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192 KEN ONO
where
A(p)(z)
:= −²(`) X ma(−mn) X qx2−m2p + X qx2`−m2p` ,
m,n≥1 x∈Z x∈Z x2≡m2p (mod 2`) x≡m (mod 2)
B(p)(z) := 2²(`) X(σ1(n) + `σ1(n/`))a(−n)Xq`x2, n≥1 x∈Z
and where ²(`) = 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∗(`)) → M2 Γ0(p`2), 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
` = −1. If f(z) = nÀ−∞ a(n)qn ∈ M0(Γ∗(`)) , with a(0) = 0, then the generating function Φ(p)(z) is in M2 Γ0(p`2), 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)(z) when f(z) = J1(z) := j(z) − 744. In terms of the classical Weber functions
(1.20) f1(z) = η(z/2) and f2(z) = √2 · η(2z),
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
Φ1,J1 (z) = η(2z)η(2pz)E4(pz)f2(2z)2f2(2pz)2 ·¡f1(4z)4f2(z)2 −f1(4pz)4f2(pz)2¢.
Although Theorem 1.3 gives a precise description of the forms Φ1,J (z), it is interesting to note that they are intimately related to Hilbert class polynomials,
the polynomials given by Y
(1.21) HD(x) = (x−j(τ)) ∈ Z[x],
τ∈CD
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) := Φ(p)1 |M. M∈Γ0(p)\SL2(Z)
SINGULAR MODULI FOR MODULAR CURVES AND SURFACES 193
If N∗(z) is the normalization of Np(z) with leading coefficient 1, then we have
∆(z)H75(j(z))
∗ E4(z)∆(z)2H3(j(z))H507(j(z)) p ∆(z)3H4(j(z))H867(j(z))
∆(z)5H7(j(z))2H2523(j(z))
if p = 5, if p = 13, if p = 17,
if p = 29,
where ∆(z) = η(z)24 is the usual Delta-function. These examples illustrate a general phenomenon in which Np (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
µµ ¶ ¶ (1.23) a(p) := + 1 ,
µµ ¶ ¶ (1.24) b(p) := + 1 ,
µ µ ¶¶ (1.25) c(p) := 6 p− p ,
and let Dp be the negative discriminants −D = −3,−4 of the form x2−4p 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∗(z) = (E4(z)H3(j(z)))a(p) ·H4(j(z))b(p) ·∆(z)c(p) ·H3·p2 (j(z))· Y HD(j(z))2. −D∈Dp
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.
194 KEN ONO
2.1. Maass-Poincar´e series. Here we give more details on the Poincar´e se-ries Fλ(−m;z) (see [Bru02, BO, BJO06, Hir73] for more on such series). Sup-
pose that λ is an integer, and that k := λ + 1. For each A = γ δ ∈ Γ0(4), let µ ¶
j(A,z) := δ ²−1(γz + δ)2
be the factor of automorphy for half-integral weight modular forms. If f : h → C is a function, then for A ∈ Γ0(4) we let
(2.1) (f |k A)(z) := j(A,z)−2λ−1f(Az).
As usual, let z = x+ iy, and for s ∈ C and y ∈ R− {0}, we let (2.2) Ms(y) := |y|−k Mk sgn(y),s−1 (|y|),
where Mν,µ(z) is the standard M-Whittaker function which is a solution to the differential equation
∂z2 + −4 + z + 1 z2µ2 u = 0. If m is a positive integer, and ϕ−m,s(z) is given by
ϕ−m,s(z) := Ms(−4πmy)e(−mx),
then recall from the introduction that X
(2.3) Fλ(−m,s;z) := (ϕ−m,s |k A)(z).
A∈Γ∞\Γ0(4)
It is easy to verify that ϕ−m,s(z) is an eigenfunction, with eigenvalue (2.4) s(1 −s) + (k2 −2k)/4,
of the weight k hyperbolic Laplacian 2 ¶ µ ¶ ∆k := −y ∂x2 + ∂y2 + iky ∂x + i∂y .
Since ϕ−m,s(z) = O yRe(s)−k as y → 0, it follows that Fλ(−m,s;z) converges
absolutely for Re(s) > 1, is a Γ0(4)-invariant eigenfunction of the Laplacian, and is real analytic.
Special values, in s, of these series provide examples of half-integral weight weak Maass forms. A weak Maass form of weight k for the group Γ0(4) is a smooth function f : h → C satisfying the following:
(1) For all A ∈ Γ0(4) we have
(f |k A)(z) = f(z).
(2) We have ∆kf = 0.
(3) The function f(z) has at most linear exponential growth at all the cusps.
In particular, the discussion above implies that the special s-values at k/2 and 1 − k/2 of Fλ(−m,s;z) are weak Maass forms of weight k = λ + 1 when the series is absolutely convergent. If λ ∈ {0,1} and m ≥ 1 is an integer for which (−1)λ+1m ≡ 0,1 (mod 4), then this implies that the Kohnen projections Fλ(−m;z), from the introduction, are weak Maass forms of weight k = λ + 1 on Γ0(4) in Kohnen’s plus space.
SINGULAR MODULI FOR MODULAR CURVES AND SURFACES 195
If λ = 1 and m is a positive integer for which m ≡ 0,1 (mod 4), then define
F1(−m;z) by
µ ¶
(2.5) F1(−m;z) := 2F1 −m, 4;z | pr1 + 24δ¤,mG(z).
The function G(z) is given by the Fourier expansion
∞ ∞
G(z) := n=0 H(n)qn + 16π√y n=−∞ β(4πn2y)q−n2,
where H(0) = −1/12 and
Z ∞
β(s) := t−2 e−stdt. 1
Proposition 3.6 of [BJO06] establishes that each F1(−m;z) is in M! . 2
Remark. The function G(z) plays an important role in the work of Hirzebruch and Zagier [HZ76] which is intimately related to Theorems 1.2, 1.3 and 1.4.
Remark. An analogous argument is used to define the series F0(−m;z) ∈ M! . 2
2.2. Exact formulas for the coefficients bλ(−m;n). Here we give exact formulas for the bλ(−m;n), the coefficients of the holomorphic parts of the Maass-Poincar´e series Fλ(−m;z). These coefficients are given as explicit infinite sums in half-integral weight Kloosterman sums weighted by Bessel functions. To define
these Kloosterman sums, for odd δ let (
1 if δ ≡ 1 (mod 4), δ i if δ ≡ 3 (mod 4).
If λ is an integer, then we define the λ+ 1 weight Kloosterman sum Kλ(m,n,c) by
(2.7) Kλ(m,n,c) := X µc¶²2λ+1eµmv¯ + nv¶.
v (mod c)∗
In the sum, v runs through the primitive residue classes modulo c, and v¯ denotes the multiplicative inverse of v modulo c. In addition, for convenience we define δ¤,m ∈ {0,1} by
(
1 if m is a square, ¤,m 0 otherwise.
Finally, for integers c define δodd(c) by (
δodd(c) := 0
if c is odd,
otherwise.
Theorem 2.1. Suppose that λ is an integer, and suppose that m is a positive integer for which (−1)λ+1m ≡ 0,1 (mod 4). Furthermore, suppose that n is a non-negative integer for which (−1)λn ≡ 0,1 (mod 4).
196 KEN ONO
(1) If λ ≥ 2, then b (−m;0) = 0, and for positive n we have bλ(−m;n) = (−1)[(λ+1)/2]π√2(n/m)2 −4 (1 −(−1)λi)
× X (1 + δodd(c/4))Kλ(−m,n,c) ·Iλ−2 4π mn .
c>0
c≡0 (mod 4)
(2) If λ ≤ −1, then
bλ(−m;0) = (−1)[(λ+1)/2]π3 −λ21−λm1 −λ(1 −(−1)λi)
× (2 −λ)Γ(2 −λ) c>0 (1 + δodd(c/4))Kλ(−m,0,c), c≡0 (mod 4)
and for positive n we have
bλ(−m;n) = (−1)[(λ+1)/2]π√2(n/m)2 −4 (1 −(−1)λi)
× X (1 + δodd(c/4))Kλ(−m,n,c) ·I2 −λ 4π mn .
c>0
c≡0 (mod 4)
(3) If λ = 1, then b1(−m;0) = −2δ¤,m, and for positive n we have
b1(−m;n) = 24δ¤,mH(n) − π 2(n/m)4 (1 + i)
× X (1 + δodd(c/4))K1(−m,n,c) ·I1 4π mn .
c>0
c≡0 (mod 4)
(4) If λ = 0, then b0(−m;0) = 0, and for positive n we have
b0(−m;n) = −24δ¤,nH(m) + π 2(m/n)4 (1 − i)
× X (1 + δodd(c/4))K0(−m,n,c) ·I1 4π mn .
c>0
c≡0 (mod 4)
Remark. For positive integers m and n, the formulas for bλ(−m;n) are nearly uniform in λ. In fact, this uniformity may be used to derive a nice duality (see Theorem 1.1 of [BO]) for these coefficients. More precisely, suppose that λ ≥ 1, and that m is a positive integer for which (−1)λ+1m ≡ 0,1 (mod 4). For every positive integer n with (−1)λn ≡ 0,1 (mod 4), this duality asserts that
bλ(−m;n) = −b1−λ(−n;m).
The proof of Theorem 2.1 requires some further preliminaries. For s ∈ C and y ∈ R− {0}, we let
(2.9) Ws(y) := |y|−k W2 sgn(y),s−2 (|y|),
where Wν,µ denotes the usual W-Whittaker function. For y > 0, we have the relations
(2.10) Mk (−y) = e2 , 2
(2.11) W k (y) = Wk (y) = e−y , 2 2
...
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