Note on mod p property of Hermitian modular forms

arXiv:1601.03506v1 [math.NT] 14 Jan 2016

NOTE ON MOD p PROPERTY OF HERMITIAN MODULAR FORMS TOSHIYUKI KIKUTA AND SHOYU NAGAOKA

Abstract. Some congruence relations related to some Hermitian modular forms are given.

1. Introduction Serre [12] developed the theory of p-adic modular forms. In his theory, the d f played an important role. The notion of such Ramanujan operator θ : f 7−→ dq operator was extended to the case of Siegel modular forms in [2] and produced several interesting results in the mod p theory of Siegel modular forms (e.g. [6], [11]). The main purpose of this paper is to extend these results to the case of Hermitian modular forms. Let Γ2 (OK ) be the Hermitian modular group of degree two with respect to an imaginary quadratic number field K. We denote the Kronecker symbol of K by χK and the class number of K by hK . Krieg [7] constructed a weight k Hermitian modular form which coincides with the weight k Hermitian Eisenstein series for Γ2 (OK ) in the case hK = 1. We denote the modular form by Fk,K in this paper (cf. § 4.1). The first main result is stated as follows: Theorem 1. Assume that a prime number p > 3 satisfies χK (p) = −1 and hK 6≡ 0 (mod p). Then the Hermitian modular form Fk,K satisfies Θ(Fp+1,K ) ≡ 0

(mod p),

where Θ is the theta operator (for the precise definition see § 3.2). The notion of mod p singular modular form is defined by B¨ocherer and Kikua [1] in the case of Siegel modular forms. The notion is extended to the case of Hermitian modular forms (cf. § 3.3). The second main result is as follows: Theorem 2.√ Assume that p > 3 is a prime number such that p ≡ 3 (mod 4) and K = Q( −p). Then the modular form F p+1 ,K is a mod p singular Hermitian 2 modular form. 2. Hermitian modular forms In this section we briefly recall some basic properties of Hermitian modular forms. The Hermitian upper half-space of degree n is defined to be   1 t Hn := Z ∈ Mn (C) | (Z − Z) > 0 i 1

2

TOSHIYUKI KIKUTA AND SHOYU NAGAOKA

where t Z denotes the transpose, complex conjugate of Z. Let K be an imaginary quadratic number field of discriminant −DK . Denote the ring of integers by OK and the order of the unit group in OK by wK . Let χK stand for the attached Kronecker symbol. We also denote the class number of K by hK . The group    0 −1n Γn (OK ) := M ∈ M2n (OK ) | t M Jn M = Jn := 1n 0

is called the Hermitian modular group of degree n associated with K. We denote the vector space of Hermitian modular forms of weight k for Γn (OK ) by Mk (Γn (OK )). Each F ∈ Mk (Γn (OK )) possesses a Fourier expansion of the form X a(F ; H)exp(2πitr(HZ)), Z ∈ Hn , F (Z) = 0≤H∈Λn (OK )

where

Λn (OK ) :=

n

H = (hµν ) ∈ Mn (K) | t H = H, hµµ ∈ Z,

For a subring R ⊂ C, we set

p

−DK hµν ∈ OK

o

.

Mk (Γn (OK ))R := { F ∈ Mk (Γn (OK )) | a(F ; H) ∈ R (∀H ∈ Λn (OK )) }. 3. Congruences among Hermitian modular forms

3.1. Generalized q-expansion. We recall the Fourier expansion of F ∈ Mk (Γn (OK )): X a(F ; H)exp(2πitr(HZ)), Z ∈ Hn . F (Z) = 0≤H∈Λn (OK )

For simplicity, we use the abbreviation

q H := exp(2πitr(HZ)). P The generalized q-expansion F = a(F ; H)q H can be considered as an element in a formal power series ring C[[q]] (cf. [9], p.248), from which we note Mk (Γn (OK ))R ⊂ R[[q]]

for a subring R ⊂ C. For a prime number p, we denote the local ring at p by Z(p) , which is the ring P of p-integral rational numbers. For two elements Fi = a(Fi ; H)q H ∈ Z(p) [[q]] (i = 1, 2), we write that F1 ≡ F2 (mod p) if a(F1 ; H) ≡ a(F2 ; H) (mod p) holds for all H ∈ Λn (OK ). 3.2. Theta operator. The theta operator over C[[q[] is defined as X X Θ:F = a(F ; H)q H 7−→ Θ(F ) := a(F ; H) · det(H) q H .

In the case that n = 1, the theta operator is equivalent to the Ramanujan operator, which produces several interesting results (cf. [12]). It should be noted that Θ(F ) is not necessarily a Hermitian modular form even if F is. We fix a prime number p. If F ∈ Mk (Γn (OK ))Z(p) satisfies Θ(F ) ≡ 0 (mod p),

then F is called an element of the mod p kernel of the theta operator.

NOTE ON MOD p PROPERTY OF HERMITIAN MODULAR FORMS

3.3. Mod p singular Hermitian modular forms. Assume that F = Mk (Γn (OK ))Z(p) . If there is an integer r (r < n) such that

P

3

a(F ; H)q H ∈

a(F ; H) ≡ 0 (mod p) holds for all H ∈ Λn (OK ) with rank(H) > r, then F is called the mod p singular Hermitian modular form. It is obvious that, if F is a mod p singular Hermitian modular form, then Θ(F ) ≡ 0 (mod p). 4. Main theorems This section considers the mod p properties of a Hermitian modular forms of degree two. 4.1. Krieg’s result. Given a prime q dividing DK define the q-factor χq of χK (cf. [8], p.80). Then χK can be decomposed as Y χq . χK = q|DK

We set aDK (ℓ) :=

Y

(1 + χq (−ℓ)).

q|DK

Let DK = mn with coprime m, n. We set Y ψm := χq ,

ψ1 := 1.

q:prime q|m

For H ∈ Λ2 (OK ) with H 6= O2 , we define

ε(H) := max{ℓ ∈ N | ℓ−1 H ∈ Λ2 (OK )}.

Krieg’s result is stated as follows: Theorem 4.1. (Krieg [7]) Asuume that k ≡ 0 (mod wK ) and k > 4. Then there exists a modular form Fk,K ∈ Mk (Γ2 (OK )) whose Fourier coefficient a(Fk,K ; H) is given by    X DK · det(H) 4k(k − 1)  k−1  d GK k − 2; if H > 0,   B · Bk−1,χK d2   0 3 is a prime number such that χK (p) = −1 and hK 6≡ 0 (mod p). Let a(Fk,K ; H) be the Fourier coefficient of Fk,K at H, If det(H) 6≡ 0 (mod p), then (∗)

a(Fp+1,K ; H) ≡ 0

(mod p).

Proof. By Theorem 4.1, the Fourier coefficient a(Fp+1,K ; H) is expressed as   X DK · det(H) 4(p + 1)p dp GK p − 1; a(Fp+1,K ; H) = Bp+1 · Bp,χK d2 0 0. First we look at the factor A :=

4(p + 1)p . Bp+1 · Bp,χK

By Kummer’s congruence relation, we obtain B2 1 Bp+1 ≡ = (mod p) • p+1 2 12 •

−2hK Bp,χK (mod p). ≡ (1 − χK (p))B1,χK = (1 − χK (p)) p wK

NOTE ON MOD p PROPERTY OF HERMITIAN MODULAR FORMS

5

Since p > 3, χK (p) = −1, and hK 6≡ 0 (mod p), the factor A is a p-adic unit. Next we shall show that, if det(H) 6≡ 0 (mod p), then the factor B :=

X

dp GK

0