Class invariants for certain non-holomorphic modular functions

Inspired by prior work of Bruinier and Ono and Mertens and Rolen, we study class polynomials for non-holomorphic modular functions arising from modular forms of negative weight. In particular, we give general conditions for the irreducibility of class polynomials. This allows us to easily generate infintely many new class invariants.


Introduction and Statement of the Main Results
Let us consider Klein's j-invariant, j(τ ), which is defined by where q := e 2πiτ and τ := x + iy ∈ H with x, y ∈ R. The function j(τ ) is a modular function and its evaluations at CM points (quadratic imaginary points in H) are called singular moduli. These distinguished numbers play a central role in explicit class field theory and the classical theory of complex multiplication. In particular, they are algebraic, and they generate so-called Hilbert class fields of imaginary quadratic fields. For a general survey of this theory, see e.g. Borel [2]. Throughout, let D ≡ 0, 1 (mod 4) be a negative integer and let Q red D be the set of SL 2 (Z)-reduced positive definite integral binary quadratic forms, i.e., those forms Q(x, y) = ax 2 + bxy + cy 2 , with −a < b ≤ a < c or 0 ≤ b ≤ a = c and with discriminant D = b 2 −4ac. Each quadratic form of discriminant D is uniquely SL 2 (Z)-equivalent to exactly one form in Q red D . The associated complex point τ Q of a given quadratic form Q is the unique point in the upper half plane which satisfies the equation Q(τ Q , 1) = 0. We call the associated value j(τ Q ) the singular modulus corresponding to τ Q . Now we introduce the Hilbert class polynomial as the following: Date: April 23, 2015. 1 Celebrated results give that it is irreducible and generates the Hilbert class field of Q( √ D) if D is fundamental. For more general discriminants, it generates ring class fields (see Borel [2]).
Analogously, we set for any (possibly non-holomorphic) modular function and where P D denotes the set of quadratic forms in Q red D with gcd(a, b, c) = 1. One can easily see as in [7] the following relation between H D and H D where h(d) is the class number of discriminant d, ε(a) = 1 if f ≡ ±1 (mod 12) and ε(a) = −1 otherwise. In pathbreaking work, Bruinier and Ono connected these polynomial to partitions, when P was an explicit non-holomorphic form of level 6, yielding a formula for the number of partitions as a finite sum of algebraic numbers. They, together with Sutherland, conjectured that the associated polynomials always generate ring class fields, and this was shown by Mertens and Rolen in [17]. Many others have studied properties of these non-holomorphic singular moduli (see [1,7,12,13,14]). Here, we consider the general problem of constructing class invariants (i.e., modular forms whose CM-values generate Hilbert class fields) from non-holomorphic modular forms of a special shape.
In particular, we show the following, where B 0 , B 1 , B 2 , B 3 and B 4 are defined in the proof of Theorem 1.1 and, in the setting of 1.1, are sums which depend on m and k, and where P is an iterated non-holomorphic derivative of F of weight 0. Theorem 1.1. Suppose F is a weakly holomorphic modular form F of weight −2k on SL 2 (Z) with principal part m n=1 a n q −n and rational Fourier coefficients. If D is a fundamental discriminant satisfying Remark 1.2. The bounds used in this paper can be easily adapted for higher levels. Therefore, once one explicitly knows the higher level Heegner points, a similar analysis could be done to generate class invariants. (For example, in level 6 for quadratic forms of discriminant −24n + 1 the Heegner points have been determined by Dewar and Murty, see [9].) Note that it is easy to show that the singular moduli in Theorem 1.1 lie in the Hilbert class field (see Lemma 4.4 in [6]). In particular, we have the following. These bounds can also be simplified to give the following.
Remark 1.5. These bounds depend on a parameter a. In Section 4, we give explicit examples and discuss how to make an optimal choice of a in a special case.
The paper is organized as following. In Section 2 we review relevant background information including Masser's formula and a convenient form of Shimura reciprocity due to Schertz and we recall the Maass-Poincaré series. The proof of Theorem 1.1 and Corollary 1.4 is subject of Section 3. In Section 4 we apply Corollary 1.3 in the specific example of F = E 10 /∆.

Acknowledgements
This project was written at the Cologne Young Researchers in Number Theory Program 2015. The authors wish to thank the organizer, Larry Rolen, for his generous support and his advisement throughout the project. Furthermore, they want to thank for the generous support covered by the DFG Grant D-72133-G-403-151001011 of Larry Rolen. They would also like to thank Michael Griffin for helpful conversations. Further, the authors are grateful to Claudia Alfes for useful comments.

Preliminaries
2.1. Differential operators. Let F be a weakly holomorphic form of weight −2k.
We apply the Maass raising operator, defined for l ∈ N by , k times to F to get a non-holomorphic modular function of weight 0 The Maass raising operator maps a (not necessarily holomorphic) modular form of weight k to a (possibly) non-holomorphic modular form of weight k + 2. It is a canonical way to raise a modular form of negative weight to a non-holomorphic modular function (see Bump [8]).

2.2.
Masser's formula. If one applies the Maass raising operator k times to a weakly holomorphic modular form of weight −2k, one gets a non-holomorphic modular function. We have to interpolate these non-holomorphic modular function by a holomorphic modular function at CM-points of certain quadratic forms to be able to apply later Schertz's theorem (which is a special case of the general theory of Shimura reciprocity). The non-holomorphic modular function P F we obtain is an almost holomorphic modular form. The ring of almost holomorphic modular forms is the ring of functions which transform as a modular form but instead of being holomorphic they are a polynomial in 1/ℑ(τ ) with coefficients which are holomorphic. It is well-known that this ring is generated by E 2 (τ ) − 3/πℑ(τ ), E 4 (τ ), E 6 (τ ) where E 2 , E 4 and E 6 are the normalized Eisenstein series of weight 2, 4 and 6. The modular form E 2 (τ ) − 3/πℑ(τ ) is also known as E * 2 (τ ). For a survey of almost holomorphic modular functions, see e.g. Zagier [22].
Hence, we can express the non-holomorphic modular function as a polynomial in these three generators. Because E 4 and E 6 are holomorphic, the only function we have to interpolate is E * 2 . Masser states a useful formula to interpolate E * 2 at quadratic imaginary points in H by a (meromorphic) modular form, see Appendix I in Masser [16]. For this purpose, let V be a system of representatives of SL 2 (Z)\Γ −D for a discriminant D < 0 with associated positive definite integral binary quadratic form Q and corresponding CM point τ Q and let Γ −D be the set of all primitive integral 2 × 2-matrices of determinant −D. Then we introduce the modular polynomial by By expanding this polynomial, we can define the numbers β µ,ν (τ Q ) by

Now we can state the relation which Masser gives.
Proposition 2.1 ( [16]). If Q is a positive definite integral binary quadratic form of discriminant D < 0 and τ Q is the associated CM point, then Thus we get the following.
This lemma reduces us to studying classical modular functions, where work of Schertz applies.

Maass-Poincaré series.
In the proof of Theorem 1.1 we will have need of careful estimates for the coefficients of weakly holomorphic modular forms. This is conveniently provided by the theory of Maass-Poincaré series due to Niebur [18] and Fay [10], as further developed by many others (see e.g. [3,4]).
For v > 0, k ∈ Z and s ∈ C we define where M ν,µ denotes the usual M-Whittaker function (see e.g. [11], p. 1014). Using this, we construct the following Poincaré series for Γ 0 (N): Here m ∈ N, τ = u + iv ∈ H, ℜ(s) > 1, k ∈ −N, and Γ ∞ := {± ( 1 n 0 1 ) | n ∈ Z}. In the special case when k < 0 and s = 1 − k 2 , the Poincaré series P m,k,N := P m,s,k,N defines a harmonic Maass form of weight k for Γ 0 (N) whose principal part at the cusp ∞ is given by q −m , and at all other cusps the principal part is 0 (which is not of importance for our purposes because we are only interested in level N = 1). For more background on Poincaré series and harmonic Maass forms see [5,19].
In this situation, we can give explicit Fourier expansions. In order to do so we agree on the following notation. By I s and J s we denote the usual I-and J-Bessel where the incomplete gamma function Γ(α, x) is defined as and the coefficients b m,k,N (l, v) are as follows.

Work of Schertz.
In this subsection we review work of Schertz which gives a convenient description of the singular moduli of modular functions.
Definition 2.4. Let N ∈ N and D = t 2 d < 0 be a discriminant with t ∈ N and d a fundamental discriminant. Moreover, let {Q 1 , . . . , Q r } be a system of representatives of primitive quadratic forms modulo SL 2 (Z). We call the set {Q 1 , . . . , Q r } an N-system mod t if the conditions gcd(c j , N) = 1 and b j ≡ b l (mod 2N), 1 ≤ j, l ≤ r are satisfied.
Theorem 2.5 (Schertz, [20]). Let g be a modular function for Γ 0 (N) for some N ∈ N whose Fourier coefficients at all cusps lie in the Nth cyclotomic field. Suppose furthermore that g(τ ) and g(− 1 τ ) have rational Fourier coefficients, and let Q(x, y) = ax 2 + bxy + cz 2 be a quadratic form with discriminant D = t 2 d, d a fundamental discriminant, with gcd(d, N) = 1 and N|a. Then, unless g has a pole at τ Q , we have that g(τ Q ) ∈ Ω t , where Ω t is the ring class field of the order of conductor t in Q( √ d). Moreover if {Q = Q 1 , Q 2 , . . . , Q h } is an N-system mod t, then where Gal D is the Galois group of Ω t /Q( √ D).
Remark 2.6. Schertz also proves that an N-system mod t always exists.
2.5. Quadratic forms. In [22] it is stated that every SL 2 (Z)-equivalence class of quadratic forms with discriminant D has a unique representative in We prove that in this set there is one representative where the absolute value of the corresponding CM-point is larger than the others. This will be an important ingredient of our proof of irreducibility of the class polynomials.
Proposition 2.7. For every discriminant D there is exactly one equivalence class whose representantive in Q red D has a = 1. Proof. We consider two cases: (1) D ≡ 0 mod 4: The form [1, 0, −D/4] certainly belongs to Q red D . Now for every representative with a = 1 in Q red D we have that b = 0. We immediately get that c = −D/4.   Then where |E F (τ )| is absolutely bounded by an explicit constant in (2).
For the constant term of the Fourier-expansion we get Continuing, by (1) we obtain Using some simple estimates and that y lies in the fundamental domain of the full modular group, therefore y ≥ √ 3/2, we get Now we use the estimates for the coefficients of Poincaré series which were established above and get E P n,−2k,1 and .

Now we estimate the first sum by
.
and a similar result also holds if f ′ (x) < 0 this last sum can be estimated against an integral The integral can be evaluated explicitly, which easily yields the following bound Using these estimates, one obtains For the second sum we first note that We can therefore estimate Thus, k which completes the proof.
We are now in position to prove Theorem 1.1.
Proof of Theorem 1.1. By Theorem 2.5 and 2.2, we immediateley see that the Galois group of Ω t /Q( √ D) acts transitively on the roots and therefore the polynomial must be a power of an irreducible polynomial. We now prove that for large enough D there cannot be multiple roots. Suppose that P (τ Q 1 ) = P (τ Q 2 ) where τ Q 1 is the CM-point of the quadratic form with a = 1. We will see that for large enough discriminant −D, the value at this point will be larger than the rest, so we cannot have multiple roots. We set e 2πiτ Q j =: q j for j = 1, 2. One then obtains that We then have using Lemma 3.1 that Hence one obtains Using some estimates in the spirit of Lemma 3.1 we get that (−1) k−r k r We also have that Hence, if the polynomial is reducible we must have that where Equivalently, we have Note that as a function of −D the right hand side of the last inequality stays bounded so for large enough −D this inequality cannot be satisfied and hence the polynomials must be irreducible.
Proof of Corollary 1.4. Since all negative fundamental discriminants D with |D| < 15 have class number one, these polynomials are automatically irreducible. Therefore we assume that −D ≥ 15. We have k r=0 k r A simple calculation gives that for c > 1 and √ −D > k mπ k 2c−1 c − 1 the following inequality holds: Therefore the polynomial is irreducible if √ −D > k mπ k 2c−1 c − 1 and √ −D > 2 π log 615cm 1+k m + k √ 3π k m n=1 |a n | |a m | .

Example
We consider F := E 10 /∆, a weakly holomorphic modular form of weight −2. Zagier previously considered this example in his foundational paper on singular moduli [21]. Since E 10 does not vanish at infinity and ∆ is a cusp form with a simple root at infinity, F has a simple pole at infinity. Thus we have Hence, in the setting of Corollary 1.4 we have m = k = b 1 = 1. Therefore, the two inequalities reduce to √ −D > max c π(c − 1) , 2 π log 615c 1 + 1 √ 3π . which is easily seen to be irreducible (in fact, its roots generate the field Q( √ 5)).