Automorphic products that are singular modulo primes

We use Rankin--Cohen brackets on O(n, 2) to prove that the Fourier coefficients of reflective Borcherds products often satisfy congruences modulo certain primes.


Introduction
This note is inspired by the paper [10], in which it was observed that most of the Fourier coefficients of the (suitably normalized) Siegel cusp form Φ 35 of degree two and weight 35 are divisible by the prime p = 23.More precisely, if one writes a(T )e 2πiTr(T Z) , Z ∈ H 2 , the sum extending over positive-definite half-integral (2 × 2)-matrices T , then the main result of [10] is that (1.1) a(T ) ≡ 0 (mod 23) ⇒ det(T ) ≡ 0 (mod 23).
This has already been generalized in several ways.In [16], similar congruences are derived for Siegel cusp forms of higher weights.The papers [9,15] prove analogous results for Hermitian modular forms of degree two over the Gaussian and Eisenstein integers.The paper [13] considers quaternionic modular forms of degree two, while [1,12,14] consider Siegel modular forms of general degree.We call modular forms satisfying congruences of type (1.1) singular modulo p.
In this note, we start with the fact that the cusp form Φ 35 is a reflective Borcherds product [2,3,6,7], which in this situation means that it vanishes only on Humbert surfaces in the Siegel upper half-space that are fixed by transformations in the Siegel modular group.A natural generalization is to consider reflective Borcherds products on general orthogonal groups O(n, 2), with Siegel modular forms appearing through the exceptional isogeny from Sp 4 (R) to O (3,2).
It turns out that reflective Borcherds products on O(n, 2) with simple zeros and of weight k are very often singular modulo primes p dividing n/2 − 1 − k.In this note, we give a general argument to prove singularity modulo p that takes a set of two or more reflective Borcherds products and proves that some of them are singular modulo specific primes, using an identity based on the Rankin-Cohen brackets on O(n, 2).This argument requires almost no computation: the presence of congruences such as (1.1) for Φ 35 can be deduced from the location of its zeros.We also give similar arguments that can be used to prove that a single reflective product is singular modulo certain primes.
This note is organized as follows.In §2 we review reflective modular forms and define what it means for a modular form to be singular modulo a prime p.In §3 we introduce the Rankin-Cohen bracket on O(n, 2) and explain how to use it to derive modular forms that are singular modulo primes.In the last two sections we work out over 50 reflective Borcherds products that are singular modulo primes.In particular, for every prime p < 60, we construct at least one mod p singular modular form.

Reflective modular forms and singular modular forms modulo primes
Let L be an even integral lattice of signature (n, 2) with n ≥ 3, and let L R = L ⊗ R and L C = L ⊗ C. The Z-valued quadratic form on L is denoted by Q and the even bilinear form is Attached to the orthogonal group O(L C ) is the Hermitian symmetric domain D, the Grassmannian of oriented negative-definite planes in L R .This is naturally identified with one of the two connected components of by identifying [X + iY] ∈ P 1 (L C ) with the plane through X and Y.We denote by O + (L) the orthogonal subgroup that fixes both D and L.
Let Γ ≤ O + (L) be a finite-index subgroup and χ : Γ → C × a character.A modular form of integral weight k, level Γ and character χ is a holomorphic function F on the cone over D, that satisfies the functional equations Reflective modular forms were introduced by Borcherds [2] and Gritsenko-Nikulin [7] in 1998 and they have applications to generalized Kac-Moody algebras, hyperbolic reflection groups and birational geometry.The above definition is somewhat stronger than that of [7], where F is called reflective if the reflections corresponding to zeros of F lie in the larger group O + (L).Bruinier's converse theorem [4] shows that, in many cases, all reflective modular forms can be constructed through the multiplicative Borcherds lift [3,2].In this case, the Fourier series of a reflective form has a natural infinite product expansion in which the exponents are the Fourier coefficients of a modular form (or Jacobi form) for SL 2 , and we refer to it as a reflective Borcherds product.
To define modular forms that are singular at a prime p we have to work in the neighborhood of a fixed cusp.Suppose c ∈ L is a primitive vector of norm 0 and c ′ ∈ L ′ is an element of the dual lattice with c, c ′ = 1.Let L c,c ′ be the orthogonal complement of c and c ′ , i.e.
Attached to the pair (c, c ′ ) we have the tube domain which is one of the two connected components of the set On H c,c ′ , any modular form F can be written as a Fourier series in which the actual values of λ range over a discrete group depending on Γ and the character χ.To be more precise: there exists a sublattice K of L c,c ′ such that λ lies in the dual K ′ of K whenever a F (λ) = 0.By definition, the level of K is the smallest positive integer A non-constant modular form F is called singular (with respect to the pair (c, c ′ )) if its Fourier series on H c,c ′ is supported on vectors λ of norm zero.By analogy, we define singular modular forms modulo a prime p as follows: Definition 2.2.Let F be a non-constant modular form and p be a prime not dividing D F .The form F is called singular modulo p (at the cusp determined by (c, c ′ )) if its Fourier coefficients are all integers and if a F (λ) ≡ 0 (mod p) for all vectors λ for which Q(λ) is nonzero modulo p.
Remark 2.3.Using the Fourier-Jacobi expansion, it is not difficult to show that a modular form is singular if and only if its weight is k = n/2 − 1.In this case, it is singular at every cusp.
The notion of mod p singular modular forms also appears to be independent of the choice of cusps, and the weight appears to satisfy the similar constraint k ≡ (n/2 − 1) (mod p).
Unfortunately we do not have a proof of this.The converse is false: most modular forms of weight k ≡ (n/2 − 1) mod p fail to be singular modulo p.
Singularity with respect to (c, c ′ ) is closely related to the holomorphic Laplace operator.If e 1 , ..., e n is any basis of L c,c ′ with Gram matrix S, and z 1 , ..., z n are the associated coordinates on L c,c ′ ⊗ C, then define where s ij are the entries of S −1 .Note that ∆ is independent of the basis e i .
Applying ∆ e 2πi λ,Z = −Q(λ)e 2πi λ,Z , to the Fourier series termwise shows that the form F is annihilated by ∆ if and only if it is singular at (c, c ′ ).Similarly, if F has integral coefficients then F is singular modulo p if and only if here we recall that D F • ∆(F ) also has integral Fourier coefficients at the cusp (c, c ′ ) and that p does not divide D F by definition.The setting of [10], i.e.Siegel modular forms of degree two, corresponds to the case of the lattice L = 2U ⊕ A 1 i.e.Z 5 with Gram matrix 0 0 0 0 1 0 0 0 1 0 0 0 2 0 0 0 1 0 0 0 1 0 0 0 0 .If we work with c = (1, 0, 0, 0, 0) and c ′ = (0, 0, 0, 0, 1) then vectors (0, z 1 , z 2 , z 3 , 0) of H c,c ′ correspond exactly to matrices ( z 1 z 2 z 2 −z 3 ) in the Siegel upper half-space in a way that is compatible with the actions of O(3, 2) and Sp 4 (R), and the Laplace operator at (c, c ′ ) becomes (up to a scalar multiple) the theta-operator . See also Section 4.1 below.

The construction of singular automorphic products modulo primes
Let L be an even lattice of signature (n, 2) with n ≥ 3 that contains a primitive vector c of norm zero and a vector c ′ ∈ L ′ with c, c ′ = 1.The Laplace operator attached to the pair (c, c ′ ) is simply denoted ∆.Let Γ ≤ O + (L) be a modular group.Note that Γ satisfies Koecher's principle: the Baily-Borel compactification of D/Γ contains no cusps in codimension one.Lemma 3.1.For modular forms F of weight k and G of weight ℓ for Γ, the bracket Proof.Up to a scalar multiple, this is the first Rankin-Cohen bracket of F and G as defined by Choie and Kim [5].The assumption of [5] that the lattice L splits two hyperbolic planes is unnecessary.This lemma can also be proved directly by analyzing how ∆(F ) transforms under the modular group.In particular, it follows from [21, Lemma 2.4] that Since F is singular modulo p if and only if all Fourier coefficients of ∆(F ) vanish modulo p, we obtain the corollary: Corollary 3.2.Let p be a prime that divides the numerator of n 2 − 1 − k.Suppose G is a modular form of weight ℓ that is not identically zero modulo p. Suppose p does not divide ℓ and that p does not divide the numerator of n 2 − 1 − ℓ.The following are equivalent: (1) F is singular modulo p; (2) The cusp form [F, G] vanishes identically modulo p.Now suppose that F is a reflective modular form for Γ ≤ O + (L) with only simple zeros, and that G is a modular form for Γ that is non-vanishing on every zero r ⊥ of F .Since the associated reflection σ r is an involution and is contained in Γ, it follows that If G also happens to be a reflective modular form for Γ, with only simple zeros that are distinct from those of F , then the above argument shows that [F, G] is divisible by both F and G and therefore the quotient [F,G]  F G is a holomorphic modular form of weight two without character.Many groups Γ do not admit holomorphic modular forms of weight two.(For example, this is always true if n > 6, and it is usually true for Γ = O + (L) if the discriminant of L is reasonably small.)In these cases, we obtain [F, G] = 0 and therefore an integral relation among ∆(F G), ∆(F )G and F ∆(G).This is summarized below: Proposition 3.3.Let L be an even lattice of signature (n, 2) with n ≥ 3. Suppose F and G are reflective modular forms for Γ ≤ O + (L) of weights k and ℓ with simple and disjoint zeros, and that Γ admits no modular forms of weight two with trivial character.Then we have the identity In particular, (1) F is singular modulo every prime dividing defines a cusp form of weight 2 + N i=1 k i for Γ.This is also a special case of the Rankin-Cohen brackets defined in [5].The identity in Proposition 3.3 generalizes to an identity involving any number of reflective products; however, this does not appear to give any information not already obtained from considering the products in pairs.It was proved in [20] that every holomorphic Borcherds product of singular weight on L can be viewed as a reflective modular form, possibly after passing to a distinct lattice in L ⊗ Q.It is amusing that the notion of reflective modular forms plays a similar role for congruences.

Examples
In this section we use Proposition 3.3 to produce a number of examples of reflective Borcherds products on orthogonal groups of root lattices or related lattices that are singular modulo certain primes.The non-existence of modular forms of weight two in the nontrivial case of n ≤ 6 can be derived from [17,18,19], where the entire graded rings of modular forms were determined.
We denote by U the hyperbolic plane, i.e. the lattice Z 2 with Gram matrix ( 0 1 1 0 ).Let A n , D n , E 6 , E 7 and E 8 be the usual root lattices.For a lattice L and d ∈ N, we write L(d) to mean L with its quadratic form multiplied by the factor d.

4.1.
Siegel modular forms of degree two.When L is the lattice 2U ⊕ A 1 with n = 3, modular forms for O + (L) are the same as Siegel modular forms of degree two and even weight for the level one modular group Sp 4 (Z).Through this identification, rational quadratic divisors become the classical Humbert surfaces defined by singular relations.There are two equivalence classes of reflective divisors: (i) The Humbert surface of invariant one, which is represented by the set of diagonal matrices ( τ 0 0 w ) in H vanishes with simple zeros on the Humbert surface of invariant four.We calculate Proposition 3.3 and the non-existence of Siegel modular forms of weight two yields: (1) Ψ 5 is singular modulo p = 3; (2) Φ 30 is singular modulo p = 59; (3) Φ 35 = Ψ 5 Φ 30 is singular modulo p = 23.
4.2.Siegel paramodular forms of degree two and level 2 and 3. Section 4.1 gives the simplest example of a number of realizations of arithmetic subgroups of Sp 4 (Q) as orthogonal groups of lattices.When L = 2U ⊕ A 1 (t), modular forms for O + (L) are the same as Siegel paramodular forms of degree two and level t that are invariant under certain additional involutions.We will work out the congruences implied by Proposition 3.3 when t = 2 or t = 3.
Remark This implies that Φ 120 M 7 is singular modulo p = 31 and also that Φ 120 is singular modulo p = 13.Note that neither M 7 nor Φ 120 M 7 is a Borcherds product.We conclude with an example of a mod p singular Borcherds product that is not reflective and also has non-simple zeros.
Let L = 2U ⊕ D 11 and consider the following two Borcherds products for O + (L): (i) Ψ 1 , a meromorphic modular form of weight 1 which vanishes precisely with multiplicity 1 on hyperplanes r ⊥ with r ∈ L ′ and Q(r) = 1/2 and whose only singularities are simple poles along hyperplanes s ⊥ with s ∈ L ′ and Q(s) = 3/8; (ii) Φ 142 , a cusp form of weight 142 which vanishes precisely with multiplicity 1 on hyperplanes λ ⊥ with λ ∈ L and Q(λ) = 1, and with multiplicity 26 on hyperplanes s ⊥ with s ∈ L ′ with Q(s) = 3/8.
The form Φ 142 is the Jacobi determinant of the generators of a free algebra of meromorphic modular forms constructed in [18].The divisors r ⊥ and λ ⊥ are reflective, i.e. the associated reflections lie in O + (L).However, the divisors s ⊥ are not reflective.By analyzing its Taylor series along the divisor s ⊥ , we find that By comparing the residues, or leading terms in the Laurent series of both sides along s ⊥ , we find that c = 1950.Therefore: Theorem 5.3.
neither k nor ℓ.More generally, under these assumptions, F is singular modulo any prime p that divides n 2 − 1− k to a greater power than any of n 2 − 1 − ℓ and n 2 − 1 − k − ℓ, and similarly for G and F G. Remark 3.4.The bracket [−, −] can be generalized to any number of modular forms.Let F 1 , ..., F N be modular forms for Γ ≤ O + (L) of weights k 1 , ..., k N .Then [F 1 , ..., F N 2 ; (ii) The Humbert surface of invariant four, which is represented by the set of matrices ( τ z z τ ) with equal diagonal entries.Both reflective Humbert surfaces occur as the zero locus of a Borcherds product for O + (L): (a) The form Ψ 5 of weight k = 5, a square root of the Igusa cusp form of weight 10, vanishes with simple zeros on the Humbert surface of invariant one; (b) The quotient Φ 30 = Φ 35 /Ψ 5 of weight ℓ = 30, where Φ 35 is the cusp form of weight 35, [Φ 142 , Ψ 1 ] Φ 142 Ψ 1is a meromorphic modular form of weight 2 with trivial character for O + (L) whose only singularities are poles of multiplicity two along the hyperplanes s ⊥ .By the structure theorem of [18, Theorem 1.2], there is a constant c such that[Φ 142 , Ψ 1 ] = c • Φ 142 Ψ Ψ 1 )Φ 142 = cΦ 142 Ψ 3 1 .