Analytic Number Theory A Tribute to Gauss and Dirichlet Part 11 ppt

192 KEN ONO

where

A(p)

,f (z)

:= −() 

m,n≥1

ma(−mn)







x∈Z

x2≡m2p (mod 2)

q

x2−m2p

4 + 

x∈Z x≡m (mod 2)

q

x2−m2p

4



 ,

B(p)

,f (z) := 2()



n≥1

(σ1(n) + σ1(n/))a(−n)



x∈Z

qx2

,

and where ()=1/2 for  = 1, and is 1 otherwise. As usual, σ1(x) denotes the

sum of the positive divisors of x if x is an integer, and is zero if x is not an integer.

Bringmann, Rouse and the author have shown [BOR05] that these generating

functions are also modular forms of weight 2. In particular, we obtain a linear map:

Φ(p)

, : M0(Γ∗

0()) → M2

Γ0(p2),

·

p



(where the map is defined for the subspace of those functions with constant term

0).

Theorem 1.2. (Bringmann, Ono and Rouse; Theorem 1.1 of [BOR05])

Suppose that p ≡ 1 (mod 4) is prime, and that  = 1 or is an odd prime with



p

 = −1. If f(z) = 

n

−∞ a(n)qn ∈ M0(Γ∗

0()) , with a(0) = 0, then the

generating function Φ(p)

,f (z) is in M2

Γ0(p2),

·

p



.

In Section 3 we combine the geometry of these surfaces with recent work of

Bruinier and Funke [BF06] to sketch the proof of Theorem 1.2. In this section

we characterize these modular forms Φ(p)

,f (z) when f(z) = J1(z) := j(z) − 744. In

terms of the classical Weber functions

(1.20) f1(z) = η(z/2)

η(z) and f2(z) = √

2 ·

η(2z)

η(z) ,

we have the following exact description.

Theorem 1.3. (Bringmann, Ono and Rouse; Theorem 1.2 of [BOR05])

If p ≡ 1 (mod 4) is prime, then

Φ(p)

1,J1 (z) = η(2z)η(2pz)E4(pz)f2(2z)2f2(2pz)2

4η(pz)6 ·



f1(4z)

4f2(z)

2 − f1(4pz)

4f2(pz)

2

.

Although Theorem 1.3 gives a precise description of the forms Φ(p)

1,J1 (z), it is

interesting to note that they are intimately related to Hilbert class polynomials,

the polynomials given by

(1.21) HD(x) = 

τ∈CD

(x − j(τ )) ∈ Z[x],

where CD denotes the equivalence classes of CM points with discriminant −D. Each

HD(x) is an irreducible polynomial in Z[x] which generates a class field extension

of Q(

√−D). Define Np(z) as the “multiplicative norm” of Φ1,J1 (z)

(1.22) Np(z) := 

M∈Γ0(p)\SL2(Z)

Φ(p)

1,J1 |M.

SINGULAR MODULI FOR MODULAR CURVES AND SURFACES 193

If N∗

p (z) is the normalization of Np(z) with leading coefficient 1, then we have

N∗

p (z) =







∆(z)H75(j(z)) if p = 5,

E4(z)∆(z)2H3(j(z))H507(j(z)) if p = 13,

∆(z)3H4(j(z))H867(j(z)) if p = 17,

∆(z)5H7(j(z))2H2523(j(z)) if p = 29,

where ∆(z) = η(z)24 is the usual Delta-function. These examples illustrate a

general phenomenon in which N∗

p (z) is essentially a product of certain Hilbert class

polynomials.

To state the general result, define integers a(p), b(p), and c(p) by

a(p) := 1

2

3

p



+ 1

(1.23) ,

b(p) := 1

2

2

p



+ 1

(1.24) ,

c(p) := 1

6

p −

3

p

 (1.25) ,

and let Dp be the negative discriminants −D = −3, −4 of the form x2−4p

16f2 with

x, f ≥ 1.

Theorem 1.4. (Bringmann, Ono and Rouse; Theorem 1.3 of [BOR05])

Assume the notation above. If p ≡ 1 (mod 4) is prime, then

N∗

p (z)=(E4(z)H3(j(z)))a(p)

·H4(j(z))b(p) ·∆(z)

c(p) ·H3·p2 (j(z))· 

−D∈Dp

HD(j(z))2.

The remainder of this survey is organized as follows. In Section 2 we compute

the coefficients of the Maass-Poincar´e series Fλ(−m; z), and we sketch the proof of

Theorem 1.1 by employing facts about Kloosterman-Sali´e sums. Moreover, we give

a brief discussion of Duke’s theorem on the “average values”

Tr(d) − Gred(d) − Gold(d)

H(d) .

In Section 3 we sketch the proof of Theorems 1.2, 1.3 and 1.4.

Acknowledgements

The author thanks Yuri Tschinkel and Bill Duke for organizing the exciting Gauss￾Dirichlet Conference, and for inviting him to speak on singular moduli.

2. Maass-Poincar´e series and the proof of Theorem 1.1

In this section we sketch the proof of Theorem 1.1. We first recall the construc￾tion of the forms Fλ(−m; z), and we then give exact formulas for the coefficients

bλ(−m; n). The proof then follows from classical observations about Kloosterman￾Sali´e sums and their reformulation as Poincar´e series.