( ρ χ ) for all χ , in order to deduce Span( ρ χ ).Our next goal is to show that EE (cid:48) ( ρ χ ) is a Hecke module with respect to the classical Hecke algebra.That is, it is closed under the action of classical Hecke operators of level coprime to M and under theaction of Atkin-Lehner involutions. For M (cid:48) coprime to M , we haveInd (cid:161) EE (cid:48) ( ρ χ ) | k T M (cid:48) (cid:162) = π Hecke (cid:161) T M (cid:48) Ind EE (cid:48) ( ρ χ ) (cid:162) = π Hecke (cid:161) T M (cid:48) EE( ρ χ ) (cid:162) ⊆ EE (cid:161) T M (cid:48) ρ χ (cid:162) .The first equality is stated in Proposition 2.15. The second equality follows from the definition of EE (cid:48) .The inclusion, employed at the end, is proved in Lemma 4.2. A similar argument shows that EE (cid:48) ( ρ χ ) isalso closed under the action of all Atkin-Lehner involutions.As a consequence of this, we can focus on newforms. The remainder of the proof is about deduc-ing a contradiction to M new k ( Γ ( M )) (cid:42) EE (cid:48) k ( ρ M ). Suppose that indeed M new k ( Γ ( M )) (cid:42) EE (cid:48) k ( ρ M ) forsome M . Then the orthogonal complement with respect to the regularized scalar product of EE (cid:48) k ( ρ M )in S new k ( Γ ( M )) in nonempty. Since both EE (cid:48) k ( ρ M ) and S new k ( Γ ( M )) are classical Hecke modules, thatorthogonal complement is too. In particular, we find a newform f such that 〈 f , g 〉 = (cid:48) ( ρ M ).We will show that this is impossible. §4.2.4 Proof of the Main Theorem: Twists of newforms. To treat the case l = k /2, we have to considertwists of newforms. We claim that ∀ g ∈ EE (cid:48) ( ρ M ) : 〈 f , g 〉 = =⇒ ∀ (cid:178) Dirichlet character mod N ∀ g ∈ EE (cid:48) ( ρ (cid:178) ) : 〈 f (cid:178) , g 〉 =
0. (4.4)To see this, we first pass to the induction: 〈 f (cid:178) , g 〉 = (cid:68) Ind( f (cid:178) ), Ind( g ) (cid:69) = (cid:68) T N Ind( f ), ι twist (cid:161) Ind( g ) (cid:162)(cid:69) = (cid:68) Ind( f ), π adj (cid:161) T N ι twist (cid:161) Ind( g ) (cid:162)(cid:69) .The second equality is the part of the statement of Proposition 2.19. The third one follows from theadjunction formula in Proposition 2.10. In order to prove that 〈 f (cid:178) , g 〉 =
0, it remains to be shown that π adj T N ι twist Ind g ∈ EE. This is consequence of Lemma 4.2, saying that EE is mapped to itself by homo-morphisms of representations and by Hecke operators. §4.2.5 The proof of the Main Theorem: Nonvanishing of L -functions. We are left with showing thatfor every newform f for Γ ( M ) there is a character (cid:178) mod N such that 〈 f (cid:178) , g 〉 (cid:54)= g ∈ EE (cid:48) ( ρ (cid:178) ). By interchanging the role of the first and second factor in the definition of EE, we may assumethat l ≤ k /2. Fix a negative fundamental discriminant D , and consider the Kronecker character (cid:178) D andits square | D | = (cid:178) D , which is a trivial, non-primitive Dirichlet character. In particular, the classicalEisenstein series E l , | D | ( τ ) = ∞ (cid:88) n = σ k − | D | , ( n ) q n – – roducts of Vector Valued Eisenstein Series — 4 Products of Eisenstein series M. Westerholt-Raum considered in Lemma 3.5 is an oldform that comes from level one. Further, recall the Eisenstein series E k − l , | D | , ∞ = (cid:88) γ ∈ Γ ∞ \SL ( (cid:90) ) e Γ ( | D | ) ⊗ e Γ ( | D | ) | k , ρ | D | ⊗ ρ | D | γ that was defined in Lemma 3.6. We can view E k − l , | D | , ∞ as an Eisenstein series of type ρ | D | ⊗ ρ M | D | bymeans of the inclusion ρ | D | (cid:44) → ρ M | D | . We have g : = π (cid:179) Ind (cid:161) E l , | D | (cid:162) ⊗ E k − l , | D | , ∞ (cid:180) ∈ EE (cid:161) | D | (cid:162) , π : ρ | D | ⊗ ( ρ | D | ⊗ ρ M | D | ) − (cid:16) ρ M | D | ; e γ ⊗ (cid:161) e γ (cid:48) ⊗ e γ (cid:48)(cid:48) (cid:162) (cid:55)−→ (cid:40) e γ (cid:48)(cid:48) , if γ = γ (cid:48) ;0, otherwise.In particular, by (4.4), we have 〈 Ind f (cid:178) , g 〉 = 〈 f (cid:178) , Ind − ( g ) 〉 = g .We now evalute 〈 Ind f (cid:178) , g 〉 using the inclusion ι adjoint to π to gain some flexibility with respect tothe classical Eisenstein series E l , | D | . 〈 Ind f (cid:178) , g 〉 = (cid:68) ι (cid:161) Ind f (cid:178) (cid:162) , Ind E l , | D | ⊗ E k − l , | D | , ∞ (cid:69) .We introduce a spectral parameter s ∈ (cid:67) for the Eisenstein series E k − l , | D | , ∞ , which we will later spe-cialize to s =
0. If Re s (cid:192)
0, then Proposition 3.7 computes the following Petersson scalar product interms of a convolution Dirichlet series, and Proposition 4.1 allows us to factor it: (cid:68) ι (cid:161) Ind f (cid:178) (cid:162) , Ind E l , | D | ⊗ E k − l , | D | , ∞ , s (cid:69) = (4 π ) − k ∞ (cid:88) n = n − k + s σ l − | D | , ( n ) c ( f (cid:178) D ; n ) = L (cid:161) f (cid:178) D , k − + s (cid:162) L (cid:161) f (cid:178) D × | D | , k − + s − ( l − (cid:162) L (cid:161) | D | | D | M | D | , 2( k − + s ) − ( l − + − k (cid:162) The left and right hand side have analytic continuations to s =
0, and from this we conclude that L (cid:161) f (cid:178) D , k − (cid:162) L (cid:161) f (cid:178) D , k − l (cid:162) =
0. (4.5)The first factor cannot vanish, since k − > ( k + L ( f (cid:178) D , · ). If l < ( k − k − ≤ l ≤ k /2, that is l = k /2. In the lattercase we consider the central value of an L -function twisted by a imaginary quadratic character. Wehave to allude to Waldspurger’s and Kohnen-Zagier’s work [KZ84; Wal81]. Their work on the Shimuracorrespondence shows that L ( f × (cid:178) D , k /2) appears, up to normalizing factors, as the coefficient of ahalf-integral weight newform.Since D was an arbitrary but fixed negative fundamental discriminant, the theorem follows if weshow that a newform h of half-integral weight h ( τ ) = (cid:88) D c ( D ) q D vanishes if c ( D ) = D that are fundamental. By the Hecke theory for half-integral modular formsin Theorem 1.7 of [Shi73], we find that c ( n D ) = n ∈ (cid:90) ≥ . This finishes our proof of the maintheorem. – – roducts of Vector Valued Eisenstein Series — 4 Products of Eisenstein series M. Westerholt-Raum §4.3 Example: Weight , level . We illustrate the Theorem for weight 12 and level 3 modular forms.Over (cid:81) we have Ind Γ (3) ∼= ⊕ ρ for a three dimension representation. The second component hasrepresentation matrices ρ ( T ) = − − − , ρ ( S ) = − − − with respect to the basis − , − , − .Since the component corresponds to level 1 modular forms, which are spanned by, say, E and E ,it suffices to find a spanning set for M ( ρ ).For ρ T : = T , we choose the basis e : = e (cid:179) (cid:180) , e : = e (cid:179) (cid:180) , e : = e (cid:179) (cid:180) , e : = e (cid:179) (cid:180) ,so that ρ T ( T ) = , ρ T ( S ) = .The tensor product ρ T ⊗ ρ T contains four copies of ρ . The subspace of vectors that are fixed by( ρ T ⊗ ρ T )( T ) is spanned by f : = e ⊗ e − e ⊗ e − e ⊗ e − e ⊗ e , f : = e ⊗ e + e ⊗ e + e ⊗ e − e ⊗ e − e ⊗ e − e ⊗ e , f : = e ⊗ e − e ⊗ e + e ⊗ e − e ⊗ e + e ⊗ e − e ⊗ e , f : = e ⊗ e − e ⊗ e − e ⊗ e + e ⊗ e + e ⊗ e − e ⊗ e .The space M ( ρ ) has dimension dim M (Ind Γ (3) ) − dim M =
3. There are many possible waysto span S ( ρ ). In order to emphasis the vector valued approach, we compute the components f , f , f , and f of ( T E ) ⊗ ( T E ), where E and E are the level 1 Eisenstein series of weight 4 and 8, re-spectively. These components, one checks, are modular forms for Γ (3) and determine the associatedmodular form in M ( ρ ) uniquely. f : = 〈 T E ⊗ T E , f 〉 = − q − q − q − q + O (cid:161) q (cid:162) , f : = 〈 T E ⊗ T E , f 〉 = + (cid:161) − ζ + (cid:162) q + (cid:161) ζ + (cid:162) q + (cid:161) ζ + (cid:162) q + (cid:161) ζ + (cid:162) q + O (cid:161) q (cid:162) , f : = 〈 T E ⊗ T E , f 〉 = + (cid:161) − ζ + (cid:162) q + (cid:161) ζ + (cid:162) q + (cid:161) ζ + (cid:162) q + (cid:161) ζ − (cid:162) q + O (cid:161) q (cid:162) , f : = 〈 T E ⊗ T E , f 〉 = (cid:161) − ζ − (cid:162) q + (cid:161) ζ + (cid:162) q + (cid:161) ζ + (cid:162) q + (cid:161) ζ + (cid:162) q + O (cid:161) q (cid:162) . – – roducts of Vector Valued Eisenstein Series — 4 Products of Eisenstein series M. Westerholt-Raum We close this example by isolating the (unique) newform in M ( Γ (3)), which has initial Fourierexpansion q + q − q + q + O ( q ). Solving for coefficients of q through q yields the ex-pression (cid:161) ζ − (cid:162) f + (cid:161) ζ + (cid:162) f + (cid:161) ζ + (cid:162) f + (cid:161) ζ − (cid:162) f . §4.4 A curious vanishing condition. As an immediate consequence of our Main Theorem, we getthe following vanishing condition, for which at the moment, though, we have no application. Forsimplicity, we focus on the case l < k . Corollary 4.5.
Let k, l , and ρ be as in the Main Theorem, and in addition assume that l < k . There is < N ∈ (cid:90) such that if for N , N (cid:48) | N ρ N , = (cid:161) ρ N ⊗ ρ N (cid:48) ⊗ ρ ∨ (cid:162) ( ) has basis (cid:88) m c , m σ ∈ ∆ N e m c ⊗ e m σ ⊗ v N , i ( m c , m σ ) , ≤ i ≤ dim ρ N , with v N , i ( m c , m σ ) ∈ V ( ρ ) ∨ , then the following vanishing condition holds: Given f ∈ M k ( ρ ) , then f = if for all i and all N | N , we have = (cid:88) m c , m σ a k − lc a l σ (cid:88) n ∈ (cid:81) e (cid:161) − b σ na σ (cid:162) σ l − (cid:161) d σ na σ (cid:162) v N , i ( m c , m σ ) (cid:161) c ( f ; n ) (cid:162) ,where m c = (cid:179) a c b c c c d c (cid:180) and m σ = (cid:179) a σ b σ c σ d σ (cid:180) .Proof. This follows when computing the Fourier expansion of T M E for Eisenstein series E and thenapplying Proposition 3.7. [And74] G. E. Andrews. “A general theory ofidentities of the Rogers-Ramanujan type”. Bull. Amer. Math. Soc.
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