TRACES OF SINGULAR MODULI

Don Zagier

Introduction. “Singular moduli” is the classical name for the values assumed by the modular invariant j(τ) (or by other modular functions) when the argument is a quadratic irrationality. These values are algebraic numbers and have been studied intensively since the time of Kronecker and Weber. In [5], formulas for their norms, and for the norms of their differences, were obtained. Here we obtain instead a result for their traces, and a number of generalizations. The results are closely related to a theorem of Borcherds [1] of which we will give a new proof and a generalization.

The formula for the traces of the singular values of j can be stated in two equivalent ways. The first one, formulated in §1, gives these traces as the Fourier coefficients of a certain modular form of weight 3/2. The second one, stated in §2 and proved in §§3–4, gives them as the solutions of a certain pair of recursion equations. The proof of one of the two recursions is a variant of the classical proof of the Kronecker class number relations and is quite amusing.

Borcherds’s theorem is recalled in §5 and the relationship to our result is explained. In a few words, it is as follows. There is a sequence of modular forms f_{d}(τ) of weight 1/2, one for each order in an imaginary quadratic field, and a sequence of modular forms g_{D}(τ) of weight 3/2, one for each order in a real quadratic field. Our formula says that the trace of a singular modulus of discriminant −d is the coefficient of q^{d} in the fixed modular form −g_{1}(τ) ; Borcherds’s theorem implies that it is the coefficient of q^{1} in the variable modular form f_{d}; and the relationship of the two results is that, for any d and D, the Dth Fourier coefficient of f_{d} is the negative of the dth Fourier coefficient of g_{D}. In fact, by a relatively straightforward application of Hecke operators one can obtain the full statement of Borcherds’s theorem (which describes the minimal polynomials of singular moduli, not just their traces, in terms of the n^{2}-th Fourier coefficients of f_{d} as n varies) from the formula for the traces. This is done in §6.

The third part of the paper describes generalizations in three different directions: §7 gives an interpretation of the other Fourier coefficients as traces (roughly speaking, the Dth Fourier coefficient of f_{d}, or the dth coefficient of g_{D}, describes the relative traces of singular moduli of discriminant −dD from the Hilbert class field of ℚ(\sqrt{−dD}) to its real quadratic subfield ℚ(\sqrt{D}) ); in §8 a generalization to the traces of singular values of Hauptmodules for certain other groups of genus 0 is described (they are equal to the Fourier coefficients of Jacobi forms of weight 2 and index > 1); and in §9 we state a generalization to modular forms of other small half-integral weights (there are modular forms of weight k + 1/2 and 3/2 − k satisfying the same sort of duality relation as the f_{d} and g_{D}, and their Fourier coefficients again can be interpreted as traces, but now of singular values of certain non-holomorphic modular functions). These results can be seen as analogues of the Shimura correspondence between modular forms of integral and half-integral weight.

We mention that the first results of this paper were found in 1994 and were quoted and applied in the two papers [8] and [9] by Masanobu Kaneko, who used Theorem 1 to obtain a closed formula for the Fourier coefficients of j(τ) in terms of traces of singular moduli.

§1. The trace of j(α) . Throughout this paper, d denotes a positive integer congruent to 0 or 3 modulo 4. We denote by Q_{d} the set of positive definite binary quadratic forms Q = [a, b, c] = aX^{2} +bXY +cY^{2} (a, b, c ∈ ℤ) of discriminant b^{2} −4ac = −d, with the usual action of the modular group Γ = PSL(2, ℤ). To each Q ∈ Q_{d} we associate its unique root α_{Q} in ℌ (= upper half-plane). The value of j(α_{Q}) depends only on the Γ-equivalence class of Q and, as is well-known, is an algebraic integer. More precisely, if h(−d) denotes the class number of −d, i.e., the number of Γequivalence classes of primitive quadratic forms in Q_{d}, then each of the corresponding h(−d) values of j(α_{Q}) is an algebraic integer of exact degree h(−d) and they form a full set of conjugates, so that the sum of these numbers is the trace. For instance, we have h(−3) = h(−4) = h(−7) = h(−8) = 1 and h(−15) = 2, and the corresponding j-values are

\begin{matrix}j(\frac{1 + i\sqrt{3}}{2}) = 0,  j(i) = 1728,  j(\frac{1 + i\sqrt{7}}{2}) = −3375,  j(i\sqrt{2}) = 8000, \\ j(\frac{1 + i\sqrt{15}}{2}) = \frac{−191025 − 85995\sqrt{5}}{2},  j(\frac{1 + i\sqrt{15}}{4}) = \frac{−191025 + 85995\sqrt{5}}{2} . \end{matrix}  (1)

We want to give a formula for the traces. Actually, it is convenient to make three small changes. First, we replace the function j by the normalized Hauptmodul for Γ = PSL(2, ℤ),

J(τ) = j(τ) − 744 = q^{−1} + 196884 q + 21493760 q^{2} + ⋯  (τ ∈ ℌ, q = e^{2πiτ}).  (2)

Secondly, we sum over all forms in Q_{d}, not just the primitive ones. Finally, we weight the number J(α_{Q}) by the factor 1/w_{Q}, where w_{Q} = |Γ_{Q}| (=2 or 3 if Q is Γ-equivalent to [a, 0, a] or [a, a, a], and 1 otherwise). We therefore define the Hurwitz-Kronecker class numbers H(d) and the modular trace function t(d) by

H(d) := ∑_{Q∈Q_{d}/Γ} \frac{1}{w_{Q}} ,  t(d) := ∑_{Q∈Q_{d}/Γ} \frac{1}{w_{Q}} J(α_{Q})  (d > 0, d ≡ 0 or 3 (mod 4)) .

For instance, from the examples in (1) we see that H(3) = \frac{1}{3}, H(4) = \frac{1}{2}, H(15) = 2 and

t(3) = \frac{0 − 744}{3} = −248, t(4) = \frac{1728 − 744}{2} = 492, t(15) = −191025 − 2 · 744 = −192513 .

Further values are given in the following small table.

\begin{matrix}d & 3 & 4 & 7 & 8 & 11 & 12 & 15 & 16 & 19 \\ H(d) & 1/3 & 1/2 & 1 & 1 & 1 & 4/3 & 2 & 3/2 & 2 \\ t(d) & −248 & 492 & −4119 & 7256 & −33512 & 53008 & −192513 & 287244 & −885480 \end{matrix}

\begin{matrix}d & 20 & 23 & 24 & 27 & 28 & 31 & 32 \\ H(d) & 2 & 3 & 2 & 4/3 & 2 & 3 & 2 \\ t(d) & 1262512 & −3493982 & 4833456 & −12288992 & 16576512 & −39493539 & 52255768 \end{matrix}

On the other hand, we define a (meromorphic) modular form of weight 3/2 by the formula

g(τ) := θ_{1}(τ) \frac{E_{4}(4τ)}{η(4τ)^{6}} = q^{−1} − 2 + 248 q^{3} − 492 q^{4} + 4119 q^{7} − 7256 q^{8} + · · · .  (3)

where θ_{1}, E_{4} and η are the classical modular forms

θ_{1}(τ) = ∑_{n=−∞}^{∞}(−1)^{n}q^{n^{2}}, E_{4}(τ) = 1 + 240 ∑_{n=1}^{∞} \frac{n^{3} q^{n}}{1 − q^{n}} , η(τ) = q^{1/24} ∏_{n=1}^{∞}(1 − q^{n})

of weights 1/2, 4, and 1/2, respectively. Comparing the first few coefficients of g with the above table suggests the following “modular” formula for the traces t(d) :

Theorem 1. Write the Fourier expansion of g(τ) as g(τ) = ∑_{d≥−1} B(d) q^{d} . Then

t(d) = −B(d)  (∀d > 0) .  (4)

The proof of this result will occupy the next three sections.

§2. A recursion for the numbers t(d) . Let θ(τ) be the theta series ∑_{n∈ℤ} q^{n^{2}} and U_{4} the operator ∑ c_{n}q^{n} ↦ ∑ c_{4n}q^{n}. The fact that g(τ) is a modular form of weight 3/2 having non-zero Fourier coefficients only for q^{d} with d ≡ 0 or 3 modulo 4 (the so-called “Kohnen pluscondition”; see §5) implies that the function (gθ)|U_{4} is a holomorphic modular form of weight 2 on the full modular group PSL(2, ℤ) and hence vanishes identically. Similarly, the image [g, θ]|U_{4} of the “Cohen bracket” [g, θ](τ) = g′(τ)θ(τ) − 3g(τ)θ′(τ) under U_{4} is a holomorphic modular form of weight 4 on PSL(2, ℤ) and hence is a multiple of E_{4}(τ). Writing these two facts out in terms of Fourier coefficients gives the two identities

∑_{r∈ℤ} B(4n − r^{2}) = 0 ,  ∑_{r>0} r^{2} B(4n − r^{2}) = 240σ_{3}(n)  (∀n ≥ 0) ,

where σ_{3}(0) = 1/240 and σ_{3}(n) for n ≥ 1 is the sum of the cubes of the positive divisors of n. These formulas can be rewritten

B(4n − 1) = 240σ_{3}(n) − ∑_{2≤r≤\sqrt{4n+1}} r^{2} B(4n − r^{2}) , B(4n) = −2 ∑_{1≤r≤\sqrt{4n+1}} B(4n − r^{2}),

and in this form obviously determine all the B(d) by recursion: B(−1) = 240σ_{3}(0) = 1, B(0) = −2B(−1) = −2, B(3) = 240σ_{3}(1) − 4B(0) = 248, B(4) = −2B(3) − 2B(0) = −492, etc. Thus to prove (4) it suffices to prove the corresponding identities for t(d); i.e., Theorem 1 follows from:

Theorem 2. For all integers n ≥ 1 we have the identities

∑_{|r|<2\sqrt{n}} t(4n − r^{2}) = \{ \begin{matrix}−4 & if n is a square, \\ 2 & if 4n + 1 is a square, \\ 0 & otherwise \end{matrix}  (5)

and

∑_{1≤r<2\sqrt{n}} r^{2} t(4n − r^{2}) = −240σ_{3}(n) + \{ \begin{matrix}−8n & if n is a square, \\ 4n + 1 & if 4n + 1 is a square, \\ 0 & otherwise. \end{matrix}  (6)

The proofs of equations (5) and (6) are given in the next two sections and are rather pretty. In particular, the proof of (5) is a refinement of the classical proof of Kronecker’s class number relation

∑_{|r|<2\sqrt{n}} H(4n − r^{2}) = ∑_{d|n} max\{d, n/d\} + \{ \begin{matrix}1/6 & if n is a square, \\ 0 & otherwise. \end{matrix}  (7)

§3. Proof of the first recursion for t(d). In this section we prove equations (5) and (7). Let Φ_{n}(X, Y) ∈ ℤ[X, Y] denote the well-known nth modular polynomial, defined by the property

Φ_{n}(X, j(τ)) = ∏_{M∈Γ\M_{n}}(X − j(M ∘ τ))  (τ ∈ ℌ),  (8)

where M_{n} denotes the set of 2 × 2 integral matrices of determinant n, with M and −M being identified, and ∘ denotes the usual action of P GL^{+}(2, ℝ) on ℌ. We exclude for the moment the case when n is a square. Then observing that the function Φ_{n}(j(τ), j(τ)) vanishes exactly at the points τ ∈ ℌ which are fixed by some M ∈ M_{n}, and that these are just the points α_{Q} with Q a positive definite quadratic form whose discriminant has the form r^{2} − 4n for some integer r (= tr(M)) satisfying |r| < 2\sqrt{n}, we get the classical identity

Φ_{n}(X, X) = const. × ∏_{|r|<2\sqrt{n}} H_{4n−r^{2}}(X) ,  (9)

where the constant factor (which is always ±1 and is easily determined) does not matter to us and the function H_{d}(X) is defined by

H_{d}(X) = ∏_{Q∈Q_{d}/Γ}(X − j(α_{Q}))^{1/w_{Q}}  (d > 0, d ≡ 0 or 3 (mod 4)) .  (10)

This function is X^{1/3} times a polynomial in X if d/3 is a square, (X −1728)^{1/2} times a polynomial in X if d is a square, and a polynomial in X otherwise. Clearly we have the q-expansion

H_{d}(j(τ)) = ∏_{Q∈Q_{d}/Γ}(q^{−1} − J(α_{Q}) + O(q))^{1/w_{Q}} = q^{−H(d)} (1 − t(d) q + O(q^{2})) .  (11)

On the other hand, taking as representatives for Γ\M_{n} the matrices ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) with ad = n, 1 ≤ b ≤ d, we get

Φ_{n}(j(τ), j(τ)) = ∏_{ad=n} ∏_{b=1}^{d}(j(τ) − j(\frac{aτ + b}{d}))

= ∏_{ad=n} ∏_{b=1}^{d}(q^{−1} − e^{−2πib/d} q^{−a/d} + O(q^{>0}))

= ∏_{ad=n}(q^{−d} − q^{−a})(1 + O(q^{>1}))

= ∏_{ad=n} ±q^{− max{a,d}} (1 − ε_{a} q + O(q^{2})) ,

where ε_{a} is 1 if |a − d| = 1 (which can happen only if 4n + 1 is a perfect square) and 0 otherwise. Comparing this with (9) and (11), we get both equations (5) and (7) for non-square n.

If n is a square, then Φ_{n}(X, Y) is divisible by Φ_{1}(X, Y) = X − Y and we must replace (9) by

\frac{Φ_{n}(X, Y)}{Φ_{1}(X, Y)}|_{Y =X} = const. × ∏_{|r|<2\sqrt{n}} H_{4n−r^{2}}(X) / ∏_{|r|<2} H_{4−r^{2}}(X) .

When we substitute X = j(τ), the left-hand side is ±q^{A}(1 + O(q^{2})) by essentially the same calculation as before, but omitting the factor corresponding to a = d = \sqrt{n}, b = 0. (Note that now 4n + 1 is not a square.) Hence ∑_{|r|<2\sqrt{n}} t(4n − r^{2}) = ∑_{|r|<2} t(4 − r^{2}) = −4.

§4. Proof of the second recursion for t(d). The method used to prove (5) was essentially geometric: we identified two different functions up to a constant by observing that they had the same divisors, and then obtained the desired identity by computing the expansions of both expressions at infinity. This does not generalize directly to the higher weight identity (6), but does if we think of the logarithmic derivatives of the previous functions as modular forms of weight 2 and replace these by modular forms of higher weight. (One could state this geometrically by working on the Kuga variety over ℌ/Γ, or with appropriate local coefficient systems over ℌ/Γ, but we will not do this.) More precisely, for each positive integer d ≡ 0 or 3 (mod 4) we define

Λ_{d}(τ) = \frac{d}{dτ} log H_{d}(j(τ)) = ∑_{Q∈Q_{d}/Γ} \frac{1}{w_{Q}} \frac{j′(τ)}{j(τ) − j(α_{Q})} .

This is a meromorphic modular form of weight 2, holomorphic at infinity and with a simple pole of residue 1/|Γ_{α}| at each point α ∈ ℌ satisfying a quadratic equation over ℤ of discriminant −d, and is uniquely characterized by these properties since there are no holomorphic modular forms of weight 2 on Γ. The analogue of the logarithmic derivative of equation (9) is then the following:

Proposition. Let n be a positive integer which is not a square, τ ∈ ℌ. Then

\frac{E_{4}(τ)E_{6}(τ)}{Δ(τ)} ∑_{M∈Γ\M_{n}} \frac{(E_{4}|M)(τ)}{j(τ) − j(Mτ)} = \frac{1}{4πi} ∑_{|r|<2\sqrt{n}}(r^{2} − n) Λ_{4n−r^{2}}(τ) ,  (12)

where Mτ := \frac{aτ + b}{cτ + d} and (E_{4}|M)(τ) := n^{3}(cτ + d)^{−4}E_{4}(Mτ) for M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{n}.

Proof. Both sides of the equality are meromorphic modular forms of weight 2, holomorphic at infinity, and with only simple poles, so it suffices to compare the residues. Let α ∈ ℌ be a fixed point of the action of M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{n}, where we can suppose that c > 0. The number λ = cα+d is a quadratic algebraic integer of trace r = tr(M) and norm n, and

Res_{τ=α}(\frac{E_{4}(τ)E_{6}(τ)}{Δ(τ)} \frac{(E_{4}|M)(τ)}{j(τ) − j(Mτ)}) = \frac{E_{4}(α)E_{6}(α)}{Δ(α)} \frac{n^{3}λ^{−4}E_{4}(α)}{(1 − nλ^{−2})j′(α)} = \frac{1}{2πi} \frac{−λ̄^{3}}{λ − λ̄} .

(We leave to the reader the modifications necessary if α is a root of E_{4} or E_{6}, i.e. a point with a non-trivial stabilizer in Γ.) By combining the matrices M and nM^{−1} = ( \begin{matrix}d & −b \\ −c & a \end{matrix} ), which have the same fixed points but conjugate values of λ, we can replace the expression −λ̄^{3}/(λ − λ̄) by \frac{1}{2}(λ^{3} − λ̄^{3})/(λ − λ̄) = \frac{1}{2}(r^{2} − n). The proposition follows.

The rest of the proof is similar to the calculation in the last section. Denote by S_{n}(τ) the expression on the left-hand side of (12) and write its expansion at infinity as C_{0} + C_{1}q + · · ·. From (12) and the expansion

−\frac{1}{2πi}Λ_{d}(τ) = ∑_{Q∈Q_{d}/Γ} \frac{1}{w_{Q}} \frac{q^{−1} + O(q)}{q^{−1} − J(α_{Q}) + O(q)} = H(d) + t(d) q + O(q^{2}) ,

we obtain

C_{0} = \frac{1}{2} ∑_{|r|<2\sqrt{n}}(n − r^{2})H(4n − r^{2}), C_{1} = \frac{1}{2} ∑_{|r|<2\sqrt{n}}(n − r^{2})t(4n − r^{2}) .

On the other hand, again taking as a set of representatives for Γ\M_{n} the matrices ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) with 0 ≤ b < d and ad = n, we find that S_{n}(τ) = ∑_{ad=n} a^{3}S_{a,d}(τ) where

S_{a,d}(τ) = \frac{E_{4}(τ)E_{6}(τ)}{Δ(τ)} · \frac{1}{d} ∑_{0≤b<d} \frac{E_{4}((aτ + b)/d)}{j(τ) − j((aτ + b)/d)} .

If a < d, then we can calculate the q-expansion of S_{a,d}(τ) as

S_{a,d}(τ) = (q^{−1} − 240 + O(q)) · \frac{1}{d} ∑_{ζ^{d}=1} \frac{1 + 240 ζq^{a/d} + 240σ_{3}(2) ζ^{2}q^{2a/d} + ⋯}{q^{−1} − ζ^{−1}q^{−a/d} + O(q^{a/d})}

= ∑_{\begin{matrix}l, m≥0 \\ l≡m (mod d) \end{matrix}} 240σ_{3}(l) q^{m+a(l−m)/d} (1 − 240q + O(q^{2}))

= 1 + (240 δ_{a,1} σ_{3}(n) + δ_{a,d−1}) q + O(q^{2}),

since the only pairs (l, m) which contribute to the powers q^{0} and q^{1} are (0,0), (1,1), (d,0) (if a = 1) and (0,d) (if a = d − 1). Similarly, for a > d we find

S_{a,d}(τ) = ∑_{\begin{matrix}l≥0, m≥1 \\ l+m≡0 (mod d) \end{matrix}} 240σ_{3}(l) q^{−m+a(l−m)/d} (1 − 240q + O(q^{2}))

= −δ_{a,d+1} q + O(q^{2}) ,

since this time only the pair (l, m) = (0, d) contributes. Summing over all divisors a of d, we find that the first two Fourier coefficients of S_{n}(τ) are given by

C_{0} = ∑_{\begin{matrix}0<a<\sqrt{n} \\ a|n \end{matrix}} a^{3} , C_{1} = 240σ_{3}(n) − \{ \begin{matrix}3n + 1 & if 4n + 1 = □, \\ 0 & otherwise, \end{matrix}

where the last term comes from the two factorizations of n as ad with a − d = ±1. Comparing the formula for C_{1} with the one given above, we obtain equation (6) (or more precisely, the difference between 2 times equation (6) and n times equation (5), but we have already proved (5)) for nonsquare n. The proof when n is a square is similar, except that we must also compute the Fourier expansion of S_{a,d}(τ) in the case when a = d; we leave this calculation to the reader. By comparing the two expressions obtained for C_{0} we also get the (well-known) identity

∑_{|r|<2\sqrt{n}}(n − r^{2})H(4n − r^{2}) = ∑_{d|n} min(d, n/d)^{3} − \{ \begin{matrix}n/2 & if n is a square, \\ 0 & otherwise, \end{matrix}

which together with (7) recursively determines the Hurwitz-Kronecker class numbers H(n).

§5. Relation to Borcherds’s theorem. By equation (11), the Hurwitz-Kronecker class number H(d) and the modified-trace-of-j function t(d) can be interpreted as the first two Fourier coefficients of the logarithmic derivative of H_{d}(j(τ)), where H_{d}(X) is the (nearly) polynomial function (10). Borcherds’s theorem describes the full Fourier expansion of the logarithmic derivative of H_{d}(j(τ)) in terms of certain Fourier coefficients of a meromorphic modular form f_{d}(τ) of weight 1/2. In this section we state his result and show how to deduce from it the formula for t(d) given in Theorem 1. In §6 we show how, conversely, Borcherds’s formula follows from Theorem 1 and some simple considerations about Hecke operators in integral and half-integral weight, and in the following section we show how similar ideas can be used to identify the remaining Fourier coefficients of f_{d}(τ) in terms of traces of singular moduli.

To define the functions f_{d}(τ), we need some basic facts from the theory of modular forms of half-integral weight. For any integer k, denote by M_{k+1/2}(Γ_{0}(4)) the space of functions which are holomorphic in ℌ and at the cusps and which transform under the action of Γ_{0}(4) like θ(τ)^{2k+1}, where θ(τ) is the Jacobi theta-function ∑_{n} q^{n^{2}} already used in §2. The “Kohnen plus-space” is the subspace consisting of functions whose Fourier expansion at infinity has the form ∑ c(n)q^{n} where c(n) is non-zero only for integers n satisfying (−1)^{k}n ≡ 0 or 1 modulo 4. We will denote this space simply by M_{k+1/2}, and consider its elements to be modular forms “of level one,” because of Kohnen’s basic result that M_{k+1/2} is isomorphic as a Hecke module to the space M_{2k} = M_{2k}(Γ) of holomorphic modular forms of weight 2k on the full modular group. (The definition of the action of the Hecke algebra on M_{k+1/2} will be recalled in the next section.)

From the isomorphism M_{k+1/2} ≈ M_{2k}, or directly, we see that the space M_{k+1/2} is onedimensional (spanned by θ itself) for k = 0 and is zero for k = 1 or k < 0. To get non-trivial spaces in these low half-integral weights, we have to allow modular forms with poles. Let M_{k+1/2}^{!}(Γ_{0}(4)) be defined like M_{k+1/2}(Γ_{0}(4)) but with the condition “holomorphic at the cusps” relaxed to “meromorphic at the cusps” (Borcherds [1] calls such forms “nearly holomorphic”) and define a “plussubspace” M_{k+1/2}^{!} of it by the same condition on the Fourier coefficients as before (where now c(n) may be non-zero for a finite number of negative values of n). Equivalently, a function f(τ) belongs to M_{k+1/2}^{!} if and only if f(τ)Δ(4τ)^{n} belongs to M_{k+12n+1/2} for some integer n > 0. The space M_{k+1/2}^{!} is infinite-dimensional for every k. In particular, for every integer d ≥ 0 with d ≡ 0, 3 (mod 4) there is a unique modular form f_{d} ∈ M_{1/2}^{!} having a Fourier development of the form

f_{d}(τ) = q^{−d} + ∑_{D>0} A(D, d) q^{D} ,  (13)

and the functions f_{0}, f_{3}, f_{4}, f_{7}, … form a basis of M_{1/2}^{!}. The existence and uniqueness of the forms f_{d} (=Lemma 14.2 of [1]) are easy to see. Indeed, the uniqueness is clear since the difference of any two functions satisfying the definition of f_{d} would be an element of M_{1/2} with constant term 0 and hence would vanish. For the existence, we first construct f_{0} and f_{3} “by hand” (f_{0}(τ) is just θ(τ) itself, and a non-trivial linear combination of f_{3} and f_{0} can be obtained e.g. as [θ(τ), E_{10}(4τ)]/Δ(4τ), where E_{10}(τ) = E_{4}(τ)E_{6}(τ) is the Eisenstein series of weight 10 on Γ and [θ(τ), E_{10}(4τ)] = θ(τ)E′_{10}(4τ) − 5θ′(τ)E_{10}(4τ) the Cohen bracket of θ(τ) and E_{10}(4τ)). Then for each d ≥ 4 we obtain f_{d}(τ) by multiplying f_{d−4}(τ) by j(4τ) to get a “plus”-form of weight 1/2 with leading coefficient q^{−d} and then subtracting off multiples of f_{d′} to successively kill the coefficients of q^{−d′} for 0 ≤ d′ < d. The process is effective and we find, for instance, that the Fourier expansions of the first few f_{d} begin as follows:

f_{0} = 1 + 2q + 2q^{4} + 2q^{9} + 2q^{16} + O(q^{25}),
f_{3} = q^{−3} − 248q + 26752q^{4} − 85995q^{5} + 1707264q^{8} − 4096248q^{9} + O(q^{12}),
f_{4} = q^{−4} + 492q + 143376q^{4} + 565760q^{5} + 18473000q^{8} + 51180012q^{9} + O(q^{12}),
f_{7} = q^{−7} − 4119q + 8288256q^{4} − 52756480q^{5} + 5734772736q^{8} + O(q^{9}),
f_{8} = q^{−8} + 7256q + 26124256q^{4} + 190356480q^{5} + 29071392966q^{8} + O(q^{9}),
f_{11} = q^{−11} − 33512q + 561346944q^{4} − 5874905295q^{5} + 2225561184000q^{8} + O(q^{9}) .

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Theorem 3 (Borcherds [Bo, p. 204]). Let d > 0, d ≡ 0 or 3 (mod 4). Then

H_{d}(j(τ)) = q^{−H(d)} ∏_{n=1}^{∞}(1 − q^{n})^{A(n^{2},d)} .  (14)

Example. From equation (14) for d = 3 and the above expansion of f_{3} we find

j(τ)^{1/3} = q^{−1/3} (1 − q)^{−248} (1 − q^{2})^{26752} (1 − q^{3})^{−4096248} · · · .

Comparing equation (14) with (11), we obtain :

Corollary. t(d) = A(1, d) for all d > 0.

This corollary and Theorem 1 give two apparently different ways to calculate t(d): Theorem 1 gives all of the t(d) as the Fourier coefficients of a single modular form g(τ) of weight 3/2, while the Corollary gives each t(d) as a coefficient of a different modular form f_{d} of weight 1/2. To see how the two formulas are related, we define a second sequence of modular forms, analogous to the f_{d} but now in weight 3/2 rather than 1/2, namely for each positive integer D congruent to 0 or 1 modulo 4 we define g_{D}(τ) as the unique form in M_{3/2}^{!} having a Fourier expansion of the form

g_{D}(τ) = q^{−D} + ∑_{d≥0} B(D, d) q^{d} .  (15)

For D = 1 this is just the function g(τ) defined in (3), so B(1, d) is the same as the coefficient which was denoted B(d) in §2. We can construct g_{4} either by applying the weight 3/2 Hecke operator T (2) to g_{1} (cf. §6) or in the same way as f_{3} was constructed from f_{0} ([g_{1}(τ), E_{10}(τ)]/Δ(4τ) is a linear combination of g_{1}(τ), g_{4}(τ), and g_{1}(τ)j(4τ) ), and then get the forms g_{D}(τ) for D > 4 inductively by multiplying g_{D−4}(τ) by j(4τ) and subtracting a suitable linear combination of g_{D′}(τ) with D′ < D; and the uniqueness of the g_{D} follows from the fact that M_{3/2} = {0}. Here are the beginnings of the Fourier developments of the first few g_{D}:

g_{1} = q^{−1} − 2 + 248q^{3} − 492q^{4} + 4119q^{7} − 7256q^{8} + 33512q^{11} − 53008q^{12} + O(q^{15}),
g_{4} = q^{−4} − 2 − 26752q^{3} − 143376q^{4} − 8288256q^{7} − 26124256q^{8} + O(q^{11}),
g_{5} = q^{−5} + 0 + 85995q^{3} − 565760q^{4} + 52756480q^{7} − 190356480q^{8} + O(q^{11}),
g_{8} = q^{−8} + 0 − 1707264q^{3} − 18473000q^{4} − 5734772736q^{7} − 29071392966q^{8} + O(q^{11}),
g_{9} = q^{−9} − 2 + 4096248q^{3} − 51180012q^{4} + 22505066244q^{7} − 125891591256q^{8} + O(q^{11}) .

Comparing these expansions with those of the f_{d} given above, we notice that the coefficients occurring are the same up to sign. The next theorem, which by virtue of the above corollary reduces to (4) for D = 1, says that this is true in general.

Theorem 4. A(D, d) = −B(D, d) for all D and d.

Proof. Consider the function

F (τ_{1}, τ_{2}) = \frac{f_{0}(τ_{1})g_{4}(τ_{2}) + f_{3}(τ_{1}) g_{1}(τ_{2})}{j(4τ_{2}) − j(4τ_{1})} (τ_{1}, τ_{2} ∈ ℌ) .

It is meromorphic, with a simple pole whenever 4τ_{1} and 4τ_{2} are Γ-equivalent. If ℑ(τ_{1}) and ℑ(τ_{2}) are big enough (to be precise, bigger than 1/4), then this is the case if and only if 4τ_{1} and 4τ_{2} differ by an integer. On the other hand, we can easily check that

f_{0}(τ + \frac{n}{4}) g_{4}(τ) + f_{3}(τ + \frac{n}{4}) g_{1}(τ) = −\frac{1 + i^{n}}{2πi} j′(4τ) (n = 0, 1, 2, 3),

so the residue of F (with respect to τ_{1}, keeping τ_{2} fixed) at τ_{1} = τ_{2}+n/4 (n ∈ ℤ) equals (1+i^{n})/8πi. This is the same as the residue of the elementary function F_{0}(τ_{1}, τ_{2}) = (q_{1}^{4} + q_{1}q_{2}^{3})/(q_{1}^{4} − q_{2}^{4}), where q_{j} = e^{2πiτ_{j}}, so F − F_{0} is a holomorphic function of (q_{1}, q_{2}) in a neighbourhood of (0, 0) ∈ ℂ^{2}, i.e.

F (τ_{1}, τ_{2}) = \frac{q_{1}^{4} + q_{1}q_{2}^{3}}{q_{1}^{4} − q_{2}^{4}} + ∑_{D>0, d≥0} C(D, d) q_{1}^{D} q_{2}^{d}

in this neighbourhood for some coefficients C(D, d). Here we can write D > 0 rather than D ≥ 0 in the summation because both F and F_{0} obviously vanish on the q_{2}-axis; moreover, it is clear that C(D, d) is 0 unless D and d satisfy the usual congruences D ≡ 0, 1 (mod 4), d ≡ 0, 3 (mod 4). Expanding F_{0} in a geometric series in the region |q_{2}| < |q_{1}|, we find that the Taylor expansion of F with respect to τ_{2} is given by

F (τ_{1}, τ_{2}) = ∑_{\begin{matrix}d≥0 \\ d≡0, 3 (mod 4) \end{matrix}}(q_{1}^{−d} + ∑_{\begin{matrix}D>0 \\ D≡0, 1 (mod 4) \end{matrix}} C(D, d) q_{1}^{D}) q_{2}^{d}  (τ_{1} ∈ ℌ, ℑ(τ_{2}) → ∞).

Since F is a (plus-)modular form of weight 1/2 with respect to τ_{1}, the expression in parentheses also is one, so it equals f_{d}(τ_{1}). Hence C(D, d) = A(D, d). Similarly, from the expansion

F (τ_{1}, τ_{2}) = ∑_{\begin{matrix}D>0 \\ D≡0, 1 (mod 4) \end{matrix}}(−q_{2}^{−D} + ∑_{\begin{matrix}d≥0 \\ d≡0, 3 (mod 4) \end{matrix}} C(D, d) q_{2}^{d}) q_{1}^{D}  (τ_{2} ∈ ℌ, ℑ(τ_{1}) → ∞)

and the fact that F is a (plus-)modular form of weight 3/2 with respect to τ_{2} we deduce that C(D, d) = −B(D, d). This completes the proof.

Notice that the Fourier expansions of F just given also yield formulas for all f_{d}(τ) (d ≥ 0) as linear combinations of f_{0}(τ) and f_{3}(τ) with coefficients in ℚ[j(4τ)], namely

f_{d}(τ) = ∑_{n=0}^{[d/4]} Coeff_{q^{d}}(\frac{g_{4}(τ)}{j(4τ)^{n+1}}) j(4τ)^{n}f_{0}(τ) + ∑_{n=0}^{[(d−3)/4]} Coeff_{q^{d}}(\frac{g_{1}(τ)}{j(4τ)^{n+1}}) j(4τ)^{n}f_{3}(τ) ,

and similarly for all g_{D}(τ) (D > 0) as linear combinations over ℚ[j(4τ)] of g_{1}(τ) and g_{4}(τ).

Remark. As Masanobu Kaneko has pointed out to me, there is an easier proof of Theorem 4. It is easily checked that the constant term of the q-expansion of fg vanishes for any f ∈ M_{1/2}^{!} and g ∈ M_{3/2}^{!} (because fg|U_{4} is a modular form of weight 2 on SL_{2}(ℤ) with a pole only at infinity, and every such form is the derivative of a polynomial in j). Applying this to f = f_{d}, g = g_{D} immediately gives the assertion. We have retained the original proof because of the supplementary information it gives about the functions f_{d} and g_{D}.

§6. Hecke operators. Theorems 1 and 4 together clearly imply the corollary to Borcherds’s theorem, since one has t(d) = −B(1, d) = A(1, d). In fact, they imply the full theorem in a more or less straightforward way, as we now explain.

To obtain the full minimal polynomial H_{d}(X) of the algebraic number j(α_{Q}), we need not just its trace but also the traces of all of its powers (or at least of all of its powers up to the class number). Instead of working with the traces of j (or of J), we work with the functions J_{m}, defined for every integer m ≥ 0 as the unique holomorphic function on ℌ/Γ with a Fourier expansion beginning q^{−m} + O(q) . For m = 0 this is the constant function 1 and for m = 1 it is the function J(τ) defined in (2), while the next three values have Fourier developments starting

J_{2}(τ) = q^{−2} + 42987520 q + 40491909396 q^{2} + 8504046600192 q^{3} + ⋯,
J_{3}(τ) = q^{−3} + 2592899910 q + 12756069900288 q^{2} + 9529320689550144 q^{3} + ⋯,
J_{4}(τ) = q^{−4} + 80983425024 q + 1605963589611520 q^{2} + 3497254878743101440 q^{3} + · · · .

The uniqueness of these functions is obvious (the difference of any two functions satisfying the definition of J_{m} would be a holomorphic cusp form of weight 0 and hence 0) and the existence also, since J_{m} can clearly be obtained as a monic polynomial of degree m in j(τ), e.g.

J_{2}(τ) = j(τ)^{2} − 1488 j(τ) + 159768 ,
J_{3}(τ) = j(τ)^{3} − 2232 j(τ)^{2} + 1069956 j(τ) − 36866976 ,
J_{4}(τ) = j(τ)^{4} − 2976 j(τ)^{3} + 2533680 j(τ)^{2} − 561444608 j(τ) + 8507424792 .

The minimal polynomials of all j(α_{Q}) are then determined if we know the numbers

t_{m}(d) := ∑_{Q∈Q_{d}/Γ} \frac{1}{w_{Q}} J_{m}(α_{Q})  (16)

for all d and all m ≥ 1 (or even just for 0 ≤ m ≤ H(d)). To get them, we must generalize Theorems 1 and 4 to include Hecke operators. For any integer m ≥ 1 let A_{m}(D, d) and B_{m}(D, d) denote the coefficient of q^{D} in f_{d}|_{\frac{1}{2}} T (m) and the coefficient of q^{d} in g_{D}|_{\frac{3}{2}} T (m), respectively, where |_{\frac{1}{2}} T (m) and |_{\frac{3}{2}} T (m) denote the action of the mth Hecke operator T (m) (sometimes denoted T (m^{2}) or T^{+}(m^{2})) on the space M_{k}^{!} , k = 1/2 or 3/2. The formulas for this action (at least for holomorphic modular forms, but nothing changes if we allow the pole at infinity) are given in many places, e.g. [10], Prop. 2 or, in the equivalent language of Jacobi forms, [4], Th. 4.5. In particular, for m = p prime we have

A_{p}(D, d) = p A(p^{2}D, d) + (\frac{D}{p} ) A(D, d) + A(p^{−2}D, d)  (17)

and

B_{p}(D, d) = B(D, p^{2}d) + (\frac{−d}{p}) B(D, d) + p B(D, p^{−2}d)  (18)

(with the convention that A(p^{−2}D, d) = 0 unless p^{−2}D is an integer congruent to 0 or 1 modulo 4, and similarly for B(D, p^{−2}d) ), while for D = 1 and m arbitrary we have

A_{m}(1, d) = ∑_{n|m} n A(n^{2}, d) .  (19)

Theorem 5. With the above notations, we have

(i) H_{d}(j(τ)) = q^{−H(d)} exp(− ∑_{m=1}^{∞} t_{m}(d) \frac{q^{m}}{m} ) for all d;

(ii) t_{m}(d) = −B_{m}(1, d) for all m and d;

(iii) A_{m}(D, d) = −B_{m}(D, d) for all m, D and d.

Note that parts (ii) and (iii) of this theorem are the generalizations of Theorems 1 and 4 to m ≥ 1, while Theorem 3 is obtained by combining (i), (ii), the case D = 1 of (iii), and equation (19).

Proof. (i) This identity follows directly from the definitions (10) and (16) of H_{d}(τ) and t_{m}(d) together with the identity

j(τ) − j(z) = q^{−1} exp(− ∑_{m=1}^{∞} J_{m}(z) \frac{q^{m}}{m} )  (q = e^{2πiτ} , ℑ(τ) ≫ 0) ,  (20)

an identity which has been discovered independently by a number of authors and which can be proved easily by noting that the logarithmic derivative with respect to τ of either side is, for fixed τ with sufficiently large imaginary part, a Γ-invariant meromorphic function of z whose only pole in the standard fundamental domain is a simple one of residue 1 at z = τ and which vanishes at infinity.

(ii) We first observe that the function J_{m} is the composite of J with the mth Hecke operator T (m) acting in degree 0, i.e., J_{m}(τ) is the sum of J(M(τ)) with M ranging over Γ\M_{n}. (This is obvious, since J_{1}|_{0}T (m) is a modular function with a pole q^{−m} + O(q) at infinity and no other poles.) It follows that t_{m}(d) is the sum of the values of J_{1}(α) summed over all α ∈ S_{d}|T (m), where S_{d} ⊂ ℌ/Γ is the set of α_{Q}, Q ∈ Q_{d}/Γ (counted with multiplicity 1/q_{Q}). But S_{d}|T (m) is a combination of sets S_{dm^{2}/n^{2}} with n ranging over the divisors of m. For example, S_{d}|T (p) for p

prime is equal to S_{dp^{2}} + (\frac{−d}{p})S_{d} + pS_{d/p^{2}}. (The proof of this is an exercise, carried out, e.g., on

pages 290–292 of [15].) Assertion (ii) for m = p then follows from Theorem 1 and equation (18). The general case can be proved similarly, or one can use the multiplicative properties of Hecke operators (which are the same in integral and half-integral weight) and induction on the number of prime factors of m.

(iii) Again we need only consider the case when m = p is prime. Formula (17) for the Fourier coefficients of f_{d}|_{\frac{1}{2}} T (p) is true for all D, negative as well as positive, so from the defining property

A(D, d) = δ_{D,−d} (D < 0) of f_{d} we deduce that A_{p}(D, d) = pδ_{D,−d/p^{2}} + (\frac{−d}{p})δ_{D,−d} + δ_{D,−dp^{2}}

for D < 0, i.e., the principal part of f_{d}|_{\frac{1}{2}} T (p) at infinity is pq^{−d/p^{2}} + (\frac{−d}{p})q^{−d} + q^{−dp^{2}} (with

the first term to be omitted unless −d/p^{2} is a discriminant). It follows that f_{d}|_{\frac{1}{2}} T (p) equals

pf_{d/p^{2}} +(\frac{−d}{p})f_{d}+f_{dp^{2}} and hence that its Dth Fourier coefficient A_{p}(D, d) is equal to pA(D, d/p^{2})+

(\frac{−d}{p})A(D, d) + A(D, dp^{2}). Comparing with (18), we find that (iii) follows from Theorem 4. One

could also give a dual proof by using (18) to show that g_{D}|_{\frac{3}{2}} T (p) = pg_{Dp^{2}} + (\frac{D}{p} )g_{D} + g_{D/p^{2}}, and

there is yet a third proof along the lines of the proof given for the special case Theorem 4 by looking at the singular part at infinity of the images of the two-variable modular form F (τ_{1}, τ_{2}) under the actions of T (p) on τ_{1} (in weight 1/2) and τ_{2} (in weight 3/2).

This completes the proof of Theorem 5 and of Borcherds’s product formula (14) from our more elementary formula (4). In the remaining sections of the paper we describe without detailed proofs generalizations of these results in various different directions.

§7. First generalization: other discriminants. The theorems we have discussed so far give an interpretation of the numbers A_{m}(D, d) = −B_{m}(D, d) as traces when D is equal to 1 or, more generally, whenever D is a perfect square. Is there a similar interpretation for the other values?

The answer is yes, but now we have to take “twisted” traces in which the conjugates of j(α_{Q}) are multiplied by a genus character before summing. We recall the definition of these characters. Let D and −d be a positive and a negative discriminant, respectively. For ease of exposition we suppose them to be fundamental and coprime. (Comments on other situations are given in Remark 3 below.) Then the genus character χ = χ_{D,−d} assigns to any quadratic form Q of discriminant −dD a value ±1 defined by χ(Q) = (\frac{D}{p} ) = (\frac{−d}{p}) where p is any prime represented by Q and not dividing Dd. (This is independent of the choice of p.)

Theorem 6. Let D > 1 and −d < 0 be coprime fundamental discriminants, and χ = χ_{D,−d} the associated genus character. Then

A(D, d) = \frac{1}{\sqrt{D}} ∑_{Q∈Q_{dD}/Γ} χ(Q) j(α_{Q}) .  (21)

Example. Take D = 5, d = 3, the first non-trivial case. There are two Γ-equivalence classes of quadratic forms of discriminant −15, represented by the forms Q_{1} = [1, 1, 4] and Q_{2} = [2, 1, 2] with χ(Q_{1}) = 1, χ(Q_{2}) = −1. The corresponding values of j(α_{Q}) are given in (1). We see that the right-hand side of (21) is (j(α_{Q_{1}}) − j(α_{Q_{2}}))/\sqrt{5} = −85995, in agreement with the value A(5, 3) = −B(5, 3) = −85995 given in §5.

Remarks. 1. In formula (21) we could omit the usual factor w_{Q}^{−1} because it is always 1 for D and d as in the theorem, and similarly we could write j(α_{Q}) rather than J(α_{Q}) because summing with the non-trivial character χ kills the additive constant by which j and J differ.

2. The right-hand side of (21) is a priori a rational integer. Indeed, the sum of j(α_{Q}) as Q ranges over the forms with χ(Q) = +1 is, by class field theory, the trace of the algebraic integer j(α_{Q}) from the Hilbert class field of ℚ(\sqrt{−dD}) to its real quadratic subfield ℚ(\sqrt{D}). If we write this number as λ = \frac{1}{2}(a + b\sqrt{D}) with a and b in ℤ, then the sum over all j(α_{Q}) is equal to Tr_{ℚ(\sqrt{D})/ℚ}(λ) = a and therefore the weighted sum ∑ χ(Q)j(α_{Q}) is equal to −a + 2λ = b\sqrt{D}.

3. If D and d are not coprime, then the statement of the theorem is unaltered but χ(Q) must be defined as 0 for imprimitive forms Q which are divisible by some prime dividing both D and d. Thus for instance the values A(8, 4) = 18473000 and A(5, 15) = −292658282496 are obtained as (j(α_{[1,0,8]}) − j(α_{[3,2,3]}))/\sqrt{8} and (j(α_{[1,1,19]}) − j(α_{[3,3,7]}))/\sqrt{5}, respectively. The cases when D and d are not assumed to be fundamental can be given by a suitable modification of (21) (the reader can find the right formulas easily by a little numerical experimentation) or can be obtained from the case of fundamental discriminants by suitable applications of Hecke operators.

We can also give a generalization of the Borcherds product expansion (14), but with a difference. For D and d as in the theorem we define a rational function H_{D,d}(X) by

H_{D,d}(X) = ∏_{Q∈Q_{Dd}/Γ}(X − j(α_{Q}))^{χ(Q)} .  (22)

For the same reasons as in Remark 2, this belongs to ℚ(\sqrt{D})(X) and is mapped by the nontrivial element of Gal(ℚ(\sqrt{D})/ℚ) to its inverse. It follows that the power series expansion of log H_{D,d}(j(τ)), which begins with 1, has a logarithm belonging to \sqrt{D} ℚ[[q]].

Theorem 7. Let D, d and H_{D,d}(X) be as above. Then

H_{D,d}(j(τ)) = ∏_{n=1}^{∞} P_{D}(q^{n})^{A(n^{2}D,d)}  (23)

where

P_{D}(t) := exp(−\sqrt{D} ∑_{r=1}^{∞} (\frac{D}{r} ) \frac{t^{r}}{r} ) ∈ ℚ(\sqrt{D})[[t]] .  (24)

The function P_{d}(t) is a rational function of t with coefficients in ℚ(\sqrt{D}).

Examples. For D = 1 we clearly have P_{1}(t) = 1 − t, so that (24) reduces to (14) in this case. The next values of P_{D} are

P_{5}(t) = \frac{1 − ϕ t + t^{2}}{1 − ϕ′t + t^{2}} ,  P_{8}(t) = \frac{1 − \sqrt{2} t + t^{2}}{1 + \sqrt{2} t + t^{2}} ,  P_{12}(t) = \frac{1 − \sqrt{3} t + t^{2}}{1 + \sqrt{3} t + t^{2}} ,

where ϕ = (1 + \sqrt{5})/2 is the golden ratio and ϕ′ = (1 − \sqrt{5})/2 its Galois conjugate. A typical example of the theorem (D = 8, d = 3) is the expansion

\frac{j(τ) − 2417472 − 1707264\sqrt{2}}{j(τ) − 2417472 + 1707264\sqrt{2}} = ∏_{n=1}^{∞}(\frac{1 − \sqrt{2}q^{n} + q^{2n}}{1 + \sqrt{2} q^{n} + q^{2n}} )^{A(8n^{2}, 3)}

of the ratio of j(τ) − j(\sqrt{−6}) and j(τ) − j(\sqrt{−3/2}), whereas Borcherds’s product expansion (14) would give the Fourier expansion of the product of the same two expressions in the form

j(τ)^{2} − 4834944 j(τ) + 14670139392 = q^{2} ∏_{n=1}^{∞}(1 − q^{n})^{A(n^{2}, 24)} ,

Theorem 7 follows from Theorem 6 in the same way as explained in §6 for the special case D = 1. If we denote by t(D, d) the sum occurring in the right-hand side of (21) (“twisted trace”) and by t_{m}(D, d) the corresponding sum with j replaced by J_{m}, then we can use Hecke operators in integral and half-integral weight to show that (21) generalizes to

t_{m}(D, d)) = A_{m}(D, d) \sqrt{D}  for all m, D and d .  (25)

From equations (20) and (25) we find

H_{D,d}(j(τ)) = exp(− ∑_{m=1}^{∞} t_{m}(D, d) \frac{q^{m}}{m} ) = exp(−\sqrt{D} ∑_{m=1}^{∞} A_{m}(D, d) \frac{q^{m}}{m} ) ,

and combining this with the formula

A_{m}(D, d) = ∑_{n|m} n (\frac{D}{m/n}) A(n^{2}D, d) ,

which is the generalization of (19) to fundamental discriminants D > 1, gives (23). For the final

statement of Theorem 7 we use the Gauss sum identity \sqrt{D} (\frac{D}{r} ) = ∑_{0<n<D}(\frac{D}{n} ) e^{2πinr/D} to get

P_{D}(t) = ∏_{0<n<D}(1 − ζ_{D}^{n} t)^{( \frac{D}{n})} = \frac{N_{ℚ(ζ_{D})/ℚ(\sqrt{D})}(1 − ζ_{D}t)^{2}}{N_{ℚ(ζ_{D})/ℚ}(1 − ζ_{D}t)}  ( ζ_{D} = exp^{2πi/D} ) .  (26)

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Digression on Hilbert modular surfaces, the Doi-Naganuma lifting, and twisted Borcherds products. To prove Theorem 6, two approaches are possible (although to be honest I have not actually carried out either one in detail). One way would be to generalize the proof which Borcherds gave in [1] for the case D = 1. This proof is based on the circle method and uses explicit calculations of the Fourier developments of Poincaré series, so that it is a kind of multiplicative analogue of the method which was introduced in [13] to construct the Doi-Naganuma lifting from elliptic to Hilbert modular forms and applied in [11] for the usual Shimura lifting from modular forms of half-integral to modular forms of integral weight. But in keeping with the spirit of the rest of this paper, one would like to give a more elementary proof, and also to see the “twisted traces” t(D, d) for a fixed D and variable d appear naturally as the Fourier coefficients of a single modular form g_{D} rather than a collection of modular forms {f_{d}}. The method used to prove Theorem 1 in this paper was to reduce it to the set of recursions given in Theorem 2 and then prove these by a geometric method. Specifically, to prove Theorem 2 we used the polynomial Φ_{n}(X, Y), which is the function on ℙ^{1} × ℙ^{1} ≅ ℌ/Γ × ℌ/Γ =: X_{1} whose zero-divisor is the graph of the Hecke correspondence T_{n} ⊂ X_{1}. The recursions (5) and (6) are then connected with the geometric fact that the intersection of T_{n} with the diagonal T_{1} = ℌ/Γ ⊂ X_{1} is the union over |r| < 2\sqrt{n} of the CM-cycles S_{4n−r^{2}} ⊂ ℌ/Γ. We would like to see what replaces this when D > 1.

The obvious replacement for the surface X_{1} when one is dealing with a non-square positive discriminant D, say the discriminant of a real quadratic field K, is the Hilbert modular surface X_{D} = ℌ × ℌ/SL(2, O_{K}), and the usual replacement for the graph of the Hecke correspondence in X_{1} is the curve (also denoted T_{n}) in X_{D} which was studied in [7]. The result T_{n} ∩ T_{1} = ⋃ S_{4n−r^{2}} on X_{1} then generalizes nicely to a similar formula on X_{D} (essentially T_{n} ∩ T_{1} = ⋃ S_{(4n−r^{2})/D}, at least for D prime, which we now assume). The problem is that T_{n} is not in general the zero-set of any function on X_{D}, because its homology class on the surface X_{D} is not in general trivial. Indeed, the main result of [7] says that this homology class as n varies is the nth coefficient of a modular form (with coefficients in H_{2}(X_{D})) of level D and Nebentypus (\frac{·}{D} ) and that this modular

form describes (i.e., is the kernel function for) the Naganuma map [12] from M_{2}(Γ_{0}(D), (\frac{·}{D} )) to

M_{2}(SL(2, O_{K})). This map is injective and the dimension of M_{2}(Γ_{0}(D), (\frac{·}{D} )) grows linearly with D, so it becomes increasingly difficult to find linear combinations of the curves T_{n} which are are homologically trivial.

However, as well as the Naganuma lifting from M_{k}(Γ_{0}(D), (\frac{·}{D} )) to M_{k}(SL(2, O_{K}), there is also

the lifting constructed earlier by Doi and Naganuma [3] from M_{k}(SL(2, ℤ)) to M_{k}(SL(2, O_{K}), and in [14, pp. 63–65] it was shown that it, too, could be described geometrically in terms of intersections numbers with certain divisors T̃_{n} on X_{D}. But since there are no cusp forms of weight 2 on SL(2, ℤ), this means that all of the divisors T̃_{n} are homologically trivial and hence (since X_{D} is known to be simply connected) principal, so that they can be obtained as the divisors of certain Hilbert modular functions Ψ_{n}^{(D)}. It is these functions which are the analogues of the functions Φ_{n}(j(τ_{1}), j(τ_{2})) in the case D = 1. The definition of the curves T̃_{n}, which we will not repeat here, is well suited to our twisted traces t(D, d) and twisted Heegner cycles S_{D,d} = ∑ χ(Q)[α_{Q}], because T̃_{n} is defined as the difference of two components of T_{nD^{2}} which are distinguished by the value of the Legendre symbol (·/D). Using this one finds easily that T̃_{n} ∩ T_{1} = ⋃_{r} S_{D,4n−r^{2}} and hence that the restriction of Ψ_{n}^{(D)} to T_{1} is ∏_{r} H_{D,4n−r^{2}}(j), where we have identified T_{1} ≈ ℌ/Γ with the j-line, and it is this which replaces the classical identity (9).

We now have an amusing calculation. Write J_{m}(τ) = q^{−m} + ∑_{n>0} c_{m}(n)q^{n}. The coefficients c_{m}(n) are given by c_{m}(n) = ∑_{d|(m,n)} md^{−1}c(mn/d^{2}), where c(n) = c_{1}(n) is the coefficient of q^{n} in J(τ). It follows that c_{m}(n) = mn^{−1}c_{n}(m) and that the function S_{m}(τ) := q^{−m} − ∑_{n>0} c_{n}(m)q^{n} is a multiple of J′_{m}(τ) and hence is the (obviously unique) modular form in M_{2}^{!}(Γ) with a Fourier expansion of the form q^{−m}+O(q). Now consider the function (g_{D}θ)|U_{4}, where D as usual is positive and congruent to 0 or 1 modulo 4. From g_{D} = q^{−D} − 2ε(D) + O(q) (ε(D) = 1 or 0 according as D is or is not a square) we find (g_{D}θ)|U_{4} = ∑_{f} q^{(f^{2}−D)/4} + O(q), where the summation runs over all integers f with f^{2} < D, f ≡ D (mod 2). Also, (g_{D}θ)|U_{4} is a modular form of weight 2 on Γ and is nearly holomorphic. Hence by our previous remark (g_{D}θ)|U_{4} = ∑_{f} S_{(D−f^{2})/4}, and comparing the coefficients of q^{n} on both sides we obtain the identity

∑_{r^{2}<4n} A(D, 4n − r^{2}) = ∑_{f^{2}<D, f≡D (mod 2)} c_{n}(\frac{D − f^{2}}{4} ) + 2ε(4n + D) .

Combining this with the above identity Ψ_{n}^{(D)}(τ, τ) = ∏_{r^{2}<4n} H_{D,4n−r^{2}}(j(τ)) and equation (23), we deduce after a short calculation that

Ψ_{n}^{(D)}(τ, τ) = ∏_{ν∈𝔡^{−1}, Tr(ν)>0} P_{D}(q^{Tr(ν)})^{c_{n}(Dνν′)} ,

where the product is over all all elements ν in the inverse different 𝔡^{−1} = D^{−1/2}O_{K} with Tr(ν) positive and Dνν′ either positive or else equal to −n. This strongly suggests the identity

Ψ_{n}^{(D)}(τ_{1}, τ_{2}) = ∏_{ν∈𝔡^{−1}/{±1}} P_{D}(e^{2πi(ντ_{1}+ν′τ_{2})})^{c_{n}(Dνν′)}  (27)

(in a suitable neighborhood of infinity), which is a kind of twisted Borcherds product expansion for the Hilbert modular function Ψ_{n}^{(D)}. This identity is similar (apart from the twisting) to an identity proved by Bruinier [2] for the usual curves T_{n}, and could doubtless be proved the same way, although I have not gone through the details. Indeed, the calculation should be much easier here because in Bruinier’s case only certain linear combinations of the T_{n} were homologically trivial and therefore representable as the divisor of a Hilbert modular function, and (correspondingly) the Poincaré series of weight 2, which here are trivial because Γ has genus 0, were more complicated because of the the presence of holomorphic cusp forms (on Γ_{0}(D) and Nebentypus) of weight 2.

If we take the logarithm of (27) and substitute the definition (24) of the power series P_{D}(t), then

we find that the coefficient of e^{2πi(ντ_{1}+ν′τ_{2})} in log(Ψ_{n}^{(D)}(τ_{1}, τ_{2})) is equal to ∑_{r} r^{−1}(\frac{D}{r} ) c_{n}(\frac{Dνν′}{r^{2}} ).

This formula is identical (with a_{n} = c_{n} and k = 0) to the formula for the νth Fourier coefficient of the classical Doi-Naganuma lifting of a holomorphic modular form f = ∑ a_{n}q^{n} ∈ M_{k}(Γ) (cf. [13]). We can thus interpret the logarithm of Ψ_{n}^{(D)} as the Doi-Naganuma lifting of the nearly holomorphic modular form J_{n} ∈ M_{0}^{!}(Γ), and (28) as saying that such a multiplicative version of the Doi-Naganuma lifting exists. Again, this is the analogue of the corresponding (but harder) result proved by Bruinier for the case of the Naganuma lifting.

An interesting problem would be to show that the Hilbert modular function Ψ_{n}^{(D)} is defined over ℚ or K and to compute its values at CM points, as was done in [5] for the case D = 1. In analogy with the results of [5], we would expect these values to factor completely into small primes in a way which can be described explicitly.

§8. Second generalization: other groups. Another evident way to generalize Theorem 1 is to replace the function j(τ) by a modular function of higher level, and in particular by the Hauptmodul associated to other groups of genus 0. We discuss examples of this in this section. We will consider only the case when the genus 0 group is the group Γ_{0}^{∗}(N) (the extension of Γ_{0}(N) by the group of all Atkin-Lehner involutions W_{p}, p|N). We will look in detail at the first five values N = 2, 3, 4, 5 and 6 (although there are many further values of N for which Γ_{0}^{∗}(N) has genus 0).

The corresponding Hauptmodules (with apologies for the mixed German/English spelling!) are

j_{2}^{∗}(z) = (\frac{η(z)}{η(2z)})^{24}+ 24 + 2^{12}(\frac{η(2z)}{η(z)})^{24}

= q^{−1} + 4372 q + 96256 q^{2} + 1240002 q^{3} + 10698752 q^{4} + 74428120 q^{5} + 431529984 q^{6} + ⋯,

j_{3}^{∗}(z) = (\frac{η(z)}{η(3z)})^{12}+ 12 + 3^{6} (\frac{η(3z)}{η(z)})^{12}
= q^{−1} + 783 q + 8672 q^{2} + 65367 q^{3} + 371520 q^{4} + 1741655 q^{5} + 7161696 q^{6} + ⋯,
j_{4}^{∗}(z) = (\frac{η(z)}{η(4z)})^{8} + 8 + 4^{4}(\frac{η(4z)}{η(z)})^{8}
= q^{−1} + 276 q + 2048 q^{2} + 11202 q^{3} + 49152 q^{4} + 184024 q^{5} + 614400 q^{6} + ⋯,

j_{5}^{∗}(z) = (\frac{η(z)}{η(5z)})^{6} + 6 + 5^{3}(\frac{η(5z)}{η(z)})^{6}
= q^{−1} + 134 q + 760 q^{2} + 3345 q^{3} + 12256 q^{4} + 39350 q^{5} + 114096 q^{6} + ⋯,
j_{6}^{∗}(z) = (\frac{η(z)η(2z)}{η(3z)η(6z)})^{4}+ 4 + 3^{4}(\frac{η(3z)η(6z)}{η(z)η(2z)})^{4} = (\frac{η(z)η(3z)}{η(2z)η(6z)})^{6}+ 6 + 2^{6}(\frac{η(2z)η(6z)}{η(z)η(3z)})^{6}
= q^{−1} + 79 q + 352 q^{2} + 1431 q^{3} + 4160 q^{4} + 13015 q^{5} + 31968 q^{6} + · · · .

Each of these functions takes on algebraic integer values at CM points in the upper half-plane, and we want to give a formula for their traces.

For N > 1 one must consider Heegner points rather than arbitrary CM points. The easiest way to describe these is to say that they are the images of the roots α_{Q} ∈ ℌ of quadratic forms Q(X, Y) = aX^{2} + bXY + cY^{2} with a ≡ 0 (mod N) and (a, b, c, N) = 1. (The last condition is automatic if we assume that the −d is not divisible as a discriminant by the square of any prime dividing N, an assumption we make from now on.) This requires in particular that the discriminant −d of Q is congruent to a square modulo 4N. The class of α_{Q} modulo Γ_{0}(N) has a second invariant, apart from its class in Q_{d}/Γ, namely the value of b (mod 2N). If we fix this class, i.e., choose an integer β (mod 2N) with β^{2} ≡ −d (mod 4N) and consider only forms Q = [a, b, c] with a ≡ 0 (mod N) and b ≡ β (mod 2N), then the number of Γ_{0}(N)-equivalence classes of such Q, counted as usual with multiplicity 1/w_{Q}, is H(d), and the natural map from the quotient Q_{d,N,β}/Γ_{0}(N) (with the obvious notation) to Q_{d}/Γ is a bijection. We could then define a trace t^{(N,β)}(d) as the sum of the values of j_{N}^{∗}(α_{Q}) with Q running over a set of representatives for Q_{d,N,β}/Γ_{0}(N). In our case, however, we are looking at the smaller quotient of ℌ by Γ_{0}(N)^{∗}, and here the value of t^{(N,β)}(d) is independent of β. (This is why we chose to restrict to this case.) We can therefore define

t^{(N)}(d) = ∑_{Q} \frac{1}{w_{Q}} j_{N}^{∗}(α_{Q}) ,

where the sum runs over Γ_{0}^{∗}(N)-representatives of forms Q = [a, b, c] satisfying N|a. Here are some numerical examples with N = 2 and N = 3:

t^{(2)}(4) = \frac{1}{2} j_{2}^{∗}(\frac{1 + i}{2}) = −52 , t^{(2)}(7) = j_{2}^{∗}(\frac{1 + \sqrt{−7}}{4}) = −23 , t^{(2)}(8) = j_{2}^{∗}(\frac{\sqrt{−2}}{2}) = 152 ,

and

t^{(3)}(3) = \frac{1}{3} j_{3}^{∗}(\frac{−3 + \sqrt{−3}}{6}) = −14 , t^{(3)}(11) = j_{3}^{∗}(\frac{1 + \sqrt{−11}}{6}) = 22 .

In analogy with Theorem 1, we want to express these traces as the coefficients of some modular forms of weight 3/2. In fact, the results will be more easily formulated in the language of Jacobi forms rather than modular forms of half-integral weight. (If we were looking at Hauptmodules for groups Γ_{0}(N) rather than Γ_{0}^{∗}(N), then we would actually have to work with Jacobi forms, since the trace would then depend on both the discriminant −d and on a choice of square-root β of −d (mod 4N), as explained above. In our case it would be possible to work just with modular forms, but using Jacobi forms makes things simpler.) We begin by recalling some basic definitions and facts from the theory of Jacobi forms, referring to [4] for a more detailed exposition.

For any integers k and N > 0, the space J_{k,N} of Jacobi forms of weight k and index N (on the full modular group) is defined as the set of holomorphic functions ϕ(τ, z) on ℌ × ℂ which are in a suitable sense both modular (with respect to the action (τ, z) ↦ ((aτ + b)/(cτ + d), z/(cτ + d)) of Γ on ℌ × ℂ) and elliptic (with respect to the translations of z by elements of the lattice ℤτ + ℤ), and which satisfy a suitable condition of holomorphy at infinity. The transformation equations imply that ϕ has a Fourier development of the form

ϕ(τ, z) = ∑_{n,r∈ℤ} c(n, r) q^{n}ζ^{r} (q = e^{2πiτ} , ζ = e^{2πiz})

in which the coefficient c(n, r) depends only on the quantity 4Nn − r^{2} and on r (mod 2N). The holomorphy condition at infinity is that c(n, r) vanishes unless 4Nn − r^{2} ≥ 0. If we relax the condition to merely requiring that c(n, r) = 0 if n < 0, we obtain the space of weak Jacobi forms, denoted J̃_{k,N} in [4], and if we drop it entirely we obtain the space of nearly holomorphic Jacobi forms, which we will denote J_{k,N}^{!}. The structure of the ring J̃_{∗,∗} of all weak Jacobi forms was determined in [4], Theorem 9.3 and 9.4: its even weight subring is the free polynomial algebra over M_{∗} = M_{∗}(Γ) = ℂ[E_{4}(τ), E_{6}(τ)] on two generators a = ϕ̃_{−2,1}(τ, z) ∈ J̃_{−2,1} and b = ϕ̃_{0,1}(τ, z) ∈ J̃_{0,1}, while the full ring is an extension of this by a further element c ∈ J̃_{−1,2} with 432c^{2} = ab^{3}−3E_{4}a^{3}b+ 2E_{6}a^{4}. It follows from this description that the rings J_{ev,∗}^{!} and J_{∗,∗}^{!} are given by a similar formula, but with M_{∗}(Γ) replaced by M_{∗}^{!}(Γ) = ℂ[E_{4}, E_{6}, Δ^{−1}]/(E_{4}^{3} −E_{6}^{2} = 1728Δ). The Fourier expansions of a and b begin

a = (ζ − 2 + ζ^{−1}) + (−2ζ^{2} + 8ζ − 12 + 8ζ^{−1} − 2ζ^{−2}) q
+ (ζ^{3} − 12ζ^{2} + 39ζ − 56 + …) q^{2} + (8ζ^{3} − 56ζ^{2} + 152ζ − 208 + …) q^{3} + ⋯
b = (ζ + 10 + ζ^{−1}) + (10ζ^{2} − 64ζ + 108 − 64ζ + 10ζ^{2}) q

+ (ζ^{3} + 108ζ^{2} − 513ζ + 808 − …) q^{2} + (−64ζ^{3} + 808ζ^{2} − 2752ζ + 4016 − …) q^{3} + ⋯,

where the “ … ” means that the coefficients of negative powers of ζ have been omitted (the transformation equation of Jacobi forms imply that c(n, −r) = (−1)^{k}c(n, r) for a Jacobi form of weight k). The above-mentioned periodicity property of the Fourier coefficients is clearly visible in these expansions.

If N = 1, then the periodicity property of the Fourier coefficients mentioned above implies that c(n, r) = C(4n − r^{2}) for some function d ↦ C(d), and also that k is even. There is then an isomorphism between J_{k,1} and M_{k−1/2}, and between J_{k,1}^{!} and M_{k−1/2}^{!}, sending ∑ c(n, r)q^{n}ζ^{r} to ∑ C(d)q^{d}. For example, the two forms g_{1} and g_{5} in M_{3/2} correspond to the Jacobi forms

ϕ(τ, z) = E_{4} a = (ζ − 2 + ζ^{−1}) + (−2ζ^{2} + 248ζ − 492 + …) q + ⋯,
ϕ_{5}(τ, z) = (\frac{5}{6} \frac{E_{4}^{4}}{Δ} − 816 E_{4}) a + \frac{1}{6} \frac{E_{4}^{2}E_{6}}{Δ} b
= (ζ + ζ^{−1}) q^{−1} + 0 q^{0} + (ζ^{3} + 85995ζ − 565760 + …) q + · · · .

in J̃_{2,1} and J_{2,1}^{!} , respectively. Generalizing the function ϕ = ϕ^{(1)} there are Jacobi forms ϕ^{(N)} ∈ J̃_{2,N} (2 ≤ N ≤ 6) uniquely characterized by the requirement that they have Fourier coefficients c(n, r) = B^{(N)}(4nN −r^{2}) which depend only on the discriminant r^{2}−4Nn (except in the case when N = 4 and r ≡ 0 (mod 4), in which case c(n, r) = (−1)^{r/4} B^{(4)}(16n − r^{2})), with B^{(N)}(−1) = 1, B^{(N)}(0) = −2. (In particular, the Fourier development of ϕ^{(N)} begins (ζ − 2 + ζ^{−1}) + O(q).) The representations of the forms ϕ^{(N)} in terms of the generators a = ϕ̃_{−2,1}, b = ϕ̃_{0,1} of J̃_{∗,∗} are as follows:

ϕ^{(2)} = \frac{1}{12} a (E_{4}b − E_{6}a) ,
ϕ^{(3)} = \frac{1}{12^{2}} a (E_{4}b^{2} − 2E_{6}ab + E_{4}^{2}a^{2}) ,
ϕ^{(4)} = \frac{1}{12^{3}} a (E_{4}b^{3} − 3E_{6}ab^{2} + 3E_{4}^{2}a^{2}b − E_{4}E_{6}a^{3}) ,
ϕ^{(5)} = \frac{1}{12^{4}} a (E_{4}b^{4} − 4E_{6}ab^{3} + 6E_{4}^{2}a^{2}b^{2} − 4E_{4}E_{6}a^{3}b + (4E_{6}^{2} − 3E_{4}^{3})a^{4}) ,
ϕ^{(6)} = \frac{1}{12^{5}} a (E_{4}b^{5} − 5E_{6}ab^{4} + 10E_{4}^{2}a^{2}b^{3} − 10E_{4}E_{6}a^{3}b^{2} + (8E_{6}^{2} − 3E_{4}^{3})a^{4}b − E_{4}^{2}E_{6}a^{5}) ,

so that each ϕ^{(N)}(τ, z) has the form ∑_{ν=1}^{N} f_{N,ν}(τ)ϕ̃_{−2,1}(τ, z)^{ν} ϕ̃_{0,1}(τ, z)^{N−ν} where f_{N,ν}(τ) is a holomorphic modular form of weight 2ν + 2 on SL(2, ℤ) with constant term \frac{(−1)^{ν−1}}{12^{N−1}} ({N−1 \atop ν−1}). Using these expansions one can compute the Fourier coefficients B^{(N)}(d). Here is a table for 0 < d < 30.

d  3 4 7  8 11 12 15  16 19 20 23  24 27 28
B^{(2)}(d)  52 23 −152  496 1 −1036  2256 94 −4400  8192
B^{(3)}(d) 14  34 −22 −52 138  116 −115 −348 482
B^{(4)}(d)  23  0 1 −52  94  0
B^{(5)}(d)  8  12  38  6 −20 −12  44
B^{(6)}(d)  10  28 10  −12 13 −44

(The empty entries correspond to −d which are not congruent to squares modulo 4N.) Comparing these numbers with the examples of traces t^{(N)}(d) given at the beginning of the section, we are led to the following result, which can be proved in a way analogous to the proof of Theorem 1.

Theorem 8. t^{(N)}(d) = −B^{(N)}(d) for all d and N.

§9. Third generalization: other weights. The results discussed so far concerned the modular forms j(τ) and J_{m}(τ) of weight 0, the modular forms f_{d} of weight 1/2, and the modular forms g_{D} of weight 3/2. It is natural to wonder whether there are similar results for other weights.

In fact our story so far involved not only the weights 0, 1/2, and 3/2, but also the weight 2, although in an invisible form: the existence of the Hauptmodul, a special modular form of weight 0, depended on the fact that the modular curve being studied had genus 0 and hence that the space of modular forms of the complementary weight 2 contained no cusp forms. We will find analogous results for all other pairs of half-integral weights (k + \frac{1}{2}, \frac{3}{2} − k) for which there are no non-zero cusp forms of weight 2k on SL(2, ℤ), i.e., for k = 2, 3, 4, 5, and 7.

We look first at the case k = 2. For each nonpositive discriminant −d there is a unique form u_{d} ∈ M_{5/2}^{!} such that u_{d} = q^{−d} + O(q) at infinity, and for each positive discriminant D there is a unique form v_{D} ∈ M_{−1/2}^{!} such that v_{D} = q^{−D} + O(1) at infinity. The Fourier expansions of the first few of these functions begin

u_{0} = 1 − 10q − 70q^{4} − 48q^{5} − 120q^{8} − 250q^{9} − 240q^{12} − 240q^{13} + O(q^{16}) ,
u_{3} = q^{−3} + 64q − 32384q^{4} + 131565q^{5} − 4257024q^{8} + 11535936q^{9} + O(q^{12}) ,
u_{4} = q^{−4} − 108q − 131976q^{4} − 656800q^{5} − 34867000q^{8} − 109046412q^{9} + O(q^{12}) ,
u_{7} = q^{−7} + 513q − 4451328q^{4} + 35655680q^{5} − 6275241984q^{8} + O(q^{9}) ,
u_{8} = q^{−8} − 808q − 12327152q^{4} − 112985280q^{5} − 27914678946q^{8} + O(q^{9}) ,

v_{1} = q^{−1} + 10 − 64q^{3} + 108q^{4} − 513q^{7} + 808q^{8} − 2752q^{11} + 4016q^{12} + O(q^{15}) ,
v_{4} = q^{−4} + 70 + 32384q^{3} + 131976q^{4} + 4451328q^{7} + 12327152q^{8} + O(q^{11}) ,
v_{5} = q^{−5} + 48 − 131565q^{3} + 656800q^{4} − 35655680q^{7} + 112985280q^{8} + O(q^{11}) ,
v_{8} = q^{−8} + 120 + 4257024q^{3} + 34867000q^{4} + 6275241984q^{7} + O(q^{8}) ,
v_{9} = q^{−9} + 250 − 11535936q^{3} + 109046412q^{4} − 27774693612q^{7} + O(q^{8}) .

These values suggest the following result, whose proof is entirely analogous to that of Theorem 4.

Theorem 9. For all D and d the coefficient of q^{D} in u_{d} is the negative of the coefficient of q^{d} in v_{D}.

The corresponding result holds also for the other cases k = 3, 4, 5, and 7. We will come back to this at the end of the section, but first we want to know whether there is any analogue of Theorem 1 for the new coefficients, and specifically whether the coefficients of the form v_{1} (and then later of the other forms v_{D} and u_{d}, in analogy with the generalization of Theorem 1 given in §7) can be interpreted as traces of some modular function.

The answer is yes, but we have to work with non-holomorphic modular functions. This is in a sense quite natural. The values k = 2, 3, 4, 5, 7 for which S_{k+1/2} = {0} are, by virtue of the Shimura-Kohnen correspondence, precisely those for which S_{2k}(Γ) = {0}, as already mentioned, so their special behaviour has to do with the pair of integral weights 2k and 2−2k, rather than 2 and 0 as in the case we studied originally. In particular, for these values of k there is a unique nearly holomorphic modular form F_{k} ∈ M_{2−2k}^{!}(Γ) having a Fourier development starting q^{−1} + O(1). We would like to take the sum of the values of this function at all points of Q_{d}/Γ, but of course this makes no sense since F_{k} is not Γ-invariant. However, there is a well-known non-holomorphic differentiation operator which increases the weight of a modular form by 2 (at the cost of making it mildly non-holomorphic), and by applying this the right number of times to the function F_{k} we get a Γ-invariant function which we then can evaluate on Q_{d}/Γ. For k = 2 the function F_{k} is given by

F_{2}(τ) = \frac{E_{4}(τ)E_{6}(τ)}{Δ(τ)} = q^{−1} − 240 − 141444 q − 8529280 q^{2} − 238758390 q^{3} − ⋯

and its non-holomorphic derivative is the function

K(τ) = −(\frac{1}{2πi} \frac{∂}{∂τ} + \frac{1}{2πℑ(τ)})F_{2}(τ) = \frac{E_{2}^{∗}(τ)E_{4}(τ)E_{6}(τ) + 3E_{4}(τ)^{3} + 2E_{6}(τ)^{2}}{6Δ(τ)} ,

where E_{4} and E_{6} are the usual Eisenstein series of weights 4 and 6 on SL_{2}(ℤ) and

E_{2}^{∗}(τ) = 1 − 24 ∑_{n=1}^{∞} \frac{n q^{n}}{1 − q^{n}} − \frac{3}{πℑ(τ)} = 1 − \frac{3}{πℑ(τ)} − 24 q − 72 q^{2} − 96 q^{3} − 168 q^{4} − ⋯

the non-holomorphic Eisenstein series of weight 2 on Γ.

It is well-known that, although E_{2}^{∗}(z) is not holomorphic, it has the same arithmetic property as a holomorphic modular form with rational Fourier coefficients, namely that its quotient by a meromorphic modular form of weight 2 has algebraic values (belonging to the usual class field) at CM points in the upper half-plane. It follows that K(τ) is algebraic at all CM points. Here are a few sample values.

K(\frac{1 + \sqrt{−3}}{2}) = −576, K(i) = 864, K(\frac{1 + \sqrt{−7}}{2}) = −3591, K(\sqrt{−2}) = 6464,

K(\frac{1 + \sqrt{−15}}{2}) = \frac{−176625 − 78939\sqrt{5}}{2} , K(\frac{1 + \sqrt{−15}}{4}) = \frac{−176625 + 78939\sqrt{5}}{2} .

Note that in all of these examples, the values of K(α_{Q}) are algebraic integers, but this is not always the case. For instance, the values of K(α_{Q}) for the three inequivalent quadratic forms of discriminant −31 are the roots of the polynomial

X^{3} + 37235619 X^{2} − \frac{61346290410}{31} X + 1143159756791823

in which the coefficients of X^{2} and X^{0} (trace and norm) are integral and even divisible by 31, but the coefficient of X has a denominator 31. This is because the values of E_{4}(z)/Δ(z)^{1/3} = \sqrt[3]{j(z)} and E_{6}(z)/Δ(z)^{1/2} = \sqrt{j(z) − 1728} for z ∈ Q_{d} are algebraic integers, but E_{2}^{∗}(z)/Δ(z)^{1/6} in general has a denominator \sqrt{−d}. For instance, the values of E_{2}^{∗}(z)/Δ(z)^{1/6} for z ∈ Q_{31} (with an appropriate choice of Δ(z)^{1/6}) are equal to 3α/\sqrt{−31} with α a root of α^{3} − 26α^{2} + 85α − 141 = 0.

Comparing the values of K(τ) just given with the coefficients of the function v_{1} given above, we are led to formulate the following theorem, which is analogous to Theorem 1 and can be proved in a similar way:

Theorem 10. For all positive integers d congruent to 0 or 3 (mod 4) we have

∑_{Q∈Q_{d}/Γ} \frac{1}{w_{Q}} K(α_{Q}) = d β(d) ,  (28)

where β(d) denotes the coefficient of q^{d} in the function v_{1} ∈ M_{−1/2}^{!} .

All of the extensions of Theorem 1 which we discussed in the previous sections carry over to the new situation. For instance, we can consider the image K_{m} = K ∘T (m) of K under the mth Hecke operators (m ≥ 1), which has the form p_{m}(j(τ))K(τ) + j(τ)(j(τ) − 1728)p′_{m}(j(τ)) for a certain monic polynomial p_{m}(j) of degree m − 1 (p_{1} = 1, p_{2} = j − 504, p_{3} = j^{2} − 1248j + 180252, …), determined by the requirement that p_{m}(j(τ))F_{2}(τ) = q^{−m} + O(1). Then (28) generalizes to

∑_{Q∈Q_{d}/Γ} \frac{1}{w_{Q}} K_{m}(α_{Q}) = d β_{m}(d) ,  (29)

where β_{m}(d) is the coefficient of q^{d} in the image of v_{1} under T (m), given for m = p prime by

β_{p}(d) = p^{3} β(p^{2}d) + p (\frac{−d}{p}) β(d) + β(p^{−2}d) .

As numerical examples of (29), for m = 2 we find the values

\frac{1}{3 · 3} K_{2}(\frac{1 + \sqrt{−3}}{2}) = 32256 = 8β(12) − 2β(3) ,
\frac{1}{2 · 4} K_{2}(i) = 132192 = 8β(16) ,
\frac{1}{7} K_{2}(\frac{1 + \sqrt{−7}}{2}) = 4450302 = 8β(28) + 2β(7) ,
\frac{1}{12}(K_{2}(\sqrt{−3}) + \frac{1}{3} K_{2}(\frac{1 + \sqrt{−3}}{2})) = 450927936 = 8β(48) + β(3) .

The results of §7 also carry over in the obvious way. For instance, the number −78939 occurring as the coefficient of \sqrt{5} in the formula for K((1+\sqrt{−15})/2) given above is 3/5 times the coefficient −131565 of q^{3} in v_{5} (or of −q^{5} in u_{3}). One can also give a generalization to higher levels along the lines of the formulas in §8, for all levels having no cusp forms of weight 4.

We now turn to the higher values 3, 4, 5, 6 and 7 of k. The situation here turns out to be similar, but with a difference according as k is even or odd. The forms F_{k} ∈ M_{2−2k}^{!} for these k are E_{4}^{2}/Δ, E_{6}/Δ, E_{4}/Δ and 1/Δ, respectively. The associated non-holomorphic modular forms

of weight 0 are Φ_{k} = (−1)^{k−1} ∂_{−2} ∘ ∂_{−4} ⋯ ∂_{2−2k}(F_{k}) , where ∂_{h} is the operator \frac{1}{2πi} \frac{∂}{∂τ} − \frac{h}{4πℑ(τ)}

which maps modular forms of weight h to modular forms of weight h + 2. The value of Φ_{2} = K was given above. The others are

Φ_{3}(τ) = \frac{3E_{2}^{∗2}E_{4}^{2} + 12E_{2}^{∗}E_{4}E_{6} + 13E_{4}^{3} + 8E_{6}^{2}}{36Δ} ,
Φ_{4}(τ) = \frac{5E_{2}^{∗3}E_{6} + 15E_{2}^{∗2}E_{4}^{2} + 27E_{2}^{∗}E_{4}E_{6} + 16E_{4}^{3} + 9E_{6}^{2}}{72Δ} ,
Φ_{5}(τ) = \frac{35E_{2}^{∗4}E_{4} + 70E_{2}^{∗3}E_{6} + 120E_{2}^{∗2}E_{4}^{2} + 130E_{2}^{∗}E_{4}E_{6} + 51E_{4}^{3} + 26E_{6}^{2}}{432Δ} ,
Φ_{7}(τ) = \frac{385E_{2}^{∗6} + 525E_{2}^{∗4}E_{4} + 280E_{2}^{∗3}E_{6} + 315E_{2}^{∗2}E_{4}^{2} + 168E_{2}^{∗}E_{4}E_{6} + 39E_{4}^{3} + 16E_{6}^{2}}{1728Δ} .

In particular, each Φ_{k} belongs to ℚ(J)[K], e.g. Φ_{3} = \frac{(6K − 5J − 264)^{2}}{12(J − 984)} + 2K − \frac{13}{12}J − 38 .

On the half-integral side we define γ_{k}(τ) to be the unique modular form in M_{−k+3/2}^{!} with a Fourier expansion beginning q^{−1} + O(1) if k is even (i.e., k = 2 or 4), and the unique modular form in M_{k+1/2}^{!} with a Fourier expansion beginning q^{−1} + O(q) if k is odd (i.e., k = 3, 5 or 7). If

α = q^{−1} − 2 + 8q^{3} − 12q^{4} + 39q^{7} − 56q^{8} + 152q^{11} − 208q^{12} + ⋯,
β = q^{−1} + 10 − 64q^{3} + 108q^{4} − 513q^{7} + 808q^{8} − 2752q^{11} + ⋯

denote the forms in M_{−5/2}^{!} and M_{−1/2}^{!} corresponding under the isomorphism M_{k−1/2}^{!} ≈ J_{k,1}^{!} to the

two generators a = ϕ̃_{−2,1}(τ, z) ∈ J̃_{−2,1} and b = ϕ̃_{0,1}(τ, z) ∈ J̃_{0,1} of the ring of weak Jacobi forms, then the γ_{k} are given by γ_{2} ( = v_{1}) = β and γ_{4} = α for k even and by

γ_{3} = \frac{5E_{6} α + E_{4} β}{6} = q^{−1} − 384 q^{3} + 1248 q^{4} − 19473 q^{7} + 40768 q^{8} − 272512 q^{11} + ⋯,
γ_{5} = \frac{5 E_{4}^{2} α + E_{6} β}{6} = q^{−1} + 312 q^{3} − 1632 q^{4} + 57351 q^{7} − 144704 q^{8} + 1503272 q^{11} + ⋯,
γ_{7} = \frac{5 E_{4}E_{6}α + E_{4}^{2}β}{6} = q^{−1} − 144 q^{3} + 1248 q^{4} − 109473 q^{7} + 340288 q^{8} − 5768752 q^{11} + ⋯

for k odd. The generalization of Theorem 10 to higher weights is then:

Theorem 11. Let k ∈ {2, 3, 4, 5, 7} and d > 0 an integer congruent to 0 or 3 modulo 4. Then

∑_{Q∈Q_{d}/Γ} \frac{1}{w_{Q}} Φ_{k}(α_{Q}) = \{ \begin{matrix}−(−d)^{k/2} · coefficient of q^{d} in γ_{k} & if k is even, \\ −(−d)^{(1−k)/2} · coefficient of q^{d} in γ_{k} & if k is odd. \end{matrix}

The reader can use the formula for Φ_{3} as a rational function of J and K and the values of J(α_{Q}) and K(α_{Q}) given earlier to check this theorem for k = 3 and the first few values of d.

The most surprising aspect of the theorem is the dependence on the parity of k, with the traces of Φ_{k}(α_{Q}) being divisible by a power of d (as in (28)) for even k and having a power of d as denominator for odd k. This dichotomy of course arises because of the definition of the Kohnen “plus space”, which forces us to look at half-integral weights congruent to 3/2 modulo 2 (since we want coefficients indexed by integers congruent to 0 or 3 modulo 4) and therefore to choose γ_{k} of weight \frac{3}{2} − k or \frac{1}{2} + k according whether k is even or odd.

Again, there are results of the same type also for forms of higher level, so long as the weight and level are such that the corresponding space of Jacobi forms, or of modular forms of integral weight, contains no cusp forms. In fact, there are also results, though not as clean, when cusp forms are present. For instance, for level 1 and k = 6, 8 or 10 (so that dim S_{2k}(Γ) = 1), there is no form in M_{2−2k}^{!} with a Fourier development beginning q^{−1} +O(1), and similarly no form q^{−1} +O(1) in M_{−k+3/2}^{!} , but there is a form in M_{2−2k}^{!} with Fourier expansion beginning q^{−2} + Cq^{−1} + O(1) and a form in M_{−k+3/2}^{!} with Fourier expansion beginning q^{−4} + C′q^{−1} + O(1), and the traces of the values at CM points of the (k − 1)st nonholomorphic derivative of the former are expressible in terms of the Fourier coefficients of the latter. Presumably (although I have not computed any examples) the twisted trace of the values at CM points of discriminant −Dd will also be given as a multiple of the dth coefficient of a half-integral weight form with Fourier expansion starting q^{−4D} + Aq^{−D} + O(1), for all d and D. If this picture is correct, then these correspondences would constitute a kind of Shimura lifting, compatible with Hecke operators, between nearly holomorphic modular forms of half-integral and integral weights. I leave this as an open problem. A possible method of proof would be to use renormalized integrals against a meromorphic theta-function kernel, as explained in the appendix of [6].

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