SUMS OF CLASS NUMBERS AND MIXED MOCK MODULAR ...

SUMS OF CLASS NUMBERS AND MIXED MOCK MODULAR FORMS KATHRIN BRINGMANN AND BEN KANE Abstract. In this paper,
we consider sums of class numbers of the type P 2 m≡a (mod p) H 4n − m , where p is an odd prime, n ∈ N, and a ∈ Z. By showing that these are coefficients of mixed mock modular forms, we obtain explicit formulas. Using these formulas for p = 5 and 7, we then prove a conjecture of Brown et al. in the case that n = ` is prime.

1. Introduction and statements of results Let H(n) denote the nth Hurwitz class number, i.e., the number of equivalence classes of positive definite binary quadratic forms of discriminant −n with the class containing x2 + y 2 weighted by 12 and the class containing x2 + xy + y 2 weighted 1 by 13 . Moreover, by convention H(0) = − 12 . Certain congruence classes appear as coefficients of weight 32 Eisenstein series. However, the generating function for all Hurwitz class numbers X (1.1) H(q) := H(n)q n n≥0

is not itself modular, but rather mock modular [9]. Roughly speaking, this means that H may be naturally “completed” to a non-holomorphic modular form (a further description is given in Section 2). Mock modular forms have since shown up in a variety of applications. To name a few examples, Ramanujan’s mock theta functions have been shown to be mock modular forms [19], they have led to asymptotic and exact formulas in partition theory [3, 5], they are related to Lie superalgebras [1, 4, 15], and they are connected to the quantum theory of black holes [2, 7]. Date: October 24, 2013. 2010 Mathematics Subject Classification. 11E41, 11F37, 11F30. Key words and phrases. class numbers, Lerch sums, mixed mock modular forms, mock theta functions, modular forms. The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation. This research was completed while the second author was a postdoc at the University of Cologne. 1

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KATHRIN BRINGMANN AND BEN KANE

In this paper, we prove conjectures reminiscent of the famous identity (cf. p. 154 of [8]) X
(1.2) H 4` − m2 = 2`, √ |m|d

n≥1

Moreover, we let SN,r denote the operator f |SN,r (τ ) = sieves coefficients congruent to r modulo N .

P

n∈Z

a(N n + r)q N n+r which

Theorem 1.4. For every a ∈ Z and odd prime p, we have that X 2 (1.4) (H(q)ϑa,p (τ )) U (4) ⊗ χp + Gp,a+b (q) Sp,a2 −b2 b (mod p) b6≡±a (mod p)

is a weight 2 holomorphic modular form on Γ0 (p2 ) ∩ Γ1 (p). Moreover, if a = 0, then (1.4) is a weight 2 modular form on Γ0 (p2 ). This paper is organized as follows. In Section 2, we recall Hirzebruch and Zagier’s completion of the class number generating function and introduce important series known as Appell-Lerch sums. In Section 3, we show how to complete the functions on either side of (1.4) to obtain non-holomorphic modular forms. We conclude the paper by proving Theorem 1.4 and Theorem 1.3 in Section 4. 2. Hurwitz class numbers, Lerch sums, and known facts As mentioned in the introduction, the generating function (1.1) for the Hurwitz class numbers is a mock modular form. More precisely, Hirzebruch and Zagier [9] proved

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KATHRIN BRINGMANN AND BEN KANE

that one can complete H by adding a certain simple sum involving the incomplete Gamma function Z ∞ ts−1 e−t dt. Γ (s; y) := y

The completed function is non-holomorphic, but belongs to a special class of functions known as harmonic weak Maass forms. To define these, we require the weight k hyperbolic Laplacian (denoting τ = x + iy with x, y ∈ R throughout)  2    ∂ ∂ ∂ ∂2 2 ∆k := −y + iky +i . + ∂x2 ∂y 2 ∂x ∂y Weight k harmonic weak Maass forms for Γ ⊂ SL2 (Z) are real analytic functions F : H → C satisfying the following properties: (1) F|k γ (τ ) = F (τ ) for every γ ∈ Γ, (2) ∆k (F) = 0, (3) F has at most linear exponential growth at each cusp of Γ. Here |k is the usual weight k slash operator. The class number generating function also belongs to a distinguished subspace, consisting of those forms whose nth coefficient 1 (in the Fourier expansion in x) vanishes unless (−1)k− 2 n ≡ 0, 1 (mod 4), known as Kohnen’s plus space [10]. We collect the modularity properties of H in the following theorem which can be easily concluded from Theorem 2 of [9]. Theorem 2.1. The function   1 X 1 1 2 2 b H(τ ) := H(q) + √ nΓ − ; 4πn y q −n + √ 2 8π y 4 π n>0

(2.1) is a weight

3 2

harmonic weak Maass form on Γ0 (4) in Kohnen’s plus space.

A number of other completions help us to prove Theorem 1.4. To this end, for u ∈ C \ (Zτ + Z) we define the multivariable Appell function [15, 18] (2.2)

πi`u

A` (u, v; τ ) := e

X (−1)`n q 2` n(n+1) e2πinv n∈Z

1 − e2πiu q n

.

Directly from the definition, we have (−1)` A` (u + 1, v; τ ) = A` (u, v + 1; τ ) = A` (u, v; τ ), (2.3)

`

A` (u + τ, v + `τ ; τ ) = (−1)` q − 2 e2πiv A` (u, v; τ ).

SUMS OF CLASS NUMBERS AND MIXED MOCK MODULAR FORMS

5

In order to add a non-holomorphic function which “completes” A` to satisfy modularity, we define X 1 2 ϑ (z; τ ) := eπin τ +2πin(z+ 2 ) , n∈ 12 +Z

R (u; τ ) :=

X 

 sgn (n) − E

n∈ 12 +Z

Z E (z) := 2

z

Im(u) n+ y



p 2y



1

2 τ −2πinu

(−1)n− 2 e−πin

,

2

e−πt dt.

0

We note that for u 6= 0, we also have the useful formula  ! −πu2 e 1 1 + √ Γ − ; πu2 . (2.4) E(u) = sgn(u) 1 − π|u| 2 2 π Theorem 2.2 of [18] yields the transformation properties of the completion of A` : b` (u, v; τ ) := A` (u, v; τ ) (2.5) A     `−1 i X 2πiku `−1 `−1 + e ϑ v + kτ + ; `τ R `u − v − kτ − ; `τ . 2 k=0 2 2 b` satisfies Theorem 2.2. The function A   πic v aτ + b u 2 b b` (u, v; τ ) . (2.6) A` , ; = (cτ + d) e cτ +d (−`u +2uv) A cτ + d cτ + d cτ + d Moreover, for every m1 , m2 , n1 , n2 ∈ Z, we have b` (u + n1 τ + m1 , v + n2 τ + m2 ; τ ) (2.7) A `

2

b` (u, v; τ ) . = (−1)`(n1 +m1 ) e2πiu(`n1 −n2 ) e−2πin1 v q 2 n1 −n1 n2 A 3. Mixed mock modular forms and non-holomorphic completions In this section, we show how to add a non-holomorphic function to “complete” each of the functions defined in the introduction to obtain a non-holomorphic modular form. In order to write down these completions, for N ∈ N it is helpful to define the function   X 1 2 2 Ra,N (τ ) := |n|Γ − ; 4πn y q −n . 2 n∈Z n≡a

(mod N )

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KATHRIN BRINGMANN AND BEN KANE

Note that Ra,N only depends on a (mod N ) and R−a,N = Ra,N . We further define the unary theta functions (3.1)

X

ϑa,N (τ ) := m≡a

2

qm .

m∈Z (mod N )

3.1. In this subsection, we complete (H(q)ϑ2a,p (τ )) U (4) ⊗ χ2p . Lemma 3.1. For every a, b ∈ Z and odd prime p, the function (3.2)

1 (H(q)ϑ2a,p (τ )) U (4) ⊗ χ2p + √ 8 π

X

1 X

R2b+kp,2p

τ 

b (mod p) k=0 b6≡±a (mod p)

4

ϑ2a+kp,2p

τ  4

ϑ2a,2p τ4 + √ 4π y




satisfies weight 2 modularity for Γ := Γ0 (p2 ) ∩ Γ1 (p). Proof. By Proposition 2.1 of [16], ϑ2a,p is modular of weight 21 on Γ0 (4p2 ) ∩ Γ1 (p). We first write X (3.3) (H(q)ϑ2a,p (τ )) U (4) ⊗ χ2p = (H(q)ϑ2a,p (τ )) U (4) Sp,r . 1≤r≤p−1 2 b By Lemma 1 of [11], (H(q)ϑ 2a,p (τ ))|U (2) fulfills weight 2 modularity on Γ0 (2p ) ∩ b Γ1 (2p). Since the nth coefficient of (H(q)ϑ 2a,p (τ ))|U (2) is zero unless n is even, Lemma b 4 of [11] implies that (H(q)ϑ2a,p (τ ))|U (4) satisfies weight 2 modularity on Γ0 (p2 ) ∩ Γ1 (p). Moreover, since modular forms on Γ1 (p) split into modular forms on Γ0 (p) with Nebentypus, rewriting χ2p = 1 − U (p)V (p) (where as usual f |V (d)(τ ) := f (dτ )), Lemma 1 of [11] implies that the level goes down with U (p) and back up with V (p), so that overall the group becomes Γ. By Theorem 2.1, to complete each summand (H(q)ϑ2a,p (τ )) U (4) Sp,r on the righthand side of (3.3), one must add (3.4)     X X 1 1 2 2 2  1  2 m −n m nΓ − ; 4πn y q + √ q  U (4) Sp,r .  √ 2 8π y 4 π n>0 m≡2a

(mod p)

m≡2a

(mod p)

SUMS OF CLASS NUMBERS AND MIXED MOCK MODULAR FORMS

7

Due to m ≡ 2a (mod p) and the congruences implied by U (4) and Sp,r , the congruence conditions on n and m are equivalent to m ≡ n (mod 2), m ≡ 2a (mod p), and n ≡ ±2b (mod p), where b satisfies a2 − b2 ≡ r (mod p). If no such b exists, then (3.4) equals zero. We may thus assume that such a b exists. Whenever b 6≡ 0 (mod p), the fact that p is odd implies that (3.4) equals ! 1 X     X 1 τ τ √ R±2b+kp,2p ϑ2a+kp,2p . 4 4 8 π k=0 ± Moreover, in the case that b ≡ 0 (mod p), (3.4) equals !
1 τ  τ  X ϑ2a,2p τ4 1 √ Rkp,2p ϑ2a+kp,2p + √ . 4 4 4π y 8 π k=0 To finish the proof, one then sums over all choices of b to obtain (3.2). P 3.2. In this section, we complete ± Gp,a±b (q)|Sp,a2 −b2 . We begin with a lemma, which gives a more useful form (` ∈ Z) for     πi`v `2 1 d − 2 . R`,p (v; τ ) := e p q 4 R p`τ − v − ; 2p τ dv 2 v=0



Lemma 3.2. If p is an odd prime and ` ∈ Z satisfies −p < ` ≤ p, then 2 !  √ X ` 2 2 ` 1 δ`=p ` n + Γ − ; 4πp2 n + R`,p (v; τ ) = π y q −p (n+ 2p ) + √ . 2p 2 2p p y 1 n∈ 2 +Z

Proof. We have   πi`v `2 1 2 − e p q 4 R p`τ − v − ; 2p τ 2     X  ` 2 ` Im(v) √ −p2 (n+ 2p ) e2πiv(n+ 2p` ) . =i sgn(n) − E 2p n + − y q 2 2p 2p y 1 n∈ 2 +Z

` 2p

6= 0, we have   ` . sgn(n) = sgn n + 2p

We now note that whenever n +

Differentiating and plugging in (2.4) yields the claim, after a straightforward calculation. 

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KATHRIN BRINGMANN AND BEN KANE

In order to complete the right-hand side of (1.4), we pair the terms b and −b and determine the associated completion. Lemma 3.3. Suppose that a, b ∈ Z and p is an odd prime with a 6≡ ±b (mod p). (1) If b 6≡ 0 (mod p), then the function Gbp,a,b (q) :=

X ±

1 τ  τ  1 X Gp,a±b (q) Sp,a2 −b2 − √ ϑkp+2a,2p Rkp+2b,2p 4 4 4 π k=0

satisfies weight 2 modularity for Γ. (2) If b ≡ 0 (mod p), then the function
1 τ τ  τ  ϑ X 1 2a,2p 4 Rkp,2p ϑkp+2a,2p − Gbp,a,0 (q) := Gp,a (q) Sp,a2 − √ √ 4 4 4π y 8 π k=0 satisfies weight 2 modularity for Γ. (3) In the case that a ≡ 0 (mod p), the function X b Gp,0,b Sp,−b2 0d

(pn + (a + b)) q (pn+(a+b))(pn+p+(a−b)+pm)

n≥0 m≥0

+

X

(pn − (a + b)) q (pn−(a+b))(pn+(b−a)+pm)

n≥1 m≥0

=

X (pn + (a + b)) q (pn+(a+b))(pn+p+(a−b)) n∈Z

1 − q p(pn+(a+b))

  
2πi(a+b)v a2 −b2  p d 2 p = A2 (a + b) pτ, v + 2apτ ; p τ e q . 2πi dv v=0

SUMS OF CLASS NUMBERS AND MIXED MOCK MODULAR FORMS

9

The analogous calculation for −b follows by (2.3). A similar calculation yields a uniform equation for b ≡ 0 (mod p) and a 6≡ 0 (mod p) as well. Overall, we obtain   
2πi(b±a)v a2 −b2  p d 2 . Gp,a±b (q) Sp,a2 −b2 = A2 (b ± a) pτ, v ± 2apτ ; p τ e p q 2πi dv v=0 We next prove the modularity of   
2πi(a+b)v a2 −b2  p d 2 bb (τ ) := b2 (a + b) pτ, v + 2apτ ; p τ e p q G . A 2πi dv v=0

(3.5) α β γ δ




∈ Γ, we use equation (2.6) followed by (2.7) with m1 = (a+b)βp, m2 = aβp,     α−1 n1 = (b + a) α−1 ∈ Z, and n = 2a ∈ Z to yield 2 p p For

 bb G

ατ + β γτ + δ



bb (τ ). = (γτ + δ)2 G

bb satisfies weight 2 modularity. Hence G P b It remains to compute the non-holomorphic part of ± G±b . By (2.5) and the b definition (3.5) of Gb , we have (3.6) X ±

"   1 XX p 1 d 2 2 b±b (τ ) = Gp,a±b (q) Sp,a2 −b2 + G ϑ v ± 2apτ + kp τ + ; 2p τ 4π dv ± k=0 2 ± #   2πi(b±a)v 1 2 2 × R 2bpτ − v − kp2 τ − ; 2p2 τ e2πi(a −b +k(b±a)p)τ e p . 2 X

v=0

Here we double count b ≡ 0 (mod p) to get a uniform formula. We now rewrite

q

1 (kp±2a)2 4

e

2πiv p

  ( kp2 ±2a) ϑ v ± 2apτ + kp2 τ + 1 ; 2p2 τ 2 X 2 2πiv kp kp =− q (mp+ 2 ±a) e p (mp+ 2 ±a) . m∈ 12 +Z

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KATHRIN BRINGMANN AND BEN KANE

However, we have that   X d X 2 2πiv kp kp  q (mp+ 2 ±a) e p (mp+ 2 ±a)  dv 1 ± m∈ 2 +Z

v=0   2 kp 2πi X X kp ± a q (mp+ 2 ±a) = 0, = mp + p ± 2 1 m∈ 2 +Z

which can be seen by making the change of variables m → −m − k. Moreover, by taking m → −m − k, we also see that τ  X X X 1 2 kp 2 q (mp+ 2 ±a) = 2 q 4 (2mp+kp+2a) = 2ϑ(1−k)p+2a,2p . 4 1 1 ± m∈ 2 +Z

m∈ 2 +Z

It follows that (3.6) equals 1 τ  p X − . R2b−kp,p (v; τ ) ϑ(1−k)p+2a,2p 2π k=0 4

By Lemma 3.2, we may rewrite this as
1 τ  ϑ2a,2p τ4 1 X √ − δb=p − ϑ √ (1−k)p+2a,2p 2π y 4 4 π k=0   X 2 1 1 2 × |2np − kp + 2b| Γ − ; π (2np − kp + 2b) y q − 4 (2np−kp+2b) 2 1 n∈ 2 +Z


1 τ  τ  ϑ2a,2p τ4 1 X R(1−k)p+2b,2p ϑ(1−k)p+2a,2p − δb=p =− √ √ . 4 4 2π y 4 π k=0 Statements (1) and (2) now follow from Lemma 3.2 with ` = 2b − kp. We next prove part (3). The claim is equivalent to showing that   X X
2πibv 4πi d 2 b2 bpτ, v; p2 τ  g(τ ) := Gbp,0,b Sp,−b2 =  e p q −b A p dv p ∗ b

0