TTWO GRADED RINGS OF HERMITIAN MODULAR FORMS
BRANDON WILLIAMS
Abstract.
We give generators and relations for the graded rings of Hermitian modular forms of degree twoover the rings of integers in Q ( √−
7) and Q ( √− Introduction
Hermitian modular forms of degree n ∈ N are modular forms that transform under an action of the split-unitary group SU( n, n ; O ) with entries in some order O in an imaginary-quadratic number field. Throughthe natural embedding of SU( n, n ; O ) in Sp n ( Z ), the Shimura variety attached to SU( n, n ; O ) parameterizescertain principally polarized (2 n )-dimensional abelian varieties, namely the abelian varieties A of Weil type ,i.e. admitting multiplication by O in such a way that the eigenvalues of O acting on A occur in complex-conjugate pairs. (These were investigated by Weil in connection with the Hodge conjecture; see for examplethe discussion in [13], which also explains the connection to orthogonal Shimura varieties when n = 2.) Tostudy such objects it is helpful to have coordinates on the moduli space; in other words, generators for gradedrings of Hermitian modular forms.In [6], [7], Dern and Krieg began a program to compute these rings in degree n = 2 based on Borcherds’[1] theory of orthogonal modular forms with Heegner divisors (and the exceptional isogeny from SU(2 ,
2) toSO(2 , , , O )where O is the maximal order in Q ( √−
3) and Q ( √−
1) (where the fourfold is rational) and in Q ( √−
2) (whereit is not). The contribution of this note is to carry out these computations for the imaginary-quadratic fieldsof the smallest two remaining discriminants: Q ( √−
7) and Q ( √− M ∗ (SL ( Z )) = C [ E , E ]. The Riemann-Roch theorem (in the form of the “ k/
12 formula”) shows that everymodular form of weight not divisible by 6 has a zero at the elliptic point ρ = e πi/ , and that the Eisensteinseries E and E have no zeros besides a simple zero at ρ and at i (and their conjugates under SL ( Z )),respectively. Now every form in M ∗ (SL ( Z )) of weight not a multiple of 6 is divisible by E , and every formof weight 6 k becomes divisible by E after subtracting some scalar multiple of E k . The claim follows byinduction on the weight, together with the fact that modular forms of weight k ≤ ,
2) case the role of E above is played by a Borcherds product; the elliptic point ρ is replacedby the Heegner divisors; and the evaluation at ρ is replaced by the pullbacks, which send Hermitian modularforms to Siegel paramodular forms of degree two. With increasing dimension and level, the Heegner divisorswhich occur as divisors of modular forms are more complicated and the pullback maps to Heegner divisorsare rarely surjective. To overcome these issues our basic argument is as follows. We construct Hermitianmodular forms (Eisenstein series, theta lifts, pullbacks from O(2 , Mathematics Subject Classification. a r X i v : . [ m a t h . N T ] J a n his note is organized as follows. In section 2 we review Hermitian and orthogonal modular forms, thetalifts and pullbacks. In section 3 we recall the structure of the graded rings of paramodular forms of degreetwo and levels 1 , ,
3. In sections 4 and 5 we compute the graded rings of Hermitian modular forms for therings of integers of Q ( √−
7) and Q ( √−
11) by reducing against distinguished Borcherds products of weight7 and 5, respectively. (The ideal of relations for Q ( √−
11) is complicated and left to an auxiliary file.) Insection 6 we compute the dimensions of spaces of Hermitian modular forms.
Acknowledgments.
I am grateful to Jan H. Bruinier, Aloys Krieg and John Voight for helpful discus-sions. 2.
Preliminaries
In this section we review some facts about Hermitian modular forms of degree two and the relatedorthogonal modular forms. For a more thorough introduction the book [10] and the dissertation [4] areuseful references.2.1.
Hermitian modular forms of degree two.
Let H denote the Hermitian upper half-space of degreetwo: the set of complex (2 × τ for which, after writing τ = x + iy where x = x T and y = y T , thematrix y is positive-definite. The split-unitary groupSU , ( C ) = (cid:110) M ∈ SL ( C ) : M T JM = J (cid:111) , J = (cid:18) − −
11 0 0 00 1 0 0 (cid:19) acts on H by M¨obius transformations: M · τ = ( aτ + b )( cτ + d ) − , M = (cid:0) a bc d (cid:1) ∈ SU , ( C ) , τ ∈ H . Fix an order O in an imaginary-quadratic number field K . A Hermitian modular form of weight k ∈ N (and degree two) is a holomorphic function F : H → C which satisfies F ( M · τ ) = det( cτ + d ) k F ( τ ) for all M = (cid:0) a bc d (cid:1) ∈ SU , ( O ) and τ ∈ H . Note that F extends holomorphically to the Baily-Borel boundary (i.e. Koecher’s principle) as this containsonly components of dimension 1 and 0. Cusp forms of weight k are modular forms which tend to zero ateach one-dimensional cusp: that is, modular forms f for whichlim y →∞ (cid:16) f (cid:12)(cid:12)(cid:12) k M (cid:17) ( iy ) = 0 for all M ∈ SU , ( K ) . Orthogonal modular forms and Hermitian modular forms.
Suppose Λ = (Λ , Q ) is an (cid:96) -dimensionalpositive-definite even lattice; that is, Λ is a free Z -module of rank (cid:96) and Q is a positive-definite quadraticform on Λ ⊗ R taking integral values on Λ. One can define an upper half-space H Λ = { ( τ, z, w ) : τ, w ∈ H , z ∈ Λ ⊗ C , Q (im( z )) < im( τ ) · im( w ) } ⊆ C (cid:96) +2 . This is acted upon by SO + (Λ ⊕ II , ) (the connected component of the identity) by M¨obius transformations.To make this explicit it is helpful to fix a Gram matrix S for Q and realize SO + (Λ ⊕ II , ) as a subgroup ofthose matrices which preserve the block matrix (cid:32) S (cid:33) ∈ Z × under conjugation. For such a matrix M and ( τ, z, w ) ∈ H Λ , one can define M · ( τ, z, w ) = (˜ τ , ˜ z, ˜ w ) ∈ H Λ by M (cid:32) Q ( z ) − τwτzw (cid:33) = j ( M ; τ, z, w ) (cid:32) Q (˜ z ) − ˜ τ ˜ w ˜ τ ˜ z ˜ w (cid:33) for some j ( M ; τ, z, w ) ∈ C × . The orthogonal modular group Γ Λ is the discriminant kernel of Λ ⊕ II , ; that is, the subgroup ofSO + (Λ ⊕ II , ) which acts trivially on Λ (cid:48) / Λ. An orthogonal modular form is then a holomorphic function f : H Λ → C which satisfies f ( M · ( τ, z, w )) = j ( M ; τ, z, w ) k f ( τ, z, w )for all M ∈ Γ Λ and ( τ, z, w ) ∈ H Λ . (There is again a boundedness condition at cusps which is automatic byKoecher’s principle.)Hermitian modular forms for SU , ( O K ) are more or less the same as orthogonal modular forms for thelattice of integers (Λ , Q ) = ( O K , N K/ Q ) of K . One way to see this is as follows. The complex space of ntisymmetric (4 × Pfaffian , a square root of thedeterminant) which is preserved under the conjugation action M · X = M T XM by SL ( C ); explicitly,pf (cid:32) a b c − a d e − b − d f − c − e − f (cid:33) = af − be + cd. The conjugation action identifies SL ( C ) with the spin group Spin(pf) = Spin ( C ). The six-dimensional realsubspace V = (cid:110) (cid:32) a b c − a d − b − b − d f − c b − f (cid:33) : a, c, d, f ∈ R , b ∈ C (cid:111) on which the Pfaffian has signature (4 ,
2) is preserved under conjugation by SU , ( C ), and this actionrealizes the isomorphism SU , ( C ) ∼ = Spin , ( R ). The lattice of O K -integral matrices (which is isometric to O K ⊕ II , ) is preserved by SU , ( O K ) and we obtain an embedding of SU , ( O K ) in the discriminant kernelΓ O K . This isomorphism induces an identification between the homogeneous spaces H and H Λ and allowsorthogonal modular forms to be interpreted as Hermitian modular forms of the same weight.The discriminant kernel Γ O K contains the involution α (cid:55)→ α of O K (in other words, α − α ∈ O K forall α in the codifferent O K ), and this involution does not come from the action of SU , ( O K ). This meansthat Hermitian modular forms which arise from orthogonal modular forms are either symmetric or skew-symmetric: Definition 1.
A Hermitian modular form F : H → C of weight k is (graded) symmetric if F ( z T ) = ( − k F ( z ) for all z ∈ H , and (graded) skew-symmetric if F ( z T ) = − ( − k F ( z ).Note that many references (e.g. [6],[7]) use the notion of (skew)-symmetry without respect to the grading,i.e. without the factor ( − k .The maximal discrete extension Γ ∗ K of Γ K (as computed in [11]) also contains a copy of the class groupCl( O K ) which is generally not contained in the discriminant kernel. We only consider the fields K = Q ( √− , Q ( √−
11) of class number one so we will not discuss this point further; however, if one were toextend the arguments below to general number fields then most instances of the discrete extension Γ O K ofΓ K below should probably be replaced by Γ ∗ K .2.3. Heegner divisors.
On orthogonal Shimura varieties there is a natural construction of
Heegner divisors .Suppose Λ is an even lattice of signature ( (cid:96), λ ∈ Λ of positive norm, consider theorthogonal complement λ ⊥ ∩ H Λ which has codimension one. The union of these orthogonal complementsas λ ranges through the (finitely many) primitive lattice vectors of a given norm D is Γ Λ -invariant anddefines an analytic cycle H D on Γ Λ \ H Λ . (If we do not take only primitive vectors then we obtain thedivisors (cid:80) f | D H D/f , which are also often called the Heegner divisors in the literature. For our purposesthis definition is less convenient.)The irreducible components H D, ± β of H D correspond to pairs ( ± β ) ∈ Λ (cid:48) / Λ of norm D/ disc(Λ). Inparticular when disc(Λ) is prime then every H D is irreducible.Each Heegner divisor is itself an orthogonal Shimura variety for a lattice of signature (2 , (cid:96) − H D may be identified with the paramodularthreefold X K ( D ) of level D modulo Atkin-Lehner involutions.) Moreover the intersection of any two Heegnerdivisors is itself a Heegner divisor in this interpretation. The intersection numbers can be computed ingeneral by counting certain lattice embeddings up to equivalence. However it seems worthwhile to mention atrick which (in the cases we will need) makes this computation quite easy and which works in some generality.A special case of Borcherds’ higher-dimensional Gross-Kohnen-Zagier theorem [2] shows that the Heegnerdivisors on Γ K \ H interpreted appropriately are coefficients of a modular form of weight 3. If K hasprime discriminant d K <
0, and we take intersection numbers with a fixed Heegner divisor of squarefreediscriminant m ∈ N and apply the Bruinier-Bundschuh isomorphism (see [3], or Remark 3 below) then this mplies that there are weights α m ( D ), D ∈ N such thatΦ m ( τ ) := − ∞ (cid:88) D =1 α m ( D ) (cid:88) f | D ( H m · H D/f ) q D ∈ M +3 (Γ ( − d K ) , χ ) , where χ is the quadratic Dirichlet character modulo d K , and where M +3 (Γ ( − d K ) , χ ) is the subspace ofweight three modular forms of level Γ ( − d K ) whose Fourier expansions at ∞ are supported on exponentswhich are quadratic residues. Moreover the sums (cid:80) f | D α m ( D/f ) themselves (for fixed m ) are coefficientsof a modular form of weight 5 / (4 m ) satisfying the Kohnen plus-condition and which hasconstant term − m = 1 , , − ∞ (cid:88) D =1 (cid:88) f | D α ( D/f ) q D = − q + 70 q + 48 q + 120 q + 250 q + ... = 6 θ (cid:48) ( τ )2 πi − E (4 τ ) θ ( τ ) , − ∞ (cid:88) D =1 (cid:88) f | D α ( D/f ) q D = − q + 22 q + 24 q + 100 q + ... = 3 θ (cid:48) ( τ )2 πi − E (8 τ ) θ ( τ ) , − ∞ (cid:88) D =1 (cid:88) f | D α ( D/f ) q D = − q + 14 q + 34 q + 24 q + ... = 2 θ (cid:48) ( τ )2 πi − E (12 τ ) θ ( τ ) , where θ ( τ ) = 1 + 2 q + 2 q + 2 q + ... is the usual theta function and where E ( τ ) = 1 − (cid:80) ∞ n =1 σ ( n ) q n .Unfortunately the spaces M +3 (Γ ( − d K ) , χ ) are two-dimensional for d K ∈ {− , − } . However one canspecify the correct modular forms more precisely by observing that the intersections in cohomology arethemselves the Fourier coefficients of a vector-valued Jacobi form of index m/ | d K | and weight three (fora particular representation of the Jacobi group) and the intersection numbers are obtained by setting theelliptic variable of that Jacobi form to zero. (More precisely these Jacobi forms occur as Fourier-Jacobicoefficients of the Siegel modular form introduced by Kudla-Millson in [12].) For m ≤ every d K ), spanned by the Eisenstein series (for which somecomputational aspects are discussed in [16]) so the generating series of intersection numbers is exactly whatwas called the Poincar´e square series of index m/ | d K | in [16]. In this way we can compute the relevantintersection numbers without computing any intersections. We find:(1) For K = Q ( √− ( τ ) = − − q + 20 q + 18 q + 70 q + 160 q + 94 q + ... and Φ ( τ ) = − q + 2 q + 48 q + 28 q + 142 q + 148 q + ... (2) For K = Q ( √− ( τ ) = − − q + 20 q − q + 20 q + 18 q + 70 q + ... and Φ ( τ ) = − q + 0 q + 14 q + 16 q + 82 q + 26 q + ... It follows that for K = Q ( √− H and H as a Heegner divisor of X K (1) is 2 H and as a Heegner divisor of X K (2) is just H itself; and for K = Q ( √−
11) the intersection of H and H in X K (1) is 2 H and in X K (2) is H . This means, for example, that if F is a Hermitian modular form for O K , K = Q ( √−
7) with a zero on H , then the pullbacks of all orders to H are Siegel modular forms of degreetwo with at least a double zero along the diagonal.2.4. Lifts.
To construct generators we make use of two lifts from elliptic modular forms: the
Maass lift (oradditive theta lift) and the
Borcherds lift (or multiplicative theta lift). Both theta lifts most naturally takevector-valued modular forms which transform under a Weil representation as inputs.Recall that if (Λ , Q ) is an even-dimensional even lattice with dual Λ (cid:48) then there is a representation ρ ∗ ofSL ( Z ) on C [Λ (cid:48) / Λ] = span( e γ : γ ∈ Λ (cid:48) / Λ) defined by ρ ∗ (cid:0)(cid:0) −
11 0 (cid:1)(cid:1) e γ = e − πi sig(Λ) / (cid:112) | Λ (cid:48) / Λ | (cid:88) β ∈ Λ (cid:48) / Λ e πi (cid:104) β,γ (cid:105) e β , ρ ∗ (( )) e γ = e − πiQ ( γ ) e γ . e consider holomorphic functions F : H → C [Λ (cid:48) / Λ] which satisfy the functional equations F (cid:18) aτ + bcτ + d (cid:19) = ( cτ + d ) k ρ ∗ (cid:0)(cid:0) a bc d (cid:1)(cid:1) for all (cid:0) a bc d (cid:1) ∈ SL ( Z ). These are called nearly-holomorphic modular forms if they have finite order at ∞ (inother words, F ( x + iy ) has at worst exponential growth as y → ∞ ), and are (holomorphic) modular forms or cusp forms if F ( x + iy ) is bounded or tends to zero in that limit, respectively. The functional equationunder T = ( ) implies a Fourier expansion of the form F ( τ ) = (cid:88) γ ∈ Λ (cid:48) / Λ (cid:88) n ∈ Z n − Q ( γ ) c ( n, γ ) q n e γ where q = e πiτ and c ( n, γ ) ∈ C . Then F is a nearly-holomorphic modular form if and only if c ( n, γ ) = 0for all sufficiently small n ; a holomorphic modular form if and only if c ( n, γ ) = 0 for all n <
0; and a cuspform if and only if c ( n, γ ) = 0 for all n ≤ k ≥ dim Λ, k ∈ Z . The Maass lift takes a vector-valuedmodular form F ( τ ) = (cid:80) γ,n c ( n, γ ) q n e γ of weight κ = k − dim Λ for ρ ∗ to the orthogonal modular formΦ F ( τ, z, w ) = − B k k c (0 , (cid:16) E k ( τ ) + E k ( w ) − (cid:17) + ∞ (cid:88) a,b =1 (cid:88) λ ∈ Λ (cid:48) λ positive Q ( λ ) ≤ ab ∞ (cid:88) n =1 c ( ab − Q ( λ ) , λ ) n k − e πin ( aτ + bw + (cid:104) λ,z (cid:105) ) for Λ ⊕ II , , where E k ( τ ) , E k ( w ) denote the Eisenstein series of weight k for SL ( Z ). (If k is odd then c (0 ,
0) = 0 so there is no need to define E k .) The Maass lift is additive and preserves the subspace of cuspforms.The second lift we use is the Borcherds lift, which takes a nearly-holomorphic vector-valued modular form F ( τ ) = (cid:80) γ,n c ( n, γ ) q n e γ of weight − dim Λ (where we again take Λ to be positive-definite) and yields amultivalued meromorphic orthogonal modular form (in general with character) which is locally representedas a convergent infinite product:Ψ F ( τ, z, w ) = e πi ( Aτ + (cid:104) B,z (cid:105) + Cw ) (cid:89) a,b,λ (1 − e πi ( aτ + bw + (cid:104) λ,z (cid:105) ) ) c ( ab − Q ( λ ) ,λ ) . There is an analogy to the formal k = 0 case of the Maass lift; however, the set over which a, b, λ is morecomplicated (depending on a Weyl chamber containing ( τ, z, w )) and the
Weyl vector ( A, B, C ) has noanalogue in the additive lift. The most important aspect of the Borcherds lift for us is not the productexpansion but the fact that the divisor of Ψ F may be computed exactly: it is supported on Heegner divisors,and the order of Ψ F on the rational quadratic divisor λ ⊥ (with Q ( λ ) <
0) isord(Ψ F ; λ ⊥ ) = (cid:88) r ∈ Q > c ( r Q ( λ ) , rλ )(where c ( r Q ( λ ) , rλ ) = 0 if rλ (cid:54)∈ Λ (cid:48) ). In particular Ψ F is an orthogonal modular form if and only if theseorders are nonnegative integers. In all cases the weight of F is c (0 , / Remark 2.
One can always compactify Γ Λ \ H Λ by including finitely many zero-dimensional and one-dimensional cusps (corresponding to isotropic one-dimensional or two-dimensional sublattices of Λ ⊕ II , upto equivalence). If K has class number one (or slightly more generally if the norm form on O K is alone inits genus) then our discriminant kernel Γ O K admits only one equivalence class each of zero-dimensional andone-dimensional cusps and both are contained in the closure of every rational quadratic divisor. In particularany Borcherds product which is holomorphic is automatically a cusp form. (This is peculiar to the latticesconsidered here; it is certainly not true in general.) Remark 3.
Let us say a few words about the input functions F . A general method to compute vector-valued modular forms for general lattices was given in [16] and [15] (the two references corresponding toeven and odd-weight theta lifts, respectively), and this is what was actually used in the computations be-low because the implementation was already available. Of course one can obtain all nearly-holomorphicmodular forms by dividing true modular forms of an appropriate weight by a power of the discriminant ( τ ) = q (cid:81) ∞ n =1 (1 − q n ) . However a few other formalisms apply to the particular lattices Λ = ( O K , N K/ Q )considered here:(i) Modular forms for the representation ρ ∗ attached to a positive-definite lattice Λ are equivalent to Jacobiforms of lattice index which are scalar-valued functions φ ( τ, z ) in a “modular variable” τ ∈ H and an “ellipticvariable” z ∈ Λ ⊗ C satisfying certain functional equations and growth conditions. The main advantage ofJacobi forms is that they can be multiplied: for example, in many cases it is possible to construct all Jacobiforms of a given weight and level by taking linear combinations of products of Jacobi theta functions atvarious arguments (i.e. theta blocks).(ii) If Λ has odd prime discriminant p and k +(dim Λ) / k for ρ ∗ can be identified with either a “plus-” or “minus-” subspaceof M k (Γ ( p ) , χ p ) (where χ p is the nontrivial quadratic character mod p ), i.e. the subspace of modular formswhose Fourier coefficients are supported on quadratic residues modulo p , or quadratic nonresidues mod p and p Z , respectively. The isomorphism simply identifies the form F ( τ ) = (cid:80) γ,n c ( n, γ ) q n e γ with (cid:88) γ,n c ( n, γ ) q pn ∈ M k (Γ ( p ) , χ p ) . This fails when k +(dim Λ) / c ( n, γ ) = − c ( n, − γ ), so the resulting sum is always zero!).To obtain any results in the the same spirit, it seems necessary to consider instead the “twisted sums” (cid:88) γ,n c ( n, γ ) χ ( γ ) q pn , where χ is an odd Dirichlet character mod p (and where an isomorphism Λ (cid:48) / Λ ∼ = Z /p Z has been fixed). Theresult is a modular form of level Γ ( p ) with character χ ⊗ χ p . These maps were studied in [14]; they areinjective and their images can be characterized in terms of the Atkin-Lehner involutions modulo p .2.5. Pullbacks.
Let λ ∈ O K have norm (cid:96) = N K/ Q λ , and consider the embedding of the Siegel upperhalf-space into H : φ : H −→ H , φ (( τ zz w )) = (cid:0) τ λzλz (cid:96)w (cid:1) = U λ · ( τ zz w ) , U λ := diag(1 , λ, , λ/(cid:96) ) . For any paramodular matrix M ∈ K ( (cid:96) ) := { M ∈ Sp ( Q ) : σ − (cid:96) M σ (cid:96) ∈ Z × } , σ (cid:96) := diag(1 , , , (cid:96) ) , we find U λ M U − λ ∈ SU , ( O K ) and φ ( M · τ ) = ( U λ M U − λ ) · φ ( τ ) , τ ∈ H , so φ descends to an embedding of K ( (cid:96) ) \ H into Γ K \ H (and more specifically into the Heegner divisor ofdiscriminant (cid:96) ). In particular if F : H → C is a Hermitian modular form then f := F ◦ φ is a paramodularform of the same weight, i.e. f ( M · τ ) = ( cτ + d ) k f ( τ ) for all M = (cid:0) a bc d (cid:1) ∈ K ( (cid:96) ) and τ ∈ H . The preprint [17] gives expressions in the higher Taylor coefficients about a rational quadratic divisorwhich yield “higher pullbacks” P N F , N ∈ N . If F is a Hermitian modular form of weight k then itspullback P H (cid:96) N F along the embedding above is a paramodular form of level K ( (cid:96) ) and weight k + N and a cuspform if N >
0. The higher pullbacks of theta lifts are themselves theta lifts and are particularly simple tocompute. One computational aspect of the higher pullbacks worth mentioning is that a form F vanishes tosome order h along the rational quadratic divisor if and only if its pullbacks P N F , N < h are identically zero,and this can be checked rigorously using Sturm bounds (or their generalizations) for the lower-dimensionalgroup under which P N F transforms.An important case is the N th pullback of a modular form F to a Heegner divisor along which it has orderexactly N . The result in this case is the well-known quasi-pullback and we denote it Q F . The quasi-pullbackis multiplicative i.e. Q( F G ) = Q F · Q G for all Hermitian modular forms F, G . . Paramodular forms of levels one, two and three
The pullbacks of Hermitian modular forms to certain Heegner divisors have interpretations as paramodularforms (as in subsection 2.5 above). Structure results for graded rings of paramodular forms are known fora few values of N . We will rely on the previously known generators for the graded rings of paramodularlevels 1,2 and 3. The first of these is now classical and was derived by Igusa [9]; the second was computedin [8] by Ibukiyama and Onodera; and the third was computed by Dern [5]. For convenience we expressthe generators as Gritsenko lifts or Borcherds products. (Igusa and Ibukiyama–Onodera expressed them interms of thetanulls.) Proposition 4. (i) There are cusp forms ψ , ψ , ψ of weights , , such that M ∗ ( K (1)) is generatedby the Eisenstein series E , E and by ψ , ψ , ψ .(ii) There are graded-symmetric cusp forms φ , φ , φ , φ of weights , , , and an antisymmetricnon-cusp form f such that M ∗ ( K (2)) is generated by the Eisenstein series E , E and by φ , φ , φ , φ , f .(iii) There are graded-symmetric cusp forms ϕ , ϕ , ϕ , ϕ , ϕ , ϕ of weights , , , , , and an an-tisymmetric non-cusp form f such that M ∗ ( K (3)) is generated by the Eisenstein series E , E and by ϕ , ϕ , ϕ , ϕ , ϕ , ϕ , f . For later use, we fix the following concrete generators. Let E , E denote the modular Eisenstein series; E k,m the Jacobi Eisenstein series of weight k and index m ; and E (cid:48) k,m its derivative with respect to z . Theinputs into the Gritsenko and Borcherds lifts are expressed as Jacobi forms following Remark 3 above.(i) ψ and ψ are the Gritsenko lifts of the Jacobi cusp forms ϕ , ( τ, z ) = E , E − E E ,
144 and ϕ , ( τ, z ) = E E , − E E , ψ is the Borcherds lift of E E , +7 E E , .(ii) φ , φ , φ , φ are the Gritsenko lifts of the Jacobi cusp forms ϕ , = E E , − E , , ϕ , = E , E − E , E , , ϕ , = E , E (cid:48) , − E , E (cid:48) , πi , ϕ , = E E , − E E , , respectively, and f is the Borcherds lift of E E , +4 E E , +5 E E , .(iii) ϕ , ϕ , ϕ , ϕ , ϕ , ϕ are the Gritsenko lifts of the Jacobi cusp forms ϕ , = ϕ , ϕ , ∆ , ϕ , = E E , − E , E , , ϕ , = ϕ , ϕ , ∆ ,ϕ , = ϕ , ϕ , ∆ , ϕ , = ϕ , ϕ , ∆ , ϕ , = E E , E , + E E , − E , E , , respectively, and f is the Borcherds lift of E E , E , +5 E , +5 E , E , . (Note that these are not quite thegenerators used by Dern; the choices used here simplify the ideal of relations somewhat.) Remark 5.
For later use we will need to understand the ideals of symmetric (under the Fricke involution τ (cid:55)→ − N τ − ) paramodular forms of level N ∈ { , , } which vanish along the diagonal. The pullback of aparamodular form to the diagonal is a modular form for the group SL ( Z ) × SL ( Z ) or in other words a linearcombination of expressions of the form ( f ⊗ f )( τ , τ ) = f ( τ ) f ( τ ), where f , f are elliptic modular formsof level one of the same weight; and if the paramodular form is symmetric then the pullback is symmetricunder swapping ( τ , τ ) (cid:55)→ ( τ , τ ). The graded ring of symmetric modular forms under SL ( Z ) × SL ( Z ) isthe weighted polynomial ring M ∗ (SL ( Z ) × SL ( Z )) = C [ E ⊗ E , E ⊗ E , ∆ ⊗ ∆]where E , E , ∆ are defined as usual. Therefore:(i) In level N = 1, the pullbacks of E , E , ψ to the diagonal are the algebraically independent modularforms E ⊗ E , E ⊗ E , ∆ ⊗ ∆, so every even-weight form which vanishes on the diagonal is a multiple of (which has a double zero). The odd-weight form ψ has a simple zero on the diagonal.(ii) In level N = 2, the pullbacks of E , E , φ to the diagonal are algebraically independent, so the ideal ofeven-weight symmetric forms which vanish on the diagonal is generated by φ (which has a fourth-order zerothere) and φ (which has a double zero). Moreover φ is itself a multiple of φ , so the ideal of even-weightmodular forms which vanish to order at least three along the diagonal is principal, generated by φ . Theodd-weight form φ has a simple zero along the diagonal.(iii) In level N = 3, the pullbacks of E , E , ϕ to the diagonal are algebraically independent, so the idealof even-weight symmetric forms which vanish on the diagonal is generated by ϕ , ϕ , ϕ (which have zerosof order 6 , , ϕ = ϕ ϕ and ϕ = ϕ ϕ , so the ideals of (even-weight,symmetric) forms which vanish to order at least 3 or at least 5 are (cid:104) ϕ , ϕ (cid:105) and (cid:104) ϕ (cid:105) , respectively. Theodd-weight forms ϕ and ϕ have order 3 and 1 along the diagonal, respectively, and satisfy the relations ϕ ϕ = ϕ ϕ , ϕ ϕ = ϕ ϕ , and ϕ and ϕ ϕ (and therefore all odd-weight symmetric forms with at least a triple zero on the diagonal)are multiples of ϕ . 4. Hermitian modular forms for Q ( √− K = Q ( √−
7) by studying the pullbacks to Heegner divisors of discriminant 1 and 2 and applying the structuretheorems of Igusa and Ibukiyama-Onodera. We first consider graded-symmetric forms and reduce against adistinguished Borcherds product b (which is also a Maass lift) whose divisor isdiv b = 3 H + H . We will express all graded-symmetric forms in terms of Maass lifts E , E , b , m , m , m (1)10 , m (2)10 , m , m in weights 4 , , , , , , , ,
12 which are described in more detail on the next page. The Maass lifts ofweight 4 , , , , m (1)10 vanisheson H and m (2)10 vanishes on H . By contrast m could have been chosen almost arbitrarily (so long as it isnot a multiple of E b , which is also a Maass lift), and similarly for m . Lemma 6.
Let F be a symmetric Hermitian modular form. There is a polynomial P such that F − P ( E , E , m , m (1)10 , m , m ) vanishes along the Heegner divisor H .Proof. This amounts to verifying that the pullbacks of E , E , m , m (1)10 , m , m generate the ring of sym-metric paramodular forms of level 2, and is clear in view of Ibukiyama-Onodera’s structure result and Tables1 and 2 below. (cid:3) Theorem 7.
The graded ring of symmetric Hermitian modular forms for O K is generated by Maass lifts E , E , b , m , m , m (1)10 , m (2)10 , m , m in weight , , , , , , , , . The ideal of relations is generated by m m = b ( m (1)10 + 12 m (2)10 ); m + 12 b m = E b + 36 m m (2)10 ; m m (1)10 = b ( E m + 12 m ); E b + 18 m (1)10 m (2)10 = E b m + 6 m m ; m (1)10 ( m (1)10 + 12 m (2)10 ) = m ( E m + 12 m ); E b m (1)10 + 6 E b m (2)10 + 72 m (2)10 m = E b m + 6 m m ;3 E m m (1)10 + 6 E b m + E b m + 72 m = E b + 3 E m + 18 m (1)10 m . n Table 1 we describe the even-weight Maass lifts used as generators. For each Maass lift of weight k we give its input form (in the conventionof Bruinier-Bundschuh; this is a modular form of weight k − (7) for the quadratic character) and its first pullbacks to the Heegnerdivisors of discriminant 1 and 2. (The pullbacks of odd order to H are always zero and therefore omitted.) Table 1.
Maass lifts in even weightName Weight Input form P H P H P H P H P H E q + 42 q + 70 q + 42 q + 210 q ± ... E E E − q − q − q − q − q ± ... E ψ E m q − q − q + 7 q + 8 q ± ... ψ ψ φ m (1)10 q − q + 16 q − q − q ± ... ψ E ψ φ ψ m (2)10 q − q − q + q − q ± ... ψ − ψ − E ψ − ψ m q + 3 q + 7 q − q − q ± ... ψ E ψ E ψ − E ψ
10 13 φ − E φ χ may be any odd Dirichlet character mod 7;the input form is then a modular form of level Γ (49) and character χ ⊗ χ where χ is the quadratic character. The Borcherds product b happensto lie in the Maass Spezialschar and is listed in this table. Table 2.
Maass lifts in odd weightName Weight Input form P H P H P H P H P H b χ (5) q + 3 χ (3) q + 2 χ (1) q − χ (5) q ± ... − ψ ψ − φ m χ (5) q − χ (3) q − χ (1) q − χ (5) q ± ... − ψ ψ − E ψ − φ m χ (3) q − χ (1) q + 11 χ (5) q − χ (3) q ± ... − ψ E ψ
10 62903 E ψ − E ψ φ φ The Borcherds products below can be shown to exist by a Serre duality argument as in [2].
Table 3.
Borcherds productsName Weight Divisor Graded-symmetric? b H + H yes b
28 7 H + H no roof. We use induction on the weight. As usual any modular form of negative or zero weight is constant.Using the previous lemma we may assume that F has a zero along H . Since H has a double intersectionwith H along its diagonal H it follows that the pullbacks of F to H of all orders have (at least) a doublezero along the diagonal; in particular, they are multiples of the Igusa discriminant ψ .Since the pullbacks of E , E , m (2)10 , m to H generate the graded ring of even-weight Siegel modularforms, and m (2)10 vanishes along H but pulls back to the Igusa form ψ on H , it follows that we cansubtract some expression of the form m (2)10 P ( E , E , m (2)10 , m )away from F to obtain a form whose pullbacks to both H and H are zero. Similarly, we can subtract someexpression of the form m P ( E , E , m (2)10 , m )away from F to ensure that the zero along H has multiplicity at least two.Now assume that F has exactly a double zero along H (in particular, it must have even weight) and azero along H . Suppose first that F has exactly a simple zero along H . Then its first pullback P H F hasodd weight and at least a double zero along the diagonal in X K (2) and is therefore contained in the idealgenerated by φ φ and φ φ . The products m m (2)10 and m (1)10 m (2)10 have (up to a constant multiple) exactlythese first pullbacks, so subtracting away some expression of the form m m (2)10 P ( E , E , m , m (1)10 , m , m ) + m (1)10 m (2)10 P ( E , E , m , m (1)10 , m , m )with polynomials P , P leaves us with a modular form with at least double zeros along both H and H .The double zero along H forces the second pullback to H to have at least a fourth -order zero along thediagonal and therefore to be a multiple of ψ . Since m has exactly this second pullback to H (up to aconstant multiple) and a double zero along H , we may subtract away some expression of the form m P ( E , E , m (2)10 , m )from F to obtain a form with a third-order zero along H and which continues to have a double zero on H .Finally, any modular form F with a triple zero along H and a zero along H is divisible by b (byKoecher’s principle), with the quotient Fb having strictly lower weight. By induction, F/b and therefore F is a polynomial expression in the generators in the claim.The relations were computed by working directly with Fourier expansions. Here the main difficulties aredetermining how many Fourier coefficients must be computed to show that a modular form is identicallyzero, and determining how many relations are needed to generate the full ideal. To verify the correctness ofthese computations in both cases it is enough to know the dimensions of spaces of Hermitian modular forms,and these are derived in section 6 below. (cid:3) Proposition 8.
There are holomorphic skew-symmetric forms h , h , h , h , h , h , which are obtainedfrom b and the Maass lifts constructed above by inverting b , such that every Hermitian modular form for O K is a polynomial in E , E , b , m , m , m (1)10 , m (2)10 , m , m , b , h , h , h , h , h , h . Proof.
As a skew-symmetric form, F has a forced zero on the Heegner divisor H . If F has even weight, thepoint will be to subtract away skew-symmetric forms from F to produce something with at least a seventh-order zero on the surface H , which will therefore be divisible by b . By contrast if F has odd weight thenit seems to be more effective to reduce first against the product b .(i) Suppose F has even weight, so its order along H is odd and its quasi-pullback to H takes the formQ F = ψ P ( ψ , ψ , ψ , ψ )for some polynomial P . The quotients h := b m b , h := b m b , h := b m b are holomorphic and skew-symmetric, with zeros along H of order 5 , , H is a onstant multiple of ψ . By subtracting from F expressions of the form { h , h , h } · P ( E , E , m (2)10 , m ) , we are able to force the first, third and fifth order pullbacks of F to H to vanish. But then F is divisibleby b with symmetric quotient, so we apply the previous proposition.(ii) Suppose F has odd weight (and therefore even order along H ). Then we will find expressions tosubtract away from F to force divisibility by b . (The reduction against b as in the even-weight case seemsimpossible, as there are no skew-symmetric modular forms of weight 29 and therefore no way to handlesixth-order zeros on H .) We will first force F to have at least a fourth-order zero along H . The quotients h := b m (2)10 m b , h := b m (2)10 m b are holomorphic and skew-symmetric, with zeros along H of orders 2 and 0, respectively, and their quasi-pullbacks to H are again constant multiples of ψ . By subtracting from F expressions of the form { h , h } · P ( E , E , m (2)10 , m ) , we can ensure that the 0 th and 2 nd pullbacks of F to H vanish, so ord H ( F ) ≥ F to H is skew-symmetric, has odd weight, and vanishes on the diagonal to orderat least four, so it is therefore a multiple of the weight 31 form φ φ f : i.e. F (cid:12)(cid:12)(cid:12) H = φ φ f P ( E , E , φ , φ , φ )for some polynomial P . But the form h := b m (2)10 b is holomorphic and skew-symmetric, with a fourth-order zero on H , and it restricts to (a multiple of) φ φ f on H . Therefore, some expression of the form F − h P ( E , E , m , m (1)10 , m )has a zero on H and continues to have at least a fourth-order zero on H . The result will be divisible by b with the quotient having even weight and therefore being covered by case (i). (cid:3) Hermitian modular forms for Q ( √− Q ( √−
11) to the results of Igusa and Dern on paramodular forms. The argumentis very nearly the same as the previous section. We first deal with symmetric Hermitian modular forms (ofall weights) by reduction against the distinguished Borcherds product b with divisordiv b = 5 H + H . The Maass lifts we take as generators are described in more detail in the tables on the next page.
Lemma 9.
Let F be a symmetric Hermitian modular form. There is a polynomial P such that F − P ( E , E , m , m , b , m (1)10 , m , m ) vanishes along the Heegner divisor H .Proof. We only need to check that the pullbacks of E , E , m , m , b , m (1)10 , m , m to H generate thegraded ring of paramodular forms of level 3. This is clear from Tables 4 and 5 below after comparing thepullbacks with the generators found by Dern as described in Section 3. (cid:3) s in the previous section, the input forms into the Maass lift in Tables 4 and 5 are expressed as component sums using the convention of [3] and [14]. The Borcherdsproducts b , b , b satisfy the Maass condition so they are listed both as Maass lifts and Borcherds products. Table 4.
Maass lifts in even weight
Name Weight Input form P H P H P H P H P H E q + 20 q + 32 q + 34 q + 52 q + ... E E E − q − q − q − q − q − ... E − ψ E − ϕ m q − q − q + 2 q + 31 q ± ... ψ ϕ m q − q + 14 q + 2 q − q ± ... ψ ϕ ϕ b q − q − q + q + q ± ... ψ − ψ − ϕ m (1)10 q + 3 q − q + 8 q − q − q ± ... ψ E ψ − E ϕ + ϕ ϕ m (2)10 q − q + 11 q − q − q + q ± ... ψ − ψ E ψ − ϕ m q + 136 q − q + 7 q + 463 q ± ... ψ E ψ E ψ + 34784 E ψ ϕ − E ϕ Table 5.
Maass lifts in odd weight
Name Weight Input form P H P H P H P H P H b χ (8) q − χ (7) q + 4 χ (2) q + 10 χ (6) q − χ (1) q ± ... ψ ϕ m χ (8) q + 7 χ (7) q − χ (2) q − χ (6) q + 19 χ (1) q ± ... ψ − ψ ϕ b χ (2) q − χ (6) q − χ (1) q + χ (8) q + χ (7) q ± ... ψ − ψ E ψ
10 12 ϕ E ϕ − ϕ m χ (8) q + 19 χ (7) q − χ (2) q + 82 χ (6) q − χ (1) q ± ... ψ − ψ E ψ ϕ m χ (2) q + 2 χ (6) q + 8 χ (1) q − χ (8) q − χ (7) q ± ... ψ − E ψ
10 89803 E ψ + E ψ ϕ
11 18 E ϕ − E ϕ − ϕ + ϕ Table 6.
Borcherds products
Name Weight Divisor Graded-symmetric? b H + H yes b H + H + H yes b H + H yes b
24 11 H + H no heorem 10. The graded ring of symmetric Hermitian modular forms for O K is generated by Maass lifts E , b , E , m , m , b , m , b , m , m (1)10 , m (2)10 , m , m in weights , , , , , , , , , , , , . The ideal of relations is considerably more complicated than the analogous ideal for K = Q ( √−
7) so it isleft to an auxiliary file for convenience.
Proof.
We use induction on the weight. Any modular form of nonpositive weight is constant.Let F be any symmetric Hermitian modular form. Using the previous lemma we assume that F has azero along H . Then the pullbacks of F to H of all orders have at least a double zero along the diagonaland are therefore multiples of ψ .The pullbacks of E , E , m (2)10 , m to H generate the ring of even-weight Siegel modular forms of degreetwo. Moreover, the forms m (2)10 , m , b , m vanish along H and their quasi-pullbacks to H are scalarmultiples of ψ . By successively subtracting away from F expressions of the form { m (2)10 , m , b , m } · P ( E , E , m (2)10 , m )with appropriately chosen polynomials P , we may set the zeroth, first, second and third order pullbacks to H equal to zero while maintaining a zero on the divisor H .Therefore, we may assume that F has at least a fourth-order zero on H and a zero on H . Suppose F has exactly a fourth-order zero on H . (In particular, F has even weight.) Then the quasi-pullback Q F of F to H is an odd-weight paramodular form of level 3 with at least a fourth-order zero on the diagonal, soQ F is a multiple of ϕ and Q F/ϕ is contained in the ideal (cid:104) ϕ , ϕ , ϕ (cid:105) of symmetric paramodular formsof even weight with a zero on the diagonal. Then we can writeQ F = ϕ ϕ P + ϕ ϕ P + (cid:16) − E ϕ + 16 ϕ (cid:17) ϕ P for some even-weight symmetric paramodular forms P , P , P . Since m b , m b and m (1)10 b have fourth-order zeros on H and are zero on H with respective quasi-pullbacks ϕ ϕ , ϕ ϕ and ( − / E ϕ + ϕ / ϕ ,we can take any symmetric forms ˜ P , ˜ P , ˜ P whose pullbacks to H are P , P , P (some polynomials in E , E , m , m , b , m (1)10 , m , m will do) and subtract away b · (cid:16) m ˜ P + m ˜ P + m (1)10 ˜ P (cid:17) from F to obtain an even-weight form with (at least) a fourth-order zero on H and (at least) a double zeroon H .Suppose still that F has order exactly four on H . Then the quasi-pullback of F to H is a Siegel modularform of even weight with at least an fourth-order zero on the diagonal (due to the double zero of F on H )and is therefore a multiple of ψ . Since b has a fourth-order zero on H with quasi-pullback (up to scalarmultiple) ψ , and it also has a double zero along H , we may subtract away some expression of the form b P ( E , E , m (2)10 , m ) from F to obtain a modular form which vanishes to at least order 5 along H andwhich has at least a double zero on H .Now if F has order at least 5 along H and a zero on H , then the quotient F/b is holomorphic (byKoecher’s principle) and has lower weight, so F/b and therefore F is a polynomial expression in the gener-ators in the claim. (cid:3) Proposition 11.
The graded ring of Hermitian modular forms of degree 2 for Q ( √− is generated by thesymmetric generators of Theorem 10 and the holomorphic quotients h N = b m N b N , ≤ N ≤ and h N +3 = b b m N b N +15 , ≤ N ≤ . roof. In the even-weight case our goal is to reduce against the skew-symmetric Borcherds product b withdivisor div b = 11 H + H . To show that the pullbacks to H of odd orders 1 ≤ N ≤ − N with exactly an N th order zero on H (whose N th pullback must then bea multiple of ψ ), since we have already produced preimages of the even-weight Siegel modular forms. It iseasy to see that the quotients h N = b ( m /b ) N are holomorphic and have order 11 − N on H .We will reduce odd-weight skew-symmetric forms F to even-weight skew-symmetric forms by reducingagainst b . (The reduction against b as in the previous paragraph fails as there are no skew-symmetricmodular forms of weight 25.) First we force at least a fifth-order zero on H using the holomorphic forms h N +3 = b b m N b N +15 , ≤ N ≤ , which have a zero of order 8 − N on H and whose quasi-pullbacks must be scalar multiples of ψ . Thereforeby subtracting away expressions of the form { h , h , h } · P ( E , E , m (2)10 , m )we may assume that F has at least a sixth-order zero on H .Now the pullback of F to H is an skew-symmetric modular form of odd weight with at least a sixth-orderzero on the diagonal and is therefore contined in the ideal generated by ϕ ϕ f and ϕ ϕ f . Up to scalarmultiple these are exactly the pullbacks of h = b b b and h = b b m b to H . Since h and h bothvanish to order at least 5 on H , we subtract away some expression h P ( E , E , m (2)10 , m ) + h P ( E , E , m (2)10 , m )from F to obtain a form (again called F ) whose divisor contains 5 H + H and which is therefore divisibleby b . The quotient F/b is skew-symmetric of even weight so the previous case applies. (cid:3) Dimension formulas
The task of computing ideals of relations is much easier if dimension formulas for the spaces of modularforms are available (for one thing, such formulas make it clear when enough relations have been found togenerate the ideal). In principle the dimensions can always be calculated via a trace formula or Riemann-Roch theorem; however this is a rather lengthy computation which does not seem to appear explicitly in theliterature. In this section we observe that those dimensions can be read off almost immediately from themethod of proof in sections 4 and 5 above.Recall that the Hilbert series of a finitely generated graded C -algebra M = (cid:76) ∞ k =0 M k isHilb M = ∞ (cid:88) k =0 (dim M k ) t k ∈ Z [ | t | ] . Dimension formulas for K = Q ( √− . We will express the Hilbert series of dimensions of Hermitianmodular forms for Γ K = SU , ( O K ) in terms of the Hilbert series for Sp ( Z ) and the symmetric paramodulargroup K (2) + = (cid:104) K (2) , V (cid:105) of level 2. Recall that the latter series are ∞ (cid:88) k =0 dim M k (Sp ( Z )) t k = 1 + t (1 − t )(1 − t )(1 − t )(1 − t )and ∞ (cid:88) k =0 dim M symk ( K (2)) t k = (1 + t )(1 + t )(1 − t )(1 − t )(1 − t )(1 − t )corresponding to the ring decompositions M ∗ (Sp ( Z )) = C [ E , E , ψ , ψ ] ⊕ ψ C [ E , E , ψ , ψ ]and M sym ∗ ( K (2)) = C [ E , E , φ , φ ] ⊕ φ C [ E , E , φ , φ ] , M sym ∗ ( K (2)) = M sym ∗ ( K (2)) ⊕ φ M sym ∗− ( K (2)) . e first consider (graded-) symmetric even weight Hermitian modular forms. Write H even ( t ) = (cid:88) k even dim M symk (Γ K ) t k , H odd ( t ) = (cid:88) k odd dim M symk (Γ K ) t k . Although we reduce against the product b whose zero on the Heegner divisor H is simple, the proof ofTheorem 7 suggests that we consider both the zeroth and first order pullbacks there; so altogether we takethe tuple of pullbacks P = ( P H , P H , P H , P H ) : M sym ∗ (Γ K ) −→ M ∗ (Sp ( Z )) ⊕ S ∗ +2 (Sp ( Z )) ⊕ M sym ∗ ( K (2)) ⊕ S sym ∗ +1 ( K (2)) . Then we obtain the exact sequences0 −→ ker (cid:16) P H : M sym ∗− (Γ K ) → M sym ∗− ( K (2)) (cid:17) × b −→ M sym ∗ (Γ K ) P −→ im P −→ −→ ψ · (cid:16) M ∗− (Sp ( Z )) ⊕ M ∗− (Sp ( Z )) (cid:17) −→ im P −→ M sym ∗ ( K (2)) ⊕ M sym ∗ +1 ( K (2)) −→ , from which we obtain the Hilbert seriesHilb im P = t + t (1 − t )(1 − t )(1 − t )(1 − t ) + (1 + t ) (1 − t )(1 − t )(1 − t )(1 − t )and H even ( t ) = Hilb im P + t (cid:16) H odd ( t ) − (1 + t ) t (1 − t )(1 − t )(1 − t )(1 − t ) (cid:17) = t H odd ( t ) + 1 + t − t − t − t + t (1 − t )(1 − t )(1 − t )(1 − t )(1 − t ) . By reducing odd-weight symmetric forms against b we obtain the exact sequences0 −→ M sym ∗− (Γ K ) × b −→ M sym ∗ +1 (Γ K ) P =( P H ,P H ) −→ im P −→ −→ ψ · M ∗− (Sp ( Z )) −→ im P −→ M sym ∗ +1 ( K (2)) −→ H odd ( t ) = t H even ( t ) + t (1 − t )(1 − t )(1 − t )(1 − t ) + (1 + t ) t (1 − t )(1 − t )(1 − t )(1 − t ) . These equations resolve toHilb M sym ∗ (Γ K ) = H even ( t ) + H odd ( t )= 1 + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t − t − t (1 − t )(1 − t )(1 − t )(1 − t )(1 − t ) . Now we compute dimensions of spaces of (graded) skew-symmetric modular forms. For even-weight formsthe first, third and fifth order pullbacks to H yield an exact sequence0 −→ M sym ∗− (Γ K ) × b −→ M skew ∗ (Γ K ) ( P ,P ,P ) −→ S ∗ +1 (Sp ( Z )) ⊕ S ∗ +3 (Sp ( Z )) ⊕ S ∗ +5 (Sp ( Z )) −→ ∞ (cid:88) k =0 dim M skew k (Γ K ) t k = t + t + t (1 − t )(1 − t )(1 − t )(1 − t ) + t ∞ (cid:88) k =0 dim M sym k (Γ K ) t k . As for odd-weight skew-symmetric forms, we use the exact sequences0 −→ M skew ∗− (Γ K ) × b −→ M skew ∗ +1 (Γ K ) P =( P H ,P H ,P H ) −→ im P −→ −→ φ φ f M sym ∗− ( K (2)) −→ im P −→ M ∗ +1 (Sp ( Z )) ⊕ M ∗ +3 (Sp ( Z )) −→ o obtain ∞ (cid:88) k =0 dim M skew k +1 (Γ K ) t k +1 = t + t (1 − t )(1 − t )(1 − t )(1 − t ) + t (1 + t )(1 − t )(1 − t )(1 − t )(1 − t )+ t ∞ (cid:88) k =0 dim M skew k (Γ K ) t k , reducing the computation to the previous paragraph. Altogether we find ∞ (cid:88) k =0 dim M k (Γ K ) t k = P ( t )(1 − t )(1 − t )(1 − t )(1 − t )(1 − t )where P ( t ) = 1 + t + t + t + t + t + t + t + t + t + 2 t + t + t + 2 t − t + t . The table below lists dimensions for the full space of Hermitian modular forms; the subspace of graded-symmetric Hermitian modular forms; and the subspace of Maass lifts.
Table 7.
Dimensions for Q ( √− k M k (Γ K ) 0 0 0 1 0 1 1 2 1 3 2 4 2 5 4 8 5 10 8 13dim M symk (Γ K ) 0 0 0 1 0 1 1 2 1 3 2 4 2 5 4 8 5 10 8 13dim Maass k (Γ K ) 0 0 0 1 0 1 1 2 1 3 2 3 2 4 3 5 3 5 4 6 k
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40dim M k (Γ K ) 10 17 14 22 17 26 23 35 28 42 37 52 44 63 57 76 66 90 84 109dim M symk (Γ K ) 10 17 14 22 17 26 23 34 28 41 36 50 43 60 54 72 63 84 78 101dim Maass k (Γ K ) 4 7 5 7 5 8 6 9 6 9 7 10 7 11 8 11 8 12 9 136.2. Dimension formulas for K = Q ( √− . The procedure we use to compute Hilbert series of Hermitianmodular forms for the field Q ( √−
11) is mostly the same as the previous subsection. Here we need thecorresponding series for symmetric paramodular forms of level three: ∞ (cid:88) k =0 dim M symk ( K (3)) t k = 1 + t + t + t + t + t (1 − t )(1 − t ) (1 − t ) . (This can be derived from Corollary 5.6 of [5] or computed directly. We remark that the series presented in[5] do not agree with this because the definition of “symmetric” there is not graded-symmetric.)Again write H even ( t ) = (cid:88) k even dim M symk (Γ K ) t k , H odd ( t ) = (cid:88) k odd dim M symk (Γ K ) t k . Let P = ( P H , P H , P H , P H , P H ) denote the tuple of pullbacks P : M sym ∗ (Γ K ) → M ∗ (Sp ( Z )) ⊕ S ∗ +2 (Sp ( Z )) ⊕ S ∗ +4 (Sp ( Z )) ⊕ M sym ∗ ( K (3)) ⊕ S sym ∗ +1 ( K (3)) . Reducing graded-symmetric even-weight forms against b yields the exact sequences0 → ker (cid:16) P H : M sym ∗− (Γ K ) → M sym ∗− ( K (3)) (cid:17) × b −→ M sym ∗ (Γ K ) P −→ im P → , → ψ · (cid:16) (cid:77) k ∈{ , , } M ∗− k (Sp ( Z )) (cid:17) −→ im P −→ M sym ∗ ( K (3)) ⊕ M sym ∗ +1 ( K (3)) −→ , from which we obtainHilb im P = t + t + t (1 − t )(1 − t )(1 − t )(1 − t ) + 1 + 2 t + 2 t + t (1 − t )(1 − t ) (1 − t ) nd H even ( t ) = Hilb im P + t (cid:16) H odd ( t ) − t + t + t (1 − t )(1 − t ) (1 − t ) (cid:17) = t H odd ( t ) + 1 + 2 t + t − t − t − t − t − t + t (1 − t )(1 − t ) (1 − t )(1 − t ) . Similarly, the reduction of odd-weight symmetric forms against b through the tuple of pullbacks P =( P H , P H , P H ) yields the exact sequences0 −→ M sym ∗− (Γ K ) × b −→ M sym ∗ +1 (Γ K ) P −→ im P −→ −→ ψ · (cid:16) M ∗− (Sp ( Z )) ⊕ M ∗− (Sp ( Z )) (cid:17) −→ im P −→ M sym ∗ +1 ( K (3)) −→ , so H odd ( t ) = t + t (1 − t )(1 − t )(1 − t )(1 − t ) + t + t + t (1 − t )(1 − t ) (1 − t ) + t H even ( t ) . Altogether we findHilb M sym ∗ (Γ K ) = H even ( t ) + H odd ( t )= 1 + t + t + 2 t + 2 t + 2 t + t + t + t + t + t + t + t + t + t + t − t (1 − t )(1 − t ) (1 − t )(1 − t ) . For skew-symmetric modular forms we argue as in the previous subsection and find ∞ (cid:88) k =0 dim M skew k (Γ K ) t k = t + t + t + t + t (1 − t )(1 − t )(1 − t )(1 − t ) + t ∞ (cid:88) k =0 dim M sym k (Γ K ) t k and ∞ (cid:88) k =0 dim M skew k +1 (Γ K ) t k +1 = t + t + t (1 − t )(1 − t )(1 − t )(1 − t ) + t + t + t (1 − t )(1 − t ) (1 − t )+ t ∞ (cid:88) k =0 dim M skew k (Γ K ) t k , and altogether ∞ (cid:88) k =0 dim M k (Γ K ) t k = P ( t )(1 − t )(1 − t )(1 − t ) (1 − t )where P ( t ) = 1 + t + 2 t + 2 t + 2 t + t − t − t − t + t + 2 t + 2 t + t − t − t − t − t + t + 2 t + 2 t + 2 t + 2 t + t − t + t . The table below lists dimensions for the full space of Hermitian modular forms; the subspace of graded-symmetric Hermitian modular forms; and the subspace of Maass lifts.
Table 8.
Dimensions for Q ( √− k M k (Γ K ) 0 0 0 1 1 2 1 3 3 5 4 8 6 10 10 15 14 21 19 28dim M symk (Γ K ) 0 0 0 1 1 2 1 3 3 5 4 8 6 10 10 15 14 21 19 28dim Maass k (Γ K ) 0 0 0 1 1 2 1 3 3 4 3 5 4 6 5 7 6 8 6 9 k
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