Ramanujan J (2016) 39:61–82 DOI 10.1007/s11139-015-9697-5

Ternary quadratic forms and Heegner divisors Tuoping Du1

Received: 27 March 2014 / Accepted: 15 April 2015 / Published online: 13 June 2015 © Springer Science+Business Media New York 2015

Abstract In this paper, we use the Siegel–Weil formula and the Kudla’s matching principle to prove some interesting identities between representation numbers (of ternary quadratic space) and the degree of Heegner divisors. Keywords Representation number · Heegner divisor · Shimura curve · Siegel–Weil formula · Kudla’s matching principle Mathematics Subject Classification

11G15 · 11F41 · 14K22

1 Introduction Fourier coefficients of some Eisenstein series have geometric meaning. In [10], Kudla obtained a useful matching principle which provides an identity between two genus theta series from two different quadratic spaces as both are special values of the same Eisenstein series. This simple identity connects two different quadratic spaces and also gives arithmetic and geometric interpretations of the coefficients. In this paper, we use this principle to prove some new identities on ternary quadratic spaces and relate the representation numbers for lattices in positive definite spaces and degrees of Heegner divisors in Shimura curves.

The author was partially supported by NSFC (Nos. 71006901), NSFC (Nos. 11171141), NSFJ (Nos. BK2010007), PAPD and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No.708044), and NSFC (11326052).

B 1

Tuoping Du [email protected] Department of Mathematics, Northwest University, Xi’an 710127, People’s Republic of China

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Let D be a square-free positive integer, and let B = B(D) be the unique quaternion algebra of discriminant D over Q, i.e., B is ramified at a finite prime p if and only if p|D. The reduced norm is denoted by det in this paper. Let V (D) = {x ∈ B(D) | tr(x) = 0} with the restriction quadratic form det, where tr is the reduced trace. For a positive integer N prime to D, let L D (N ) = O D (N ) ∩ V (D), where O D (N ) is an Eichler order in B of conductor N . We can view L = (L D (N ), det) as an even integral lattice in V (D). The quaternion B is definite if and only if D has an odd number of prime factors. When V (D) is positive definite, there is a very interesting but hard question to compute the representation number (for a positive integer m) r L (m) = |{x ∈ L D (N ) : det x = m}|.

(1.1)

However, there is an available method to compute the average over the genus gen(L), which we denote by ⎛



r D,N (m) = rgen(L) (m) = ⎝

L 1 ∈gen(L)

⎞−1 1 ⎠ | Aut(L 1 )|

 L 1 ∈gen(L)

r L 1 (m) . | Aut(L 1 )|

(1.2)

Indeed, it is a product of local densities, first discovered by Siegel in 1930s—Siegel’s formula [22]. These densities are computable (see for example [23,24]). When V (D) is indefinite, the representation number does not make sense anymore since a number can be represented by infinitely many vectors in L(D). The number is related to the degree of a Heegner divisor in a Shimura curve, which we will now describe. Fix an embedding i : B(D) → M2 (R) such that B(D)× is invariant under the automorphism x → x ι = t x −1 of GL2 (R). Let 0D (N ) = O D (N )1 be the group of (reduced) norm 1 elements in O D (N ) and let X 0D (N ) = 0D (N )\H be the associated Shimura curve. For a positive integer m, let Z D,N (m) be the Heegner divisor in X 0D (N ) associated to the lattice L D (N ) which we will define in Sect. 4. The degree 

deg Z D,N (m) = 2

x∈0D (N )\L D (N )m

1 , | x |

(1.3)

where L D (N )m = {x ∈ L D (N )| det(x) = m} and x = {γ ∈ 0D (N )|γ · x = x} is the stabilizer subgroup of x in 0D (N ). 1 −2 y dx ∧ dy be the normalized differential form on X 0D (N ), and let Let  = 2π vol

123





X 0D (N ), 

=

X 0D (N )



(1.4)

Ternary quadratic forms and Heegner divisors

63

be the volume of X 0D (N ) with respect to , which is a positive rational number (see (4.2)). Finally, we define the normalized degree (when D has an even number of prime factors) r D,N (m) =

deg Z D,N (m) . vol(X 0D (N ), )

(1.5)

So r D,m has two distinct meanings in this paper depending on whether D has an even or odd number of prime factors. When D has an odd number of prime factors, r D,N (m) denote average representation numbers (1.2), and when D has an even number of prime factors, r D,N (m) denote normalized degrees (1.5). In this paper, we study the relations between different r D,m using Kudla’s matching principle, following the idea in [2]. The main difference is that we need to deal with half integral modular forms and locally metaplectic covering of SL2 , instead of integral modular forms and local Weil representation of SL2 . It makes the local calculation and matching at 2 more technical and complicated. One of the main theorems, involving the same type of r D,m (all are representation numbers or degrees of Heegner divisors), is Theorem 1.1 Let D be a square-free positive integer, let p = q be two different odd primes not dividing D, and let N be a positive integer prime to Dpq. Then −

q +1 2 p+1 2 r Dp,N (m) + r Dp,N q (m) = − r Dq,N (m) + r Dq,N p (m) q −1 q −1 p−1 p−1

for every positive integer m. Remark 1.2 If p or q is 2, then one side of the above equality is a sum of five representation numbers or degrees, which is deduced from Proposition 3.3. In their work [1], Berkovich and Jagy obtained a similar result s( p 2 n) − ps(n) = 48

 f˜∈TG1, p

R f˜ (n) | Aut( f˜)|

− 96

 f˜∈TG2, p

R f˜ (n) | Aut( f˜)|

,

(1.6)

where s(n) is the representation number of x 2 + y 2 + z 2 = n, TG1, p , TG2, p are two genera of quadratic forms, and Aut( f˜) is the automorphism group of quadratic form f˜. Indeed, two terms on the right side are average representation numbers of the respective genera of quadratic forms. The work of Berkovich and Jay is concrete and they compute local density explicitly which is different from that of ours. Katsurada and Schulze-Pillot studied the action of Hecke operator on genus theta functions [8] and obtained some very interesting formulas between different genus theta functions. Essentially, these identities should also be examples of Kudla’s matching principle as pointed out in ([8, Sect. 6]) (at least for D > 1). When the space V (D) is indefinite, Theorem 1.1 is geometric and can be restated as follows.

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Corollary 1.3 Let D be a square-free positive integer with an odd number of prime factors. Let p = q be two different odd primes not dividing D, and let N be a positive integer prime to Dpq; then −2 deg Z Dp,N (m) + deg Z Dp,N q (m) = −2 deg Z Dq,N (m) + deg Z Dq,N p (m). At the end of this paper, we will give some examples of this result. Theorem 1.1 is actually a consequence of the following theorem which deals with two different types of r D,m . Theorem 1.4 Let D > 1 be a square-free positive integer, let p  D be an odd prime, and let N be a positive integer prime to Dp. Then r Dp,N (m) = −

2 p+1 r D,N (m) + r D,N p (m). p−1 p−1

Notice that, in this paper, we do not consider the case when D = 1. By Theorem 1.4, representation numbers of a positive ternary quadratic form are closely related to degrees of Heegner divisors in a Shimura curve. So one type of quantity can be used to compute the other type, see Example 5.2. This paper is organized as follows: In Sect. 2, we recall the Weil representation and Kudla’s matching principle in the general case. In Sect. 3, we prove some explicit local matchings between Schwartz functions on two local quadratic spaces and also construct global matching pairs. In Sect. 4, we first study the definite case and show that r D,N (m) is the m-Fourier coefficient of some theta integral. Then we introduce the Shimura curve and Heegner divisors. In Sect. 5, we prove that Fourier coefficients of the theta integral are connected to the degrees of Heegner divisors for the indefinite space. At the end, we prove Theorems 1.1, 1.4, and Corollary 1.3. We end this paper with some examples.

2 Preliminaries This section mainly follows Kudla’s work [10]. Let (V, Q) be a non-degenerate quadratic space over Q, and let G = SL2 . Fix the unramified canonical additive character ψ:

A/Q → C× , ψ∞ (x) = e2πi x ,

A be a double cover of SL2 (A). We identify G

A = where A is the Adele ring of Q. Let G SL2 (A) × {±1} using Rao’s normalized cocycle c(g1 , g2 ) [20], where multiplication on the right is given by [g1 , 1 ][g2 , 2 ] = [g1 g2 , 1 2 c(g1 , g2 )].

A the full inverse image in G

A . For subgroup P of G, we denote by P

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Ternary quadratic forms and Heegner divisors

65

In particular, we have subgroups N (A) = {n = [n(b), 1]|b ∈ A},  1 b n(b) = , 1 and

M(A) = {m = [m(a), ε]|a ∈ A× , ε = ±1},  a . m(a) = a −1 Let χ = χV be the quadratic character of A× /Q× associated with V defined by χ (x) = (x, (−1)m(m−1)/2 det(V )), where m = dim(V ), (, ) is the Hilbert symbol, and det(V ) ∈ Q× /Q×,2 is the determinant of the matrix for the quadratic form Q on V . χ determines a character χ ψ on

M(A) by χ ψ ([m(a), ε]) = εχ (a)γ (a, ψ)−1 , where γ (a, ψ) is the Weil index ([10]).

The group G(A) acts on the Schwartz space S(V (A)) via the Weil representation ω = ωψ determined by our fixed additive character ψ of A/Q, and this action commutes with the linear action of O(V )(A). The G˜ A -action is determined by (see for example [9]) ω((n(b), 1))ϕ(x) = ψ(bQ(x))ϕ(x), m ω((m(a), ε)ϕ(x) = χ ψ (a, ε) | a | 2 ϕ(ax),  ω((w, 1))ϕ = γ (V ) ϕ = γ (V ) V (A) ϕ(y)ψ((x, y))dy, 

(2.1)

, dy is the Haar measure on V (A) self-dual with respect to −1  ψ((x, y)), and γ (V ) = p γ (V p ) = 1, where γ (V p ) is an 8-th root of unity associated to the local Weil representation at place p (local Weil index). Let P = N M be the standard Borel subgroup of SL2 , where N and M are subgroups of n(b) and m(a), respectively.

A consisting of For s ∈ C, let I (s, χ ) be the principal series representation of G

smooth functions (s) on G A such that  ψ if m is odd, χ (m(a))|a|s+1 (g , s) (2.2) (n(b)m(a)g , s) = if m is even. χ (m(a))|a|s+1 (g , s) where w =

1

A intertwining map There is a G λ = λV : S(V (A)) → I (s0 , χ ),

(2.3)

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λ(ϕ)(g ) = ω(g )(0), ˆ in G(A).

A . Let K

∞ K

be the full inverse image of S O(2) × SL2 (Z) where g ∈ G A



section (s) ∈ I (s, χ ) is called standard if its restriction to K ∞ K is independent of

A N K



A = M s. By the Iwasawa decomposition G A ∞ K , the function λ(ϕ) ∈ I (s0 , χ ) has a unique extension to a standard section (s) ∈ I (s, χ ), where (s0 ) = λ(ϕ).

A , h ∈ O(V )(A), and ϕ ∈ S(V )(A), the associated theta series is For g ∈ G defined as θ (g , h; ϕ) =



ω(g )ϕ(h −1 x).

x∈V (Q)

For an algebraic group G over Q, we write [G] = G(Q)\G(A). The theta integral I (g , ϕ) =

1 vol([O(V )])

[O(V )]

θ (g , h, ϕ)dh

(2.4)

˜ is an automorphic form on G(Q)\ G˜ A if it is convergent. Recall the Eisenstein series 

E(g , s; ) = E(g , s; ϕ) =

(γ g , s).

(2.5)

Q)\G(

Q) γ ∈ P(

It is absolutely convergent for Re(s) > 1 and has a meromorphic continuation to the whole s-plane with finitely many poles [17]. The following is the well-known Siegel–Weil formula ([15,16], and [21, Chapter 4]). Theorem 2.1 (Siegel–Weil formula) Assume that V is anisotropic or that dim(V ) − r > 2, where r is the Witt index of V, so that the theta integral is absolutely convergent. Then (1) E(g , s; ) is holomorphic at the point s0 = m/2 − 1, where m = dim(V ), and E(g , s0 ; ) = κ I (g , ϕ), where κ = 2 when m ≤ 2 and κ = 1 otherwise. (2) If m > 1, then E(g , s0 ; ) = κ I (g , ϕ) =

κ 2

[S O(V )]

where dh is Tamagawa measure on S O(V )(A).

123

θ (g , h; ϕ)dh,

Ternary quadratic forms and Heegner divisors

67

Let V (1) , V (2) be two quadratic spaces with the same dimension and the same quadratic character χ , and are represented as follows: S(V (1) (A)) P PλP V (1) PP S(V (2) (A))

λV (2)   

PP q P 1 I (s0 , χ )     .

(2.6)

There are analogous local maps λ p : S(V p ) → I p (s0 , χ p ). Following Kudla [10], we make the following definition. (i)

(i)

Definition 2.2 For a prime p ≤ ∞, ϕ p ∈ S(V p ), i = 1, 2, are said to be matching if (2) λV (1) (ϕ (1) p ) = λV (2) (ϕ p ). p

ϕ (i) =

 p

p

(i) ϕ (i) p ∈ S(V (A)) are said to be matching if they match at each prime p.

By the Siegel–Weil formula, we have the following Kudla’s matching principle [10, Sect. 4]: Under the assumption of Theorem 2.1 for both V (1) and V (2) , one has the following identity for any matching pair (ϕ (1) , ϕ (2) ): I (g , ϕ (1) ) = I (g , ϕ (2) ).

(2.7)

It implies that their Fourier coefficients are equal. Comparing the coefficients of both sides, we will derive the main results of this paper.

3 Matchings on quadratic spaces There are two quaternion algebras over a local field Q p , the matrix algebra B sp = M2 (Q p ) (split quaternion) and the division quaternion B ra (ramified quaternion). Let sp

B0 = {x ∈ B sp | tr(x) = 0},

B0ra = {x ∈ B ra | tr(x) = 0},

sp

and let V sp = B0 and V ra = B0ra be the associated three-dimensional quadratic spaces with reduced norm. Both spaces have the same quadratic character χ p = (x, −1) p .

Q -intertwining operators for V = V sp or V ra We have G p λ : S(V ) → I (1/2, χ p ), λ(ϕ)(g ) = ω(g )ϕ(0). Here G Q p is the metaplectic double cover of G(Q p ).

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T. Du sp

We will use superscript sp and ra to indicate the association with B0 and B0ra , respectively. It is known [10, Sect. 4] that λsp is surjective. So every function ϕ ra in S(V ra ) has a matching function. The purpose of this section is to give some explicit matching pairs and to obtain some interesting local and global identities. 3.1 The finite prime case p < ∞ We assume p < ∞ in this section. Let O B ra be the maximal order in B ra , which consists of all elements of B whose reduced norm is in Z p . We drop the subscript p for simplicity in this section. Let k be the unique unramified quadratic field extension of Q p , and let Ok = Z p + Z p u be the ring of integers of k with u ∈ Ok× . Fix one optimal embedding k → B ra , then there is a uniformizer π of B ra such that πr = r¯ π for r ∈ k and π ι = −π and π 2 = p (see [5]). Then one has O B ra = Ok + Ok π = Z p + Z p u + Z p π + Z p uπ. Let L sp = M2 (Z p )



V sp and 

L ra =

 O B ra V ra  (Z p + 2O B ra ) V ra

if p is odd, if p = 2.

There is a sublattice of L sp   b   sp L 1 = A = ac −a ∈ L sp : c ≡ 0(mod p) . The dual lattices are given by  sp,

L1

=

a

b c −a



∈ V sp : c ∈ Z p , b ∈

 1 1 Zp, a ∈ Zp , p 2

and  L ra, =

  O B ra V ra  (Z p + 2O B ra ) V ra

if p is odd, if p = 2,

where   O B ra

=

1 ra π OB π ra 4 OB

if p is odd prime, if p = 2,

and L  = {x ∈ V : (x, L) ⊂ Z p }.

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Ternary quadratic forms and Heegner divisors

69

So one has isomorphisms  O B ra /O B ra ∼ L ra, /L ra ∼ = (Z/ pZ)2 , = (Z/ pZ)2 if p is odd prime,  O B ra /O B ra ∼ = (Z/2Z)2 ⊕ (Z/4Z)2 L ra, /L ra ∼ = Z/4Z ⊕ (Z/8Z)2 , if p = 2.

We denote ϕ ra = char(L ra ), ϕ ra, = char(L ra, ),

sp

sp

ϕ sp = char(L sp ), ϕ1 = char(L 1 ),

sp,

and ϕ1

(3.1)

sp,

= char(L 1 ).

Lemma 3.1 [14, Lemma 14.3] For the character ψ, the Weil index γ (V ra ) = −1,

γ (V sp ) = 1, when p = 2,

and γ (V ra ) = −ζ8−1 ,

γ (V sp ) = ζ8−1 , when p = 2.

The following are local matching pairs which are needed in this paper: Proposition 3.2 Let p be an odd prime and other notations be as above; then p+1 sp −2 sp sp p−1 ϕ + p−1 ϕ1 ∈ S(V ). sp, 2 p sp with p−1 ϕ − p+1 ∈ S(V sp ). p−1 ϕ1

(1) ϕ ra ∈ S(V ra ) matches with (2) ϕ ra, ∈ S(V ra ) matches Proof (1) Since

SL2 (Z p ) = K 0 ( p) ∪ N (Z p )wK 0 ( p), one has a splitting homomorphism  SL2 (Z p ) → SL 2 (Z p ), k → k = [k, 1]. Let K0 ( p) = {[k, 1] | k ∈ K 0 ( p)} and w = [w, 1], where   ab K 0 ( p) = ∈ SL2 (Z p ) : c ≡ 0(mod p) . cd The dimension of K0 ( p)-invariant subspace I (1/2, χ p )K0 ( p) is 2, since  ∈ I (1/2, χ p )K0 ( p) is determined by (1) and (w ).

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Notice that K 0 ( p) is generated by m(a),n(b), and n − (c) = w −1 n(−c)w, where a ∈ Z× p , b ∈ Z p , and c ∈ pZ p , so K0 ( p) is generated by [m(a), 1], [n(b), 1], and sp [n − (c), 1]. Then one can check that ϕ ra , ϕ sp , ϕ1 are all K0 ( p)-invariant for the Weil representation. We only check ω([n − (−c), 1])ϕ ra = ϕ ra and leave others to the reader. One has ωra (w )ϕ ra (x) = γ (V ra ) vol(L ra )ϕ ra, (x). So ωra ([n(−c), 1]w )ϕ ra (x) = γ (V ra ) vol(L ra )ψ p (−c det(x))ϕ ra, (x) = γ (V ra ) vol(L ra )ϕ ra, (x), i.e., ωra ([n(−c), 1]w )ϕ ra = ωra (w )ϕ ra . Then ωra ([n − (c), 1])ϕ ra = ωra (w ,−1 )ωra ([n(−c), 1]w )ϕ ra = ϕ ra . Notice that w ,−1 = [w −1 , 1] when p is odd and w ,−1 = [w −1 , −1] when p is 2 [6]. It is easy to see ωra ([m(a), 1])ϕ ra = ϕ ra and ωra ([n(b), 1])ϕ ra = ϕ ra , so as claimed λra (ϕ ra ) ∈ I (1/2, χ p )K0 ( p) . sp Now we have λra (ϕ ra ), λsp (ϕ sp ), λsp (ϕ1 ) ∈ I (1/2, χ p )K0 ( p) . Direct calculation gives λra (ϕ ra )(1) = 1, λra (ϕ ra )(w ) = γ (V ra ) p −1 , λsp (ϕ sp )(1) = 1, λsp (ϕ sp )(w ) = γ (V sp ), λsp (ϕ1 )(1) = 1, λsp (ϕ1 )(w ) = γ (V sp ) p −1 . sp

sp

From Lemma 3.1, one has λra (ϕ ra ) =

−2 sp sp p + 1 sp sp λ (ϕ ) + λ (ϕ1 ). p−1 p−1

Then we get (1). Claim (2) is similar and is left to the reader. One just needs to replace K 0 ( p) by K 0+ ( p)

 ab = ∈ SL2 (Z p ) : cd

 b ≡ 0(mod p) .  

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Ternary quadratic forms and Heegner divisors

71

When p = 2, one has SL2 (Z p ) = K 0 (16) ∪ [n − (2), 1]K 0 (16) ∪ [n − (4), 1] ×K 0 (16) ∪ [n − (8), 1]K 0 (16) ∪ N (Z p )wK 0 (16),

(3.2)

where   ab K 0 (16) = ∈ SL2 (Z p ) : c ≡ 0(mod 16) . cd   ab Let ∈ SL2 (Z p ) : a ≡ d ≡ 1(mod 4), c ≡ 0(mod 16) , which is = cd a subgroup of K 0 (16) with index 2. Let K 0 (16)

  b  sp L 2 = A = ac −a ∈ L sp :    sp b ∈ L sp : L 3 = A = ac −a    sp b ∈ L sp : L 4 = A = ac −a    sp b ∈ L sp : L ul = A = ac −a    sp b ∈ L sp L even = A = ac −a

 c ≡ 0(mod 4) ,  c ≡ 0(mod 8) ,  c ≡ 0(mod 16) ,  a ≡ 0(mod 2) ,

 : a ≡ b ≡ c ≡ 0(mod 2) .

We also denote sp

sp

sp

sp

ϕ2 = char(L 2 ), ϕ3 = char(L 3 ),

sp

sp

sp

sp

sp

sp

ϕ4 = char(L 4 ), ϕul = char(L ul ), and ϕeven = char(L even ). Proposition 3.3 Let notations be as above and p = 2; then ϕ ra ∈ S(V ra ) matches with 5 3 sp 8 sp 2 sp sp − ϕ sp + ϕ1 + 4ϕ4 − ϕul + ϕeven ∈ S(V sp ). 2 2 3 3

Proof From the group decomposition (3.2), one knows that dim I (1/2, χ p )K0 (16) = 5, sp sp sp sp where K0 (16) = {[k, 1] | k ∈ K 0 (16)}. We have a basis {ϕ sp , ϕ1 , ϕ4 , ϕul , ϕeven } for this subspace. It is easy to check that ϕ ra ∈ I (1/2, χ p )K0 (16) as in Proposition 3.2. In order to get the matching pair, it suffices to compute the values of these five functions at places 1, [n − (2), 1], [n − (4), 1], [n − (8), 1], and w . 5

λra (ϕ ra )(w ) = γ (V ra )2− 2 ,

λra (ϕ ra )([n − (2), 1]) = 0,

λra (ϕ ra )([n − (4), 1]) = 0,

λra (ϕ ra )([n − (8), 1]) = γ (V ra )2 ζ4 ,

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T. Du 1

λsp (ϕ sp )(w ) = γ (V sp )2− 2 ,

λra (ϕ sp )([n − (2), 1]) = 0,

λsp (ϕ sp )([n − (4), 1]) = γ (V sp )2 ζ4 ,

λsp (ϕ sp )([n − (8), 1]) = γ (V sp )2 ζ4 ,

3

λsp (ϕ1 )(w ) = γ (V sp )2− 2 , sp

sp

λsp (ϕ1 )([n − (2), 1]) = 0,

sp

sp

λsp (ϕ1 )([n − (4), 1]) = γ (V sp )2 ζ4 ,

λsp (ϕ1 )([n − (8), 1]) = γ (V sp )2 ζ4 ,

5

λsp (ϕ2 )(w ) = γ (V sp )2− 2 , sp

sp

λsp (ϕ2 )([n − (2), 1]) = 0,

sp

sp

λsp (ϕ2 )([n − (4), 1]) = γ (V sp )2 ζ4 ,

λsp (ϕ2 )([n − (8), 1]) = γ (V sp )2 ζ4 ,

7

λsp (ϕ3 )(w ) = γ (V sp )2− 2 ,

λsp (ϕ3 )([n − (2), 1]) = 0,

λsp (ϕ3 )([n − (4), 1]) = γ (V sp )2 2−1 ζ4 ,

λsp (ϕ3 )([n − (8), 1]) = γ (V sp )2 ζ4 ,

sp

sp

sp

sp

9

λsp (ϕ4 )(w ) = γ (V sp )2− 2 ,

λsp (ϕ4 )([n − (2), 1]) = 0,

λsp (ϕ4 )([n − (4), 1]) = γ (V sp )2 2−2 ζ4 ,

λsp (ϕ4 )([n − (8), 1]) = γ (V sp )2 2−1 ζ4 ,

sp

sp

sp

sp

3

λsp (ϕul )(w ) = γ (V sp )2− 2 , sp sp

λsp (ϕul )([n − (2), 1]) = γ (V sp )2 2−4 ζ8 ζ4 , sp

λsp (ϕul )([n − (4), 1]) = γ (V sp )2

1+i ζ4 , 23

sp

λsp (ϕul )([n − (8), 1]) = 0.

7

λsp (ϕeven )(w ) = γ (V sp )2− 2 , sp sp

λsp (ϕeven )([n − (2), 1]) = γ (V sp )2 2−2 ζ8 ζ4 , sp

λsp (ϕeven )([n − (4), 1]) = γ (V sp )2

1+i ζ4 , 2

sp

λsp (ϕeven )([n − (8), 1]) = 0.

So we have 5 3 8 2 sp sp sp sp λra (ϕ ra ) = − λsp (ϕ sp ) + λsp (ϕ1 ) + 4λsp (ϕ4 ) − λsp (ϕul ) + λsp (ϕeven ). 2 2 3 3   Remark 3.4 The coset of subgroup K 0 (16) in K 0 (16) is {1, m(3)}, so dim I (1/2, χ p )K0 (16) is 5. 3.2 The case p = ∞ In this section, we consider the case Q p = R and recall a matching pair given in [10]. Notice that B ra in this case is the Hamilton division algebra, and V ra has signature ra (x) = e−2π det(x) ∈ S(V ra ); then ϕ ra is of weight 3/2 in the sense (3, 0). Let ϕ∞ ∞ ω

ra

ra (kθ )ϕ∞

=e

3 2 iθ



ra ϕ∞ ,

cos θ sin θ kθ = , kθ = [kθ , 1]. − sin θ cos θ

On the other hand, Kudla and Millson constructed a family of weight 3/2 Schwartz sp functions ϕ∞ ∈ S(V sp ) in [12], and we follow [10, Sect. 4]. Recall that V sp = {x ∈ M2 (R) | tr(x) = 0}.

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Given an orthogonal decomposition, V sp = V + ⊕ V − , x = x + + x − ,

(3.3)

with V + of signature (1, 0) and V − of signature (0, 2). One defines (Kudla used the notation ϕ(x, ˜ z)) ϕ∞ (x, V − ) = −(4π(x + , x + ) − 1)e−π(x sp

+ ,x + )+π(x − ,x − )

.

Kudla proved the following lemma [10, Lemma 4.7]. ra , ϕ (x, V − )) is a matching Lemma 3.5 For any orthogonal decomposition (3.3), (ϕ∞ ∞ sp

3

2 given pair, and their (same) image in I (1/2, χ∞ ) is the unique weight 3/2 section ∞ by 3

3

3

2 ∞ (n(b)m(a)kθ ) = |a| 2 χ∞ (a)e 2 iθ .

Because of this matching pair, we will simply write ϕ∞ for ϕ∞ ( , V − ). sp

sp

3.3 Global matching Let D1 , D2 > 1 be two square-free integers, and let V (Di ) be the ternary quadratic spaces associated to the quaternion algebras B(Di ) over Q (with reduced norm as the quadratic form), i = 1, 2. The following global matching result is clear from Kudla’s matching principle (2.7), Proposition 3.2, and Lemma 3.5. Proposition 3.6 Assume that ϕ (i) = conditions: (i)

 p

(i)

ϕ p ∈ S(V (Di )(A)) satisfy the following

sp

ra depending on whether V (D ) is split or non(1) When p = ∞, ϕ∞ is ϕ∞ or ϕ∞ i ∞ split. (2) When p  D1 D2 ∞ or p|gcd(D1 , D2 ), we identify V (D1 ) p = V (D2 ) p and take (1) (2) any ϕ p = ϕ p ∈ S(V (D1 ) p ). sp (3) When p|lcm(D1 , D2 ) but p  gcd(D1 , D2 ), one of V (Di ) p is V p and the other (1) (2) one is V pra , we take (ϕ p , ϕ p ) to be a matching pair in Propositions 3.2 and 3.3.

Then (ϕ (1) , ϕ (2) ) is a global matching pair, and

I (g , ϕ (1) ) = I (g , ϕ (2) ), g ∈ G(A). Recall that an Eichler order of conductor N denoted by O D (N ) is an order of B(D) such that: (1) When p|D, O D (N ) p := O D (N ) ⊗Z Z p is the maximal order in the division quaternion algebra B(D) p = B ra p.

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(2) When p  D∞, there is an identification B(D) p ∼ = M2 (Q p ) under which O D (N ) p :=

  ab ∈ M2 (Z p ) : c ≡ 0 mod N . cd

Considering the lattice L D (N ) = O D (N )



V (D) in the space V (D), we know

(1) When p|D, L D (N ) p := Ł D (N ) ⊗Z Z p is a sublattice of the maximal order in ra B(D) p = B ra p , denoted by L p . (2) When p  D∞,   ab L D (N ) p := x = ∈ M2 (Z p ) : tr(x) = 0, c ≡ 0 cd

 mod N ,

sp

denoted by L p . Let p,q be different odd primes, V (1) = V (Dp), V (2) = V (Dq). Define ϕ (1) = (1) (1) (A)) as follows: l ϕl ∈ S(V



(1)

ϕl

⎧ ra sp ϕ∞ (ϕ∞ ) ⎪ ⎪ ⎪ ⎨char(L sp ) l = ra ⎪ ϕ ⎪ l ⎪ ⎩ −2 sp l+1 sp l−1 ϕl + l−1 ϕl,1 sp

if l = ∞, D has an even (odd) number primes, if l  Dpq, if l|Dp, if l = q is odd,

sp

where ϕl , ϕl,1 , and ϕlra are the functions defined in (3.1) with added subscript l. One has (1)

ϕf =

−2 q +1  char( L char( L Dp (N q)), Dp (N )) + q −1 q −1

where L Dp (N ) = L Dp (N )  char( L Dp (N q)). So I (τ, ϕ (1) ) =



char is characteristic function, and so also is

Z Z,

−2 q +1 I (τ, L Dp (N )) + I (τ, L Dp (N q)), q −1 q −1

(3.4)

where I (τ, ϕ (1) ) will be introduced in the next section. Let ϕ (2) be defined as ϕ (1) with the roles of p and q switched. Then ϕ (1) and ϕ (2) form a matching pair by Proposition 3.2. So Proposition 3.6 implies I (τ, ϕ (1) ) = I (τ, ϕ (2) ), that is

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75

Proposition 3.7 Let D be a square-free positive integer, let p = q be two different odd primes not dividing D, and let N be a positive integer prime to Dpq. Then q +1 −2 I (τ, L Dp (N )) + I (τ, L Dp (N q)) q −1 q −1 p+1 −2 I (τ, L Dq (N )) + I (τ, L Dq (N p)). = p−1 p−1

4 Representations numbers and Heegner divisors 4.1 Definite quadratic space and representations numbers In this section, we show a general fact about positive definite quadratic forms for the convenience of the readers. We will see that Fourier coefficients of the theta integrals associated to definite quadratic space are closely related to representation numbers over genus. Let (V, Q) be a positive definite quadratic space of dimension m. Define the Gaussian ϕ∞ (x) = e−2π Q(x) ∈ S(V (R)). Then we know

m

ϕ∞ (hx) = ϕ∞ (x), ω(kθ )ϕ∞ = e 2 iθ ϕ∞ for h ∈ O(V )(R) and kθ ∈ SO2 (R) ⊂ SL2 (R). ˆ define the theta kernel For any ϕ f ∈ S(Vˆ ), where Vˆ = V ⊗Z Z, m

θ (τ, h, ϕ f ϕ∞ ) = v − 4 θ (gτ , h, ϕ f ϕ∞ ), m which is a holomorphic √ modular form of weight 2 for some congruence subgroup. √ Here gτ = n(u)m( v) for τ = u + iv ∈ H, gτ = (gτ , 1), n(u), and m( v) are introduced in Sect. 2. So m

I (τ, ϕ f ϕ∞ ) = v − 4 I (gτ , ϕ f ϕ∞ ) is also a modular form of weight m2 . For an even integral lattice L in V , we let ˆ ∞ ), θ (τ, L) = θ (τ, char( L)ϕ

ˆ ∞ ), I (τ, L) = I (τ, char( L)ϕ

(4.1)

ˆ Notice that two lattices L 1 and L 2 in V are in the same class if where Lˆ = L ⊗Z Z. there is h ∈ O(V )(Q) such that h L 1 = L 2 . Two lattices L 1 and L 2 are in the same ˆ such that genus if they are equivalent locally everywhere, i.e., there is h ∈ O(V )(Q)

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ˆ h L 1 = L 2 . The group O(V )(A) acts on the set of lattices as follows: h L = (h f L)∩V , where h f is the finite part of h = h f h ∞ . Let gen(L) be the genus of L (the set of all equivalence classes of lattices in the same genus of L). Then from the above discussion, there is a bijective map O(V )(Q)\O(V )(A)/K (L)O(V )(R) ∼ = gen(L), [h] → h L , ˆ The following result is well where K (L) is the stabilizer subgroup of Lˆ in O(V )(Q). known for experts: Proposition 4.1 [7, Sect. 5.1] Let r L (n) = |{x ∈ L : Q(x) = n}|, ⎞−1   1 r L (n) ⎠ rgen(L) (n) = ⎝ , |O(L )| |O(L )| ⎛

L ∈gen(L)

L ∈gen(L)

where O(L) is the stabilizer of L in O(V ). Then, for q = e(τ ), θ (τ, h, L) = I (τ, L) =

∞ 

rh L (n)q n ,

n=0 ∞ 

rgen(L) (n)q n .

m=0

In particular, the modular form I (τ, L) is a genus theta function. 4.2 Shimura curve and Heegner divisors In this section, we assume that D > 0 has an even number of prime factors, and then V = (V (D), det) is of signature (1, 2) and is anisotropic when D > 1. According to [10, Theorem 4.23], the theta integral I (g, ϕ) is a generating function of degrees of some divisors with respect to the tautological line bundle over the Shimura curve associated to V . In this paper, these divisors are Heegner divisors in Shimura curves. Let H = GSpin(V ). It is known that there is an isomorphism H ∼ = B × . The action on V is explicit, g.v = gvg −1 , g ∈ B × , v ∈ V. There is an exact sequence 1 → Gm → H → SO(V ) → 1. Let D be the Hermitian domain of oriented negative 2-planes in V (R), and L = {w ∈ VC = V (C) : (w, w) = 0, (w, w) ¯ < 0}.

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77

H (R) acts naturally on both. The map f : L/C× ∼ = D, w = u + iv → R(−u) + Rv gives an H (R)-equivariant isomorphism between L/C× and D. So L is a (tautological) line bundle over D. The Hermitian domain also has a tube representation which we need. Indeed, we know     x 2 −y 2 + R y −2x y . H ∪ H = D, z = x + i y → R −x 0 −y −1 x Thus under this identification, the action of H(R) on D becomes the usual linear fractional action. ˆ is a Shimura curve X K over Associated to a compact open subgroup K of H (Q) Q such that ˆ . X K (C) = H (Q)\D × H (Q)/K Moreover, L descends to a line bundle on X K , which we continue to denote by L. In this section, we always assume ˆ K = Oˆ D (N )× ⊂ H (Q), which preserves the lattice L D (N ), where O D (N ) is the Eichler order of conductor ˆ = H (Q)K , one has N. By the Strong Approximation theorem H (Q) X K = X 0D (N ). 1 −2 y dx ∧ dy be the differential form on X 0D (N ); then from [14, (2.7)] Let 0 = 2π and [18, Lemma 5.3.2], we know D vol(X 0 (N ), ) := 0 = −2[O1D : 0D (N )]ζ D (−1) X 0D (N )

=

 DN  1 (1 + p −1 ) (1 − p −1 ) ∈ Z, 6 6 p|N

(4.2)

p|D

 where ζ D (s) = pD (1 − p −s )−1 is the partial zeta function, and O D is a maximal order of B containing O D (N ). ˆ x⊥ Recall the Kudla cycle on X K . Fix a x ∈ V (Q) with det(x) > 0 and h ∈ H (Q), is a subspace of signature (0, 2) and defines a sub-Shimura variety Z (x) of X h K h −1 . Its right translation by h gives a divisor Z (x, h) in X K . Let ϕ f ∈ S(Vˆ ) K and m ∈ Q>0 . If there is a x0 ∈ V (Q) such that det(x0 ) = m, we define the associated Kudla cycle Z (m, ϕ f ) [11] as Z (m, ϕ f ) =

r 

ϕ f (h −1 j x 0 )Z (x 0 , h j ),

j=1

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where ˆ : det x = m} = Supp(ϕ f ) ∩ {x ∈ V (Q)

r 

K h −1 j x0 .

j=1

Otherwise, we define Z (m, ϕ f ) = 0. Let L be a lattice in V and L m = {x ∈ L| det(x) = m}. By the Strong Approxˆ = H (Q)K . So we have the decomposition with imation theorem, one has H (Q) Q(x0 ) = m, Lˆ m =



K h −1 j x0 ,

where h j ∈ H (Q). Then we know Lm =



 K h −1 j x0 =



 K x j , x j = h −1 j x0 ∈ L ,

where  K = K ∩ H (Q), and Z (m, ϕ f ) =



Z (x, h j ) =



j

Z (h −1 j x0 ) =

j



Z (x j ).

j

where Z (x j ) are Heegner cycles in this paper. Define deg Z (m, ϕ f ) = 2

 x∈ K \L m

1 , | x |

and Z D,N (m) := Z (m, char( L D (N )), where x is the stabilizer subgroup of x in  K . Let r D,N (m) =

deg Z D,N (m) vol(X 0D (N ), )

be as in the introduction.

5 Main results In Sect. 4, we introduced the number r D,N for both the definite and indefinite spaces V (D). These numbers are related to the coefficients of theta integral for both spaces as follows:

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79

Proposition 5.1 Let D > 1 for quadratic space V (D), and letting ϕ f char( L D (N )), one has

=

3

I (τ, ϕ f ϕ∞ ) = v − 4 I (gτ , ϕ f ϕ∞ ) = sp

sp

∞ 

r D,N (m)q m ,

m=0

where r D,N (0) = 1, for m > 0. Proof When V (D) is definite, this is Proposition 4.1. When V (D) is indefinite, write sp

I (τ, ϕ f ϕ∞ ) =

∞ 

c(m)q m .

m=0

By [10, Theorem 4.23], [4, Theorem 3.4], and [13, Theorem 4.1], one has c(0) = 1 and for m > 0, c(m) =

deg Z D,N (m) . vol(X K , )

So c(m) = r D,N (m) as claimed.

 

Proof of Theorem 1.1 and Corollary 1.3: Theorem 1.1 now follows from Proposition 3.7 and Proposition 5.1. From this theorem, the definition of normalized degree of Heegner divisors (1.5), and the volume formula (4.2), one obtains the corollary easily. Proof of Theorem 1.4 : Let V (1) = V (D) and V (2) = V (Dp), and let ϕ (i) =  (i) (i) l ϕl ∈ S(V (A) be as follows. For l  p∞, we identify L D (N )l with L Dp (N )l (i) and denote ϕl = char(L D (N )l ). If D has an odd number of primes, let (1) ra (2) ϕ∞ = ϕ∞ , ϕ∞ = ϕ∞ . sp

On the other hand, if D has an even number of primes, let (1) (2) ra ϕ∞ = ϕ∞ , ϕ∞ = ϕ∞ . sp

Finally, let ϕ (1) p =−

2 p + 1 sp sp ra ϕp + ϕ , ϕ (2) p = ϕp . p−1 p − 1 p,1

Then ϕ (1) and ϕ (2) match by the results in Sect. 3. So one has by Proposition 3.6 I (τ, ϕ (1) ) = I (τ, ϕ (2) ). Comparing the mth coefficients of the both sides, and applying Propositions 5.1 and 4.1, one obtains Theorem 1.4.

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We will give two examples of Corollary 1.3, for details see [19, section 4.2]. Example 5.2 Firstly, we consider the case D > 1 with an odd number of prime factors. Let p, q be different odd primes, and let m, N be positive integers. We assume that: (1) ( pq, √ √ N D) = 1 and (N , D) = 1, (2) Z[ −m] is the ring of integers of the imaginary quadratic field √ Q( −m), (3) p, q and prime factors of N and D are all split in the field Q( −m). Then this example √ is related to quaternion algebras B( p D) and B(q D). For any order O,  in Q( −m), B( p D) respectively, consider the optimal embedding  √ √ φ : Q( −m) −→ B( p D) such that φ(Q( −m))  = φ(O). √ Now we take the special orders O = Z[ −m],  = O p D (N ) and denote the lattice L p D (N ) by L. √ For √ x ∈ L m , consider the embedding i x : Z[ −m] −→ O p D (N ) such that i x ( −m) = x. Consider the set √ √ E(Z[ −m], O p D (N )) = {the optimal embedding i x o f Z[ −m] in O p D (N )}. √ From the assumption 2), E(Z[ −m],√ O p D (N )) consists of optimal embeddings. We define an action of B( p D)× on E(Z[ −m], O p D (N )) in the following way: i xσ := σ −1 i x σ, for σ ∈ B( p D)× . pD

Let  = 0 (N ), and we call i x and i x -equivariant if there exists a σ ∈  such that i x = i xσ . There is a natural surjective map: √ i : L m  E(Z[ −m], O p D (N ))/ . It is easy to check that i induces a bijective map √ i : L m /  −→ E(Z[ −m], O p D (N ))/ .

(5.1)

√ The number | L m /  |=| E(Z[ −m], O p D (N ))/  | is given by [3, Chapter 4]. For any x ∈ L m , define the stabilizer subgroup x = {σ ∈  | σ (x) = x}. √ σ ∈ x ⇐⇒ σ (x) = x ⇔ σ xσ −1 = x ⇒ σ ∈ Q( −m). √ √ × × Since √ σ ∈ , we know that σ ∈ Z[ −m] , where Z[ −m] is the unit group of Z[ −m]. deg Z p D,N (m) = 2

 x∈\L m

123

| Lm/  | 1 =2 . √ | x | | Z [ −m]× |

(5.2)

Ternary quadratic forms and Heegner divisors

81

Following the result [19, Theorem 4.19], we know that √ | E(Z[ −m], O p D (N ))/  |= 2t h Q(√−m) ,

(5.3)

where t is the number of prime factors of N and h Q(√−m) is the class number of √ 2t+1 h √−m) . Q( −m). We then obtain deg Z p D,N (m) = |Z [√Q( −m]× | Similarly, we have deg Z p D,N q (m) = and deg Z q D,N p (m) = Then we could get

2t+2 h Q(√−m) 2t+1 h Q(√−m) (m) = , deg Z , √ √ q D,N | Z [ −m]× | | Z [ −m]× |

2t+2 h Q(√−m) √ . |Z [ −m]× |

−2 deg Z Dp,N (m) + deg Z Dp,N q (m) = 0 = −2 deg Z Dq,N (m) + deg Z Dq,N p (m). Example 5.3 We assume that D > 1 with √ an odd number of prime √ factors, N > 0, and (N , D) = 1. Fix a quadratic field Q( −m) such that Z[ −m] is the ring of integers of this field. Let √ √ s =| { p | D : p is inert in Q( −m)} |, t =| { p | N : p splits in Q( −m)} | . Two odd primes p, q are chosen such that ( pq, N D) = 1. Then from [19, Corollary 4.20], we have  2t+1+s+a p h Q(√−m) 1 , where a p = deg Z p D,N (m) = √ 0 | Z [ −m]× |

√ if p is inert in Q( −m) if otherwise,

similarly for other degrees. So we have another example of Corollary 1.3. Remark 5.4 The above two examples are special cases of Corollary 1.3, for which we give another proof. This optimal embedding method is useful for simple cases but not for general cases. Acknowledgments This paper was inspired by Kudla’s matching principle. The author thanks him for his influence. The author thanks Tonghai Yang for his good suggestion, useful discussion, and his encouragement. The author also thanks the referees for their constructive suggestions and careful listing of the typos in early version. The author is grateful to the Mathematical Science Center of Tsinghua University, for providing him a good opportunity to visit and a good research environment in the summer 2013.

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