Mock modular forms and class number relations
© Mertens; licensee Springer. 2014
Received: 16 January 2014
Accepted: 22 January 2014
Published: 12 August 2014
Almost 40 years ago, H. Cohen formulated a conjecture about the modularity of a certain infinite family of functions involving the generating function of the Hurwitz class numbers of binary quadratic forms.
We use techniques from the theory of modular, mock modular, and Jacobi forms.
In this paper, we prove a slight improvement of Cohen’s original conjecture.
From our main result, we derive so far unknown recurrence relations for Hurwitz class numbers.
11E41; 11F37; 11F30
Since the days of C.F. Gauß, it has been an important problem in number theory to determine the class numbers of binary quadratic forms. One aspect of this, which is also of interest regarding computational issues, is the so-called class number relations. These express certain sums of class numbers in terms of more elementary arithmetic functions which are easier to understand and computationally more feasible. The first examples of these relations are due to Kronecker  and Hurwitz [2, 3].
and is the usual k-th power divisor sum.
Other such examples of class number relations can be obtained, e.g., from the famous Eichler-Selberg trace formula for cusp forms on .
is a modular form of weight on Γ0(4) (, Theorem 3.1). This yields many interesting relations in the shape of (1) and (3) for H (r,n).
The case r = 1, where H(1,n) = H (n), was treated around the same time by Zagier (, see Chapter 2 in ): He showed that the function is in fact not a modular form but can be completed by a non-holomorphic term such that the completed function transforms like a modular form of weight on Γ0(4).
In more recent years, this phenomenon has been understood in a broader context: The discovery of the theory behind Ramanujan’s mock theta functions by Zwegers , Bruinier and Funke , Bringmann and Ono  and many, many others has revealed that the function is an example of a weight mock modular form, i.e., the holomorphic part of a harmonic weak Maaß form (see ‘Methods’ section for a definition). Note that in the literature, the spelling ‘Maass form’ is more common, although these functions are named after the German mathematician Hans Maaß (1911 to 1992). Using this theory, some quite unexpected connections to combinatorics occur, as for example in , where class numbers were related to ranks of so-called overpartitions.
From Zagier’s and his own results, as well as computer calculations, he conjectured that the following should be true.
(). The coefficient of X ℓ in the formal power series in (4) is a (holomorphic) modular form of weight ℓ + 2 on Γ0(4).
The goal of this paper is to prove the following result.
Conjecture 1 is true. Moreover, for ℓ > 0, the coefficient of X ℓ in (4) is a cusp form.
This obviously implies new relations for Hurwitz class numbers which to the author’s knowledge have not been proven so far. We give some of them explicitly in Corollary 2.
The main idea of the proof of Theorem 1 is to relate both summands in the coefficient of the above power series to objects which in accordance to the nomenclature in  should be called quasi mixed mock modular forms, complete them, such that they transform like modular forms and show that the completion terms cancel each other out. The same idea is also used in a recent paper by Bringmann and Kane  in which they also prove several identities for sums of Hurwitz class numbers conjectured by Brown et al. in .
The outline of this paper is as follows: The preliminaries and notations are explained in ‘Methods’ section. ‘Results and discussion’ section contains some useful identities and other lemmas which will be used in ‘Conclusions’ section to prove Cohen’s conjecture.
Since many of the proofs involve rather long calculations, we omit some of them here. More detailed proofs will be available in the author’s PhD thesis .
Note that, e.g., in , the letter ϑ stands for the Jacobi theta series.
Mock modular forms
- 1.The weight k slash operator by
- 2.The weight k hyperbolic Laplacian by (τ = x + i y)
(f| k γ)(τ) = f (τ) for all γ ∈ Γ and .
(Δ k f)(τ) = 0 for all .
f grows at most linearly exponentially in all cusps of Γ.
is the incomplete gamma function.
A mock modular form is the holomorphic part of a harmonic weak Maaß form.
There are several generalizations of mock modular forms, e.g., mixed mock modular forms, which are essentially products of mock modular forms and usual holomorphic modular forms. For details, we refer the reader to Section 7.3 in .
where of course acts via (Q,γ) ↦ γ tr Q γ .
We have the following result concerning a modular completion of the function which was already mentioned in the introduction (cf., Chapter 2 and Theorem 2).
transforms under Γ0(4) like a modular form of weight.
The idea of the proof is to write as a linear combination of Eisenstein series of weight , in analogy to the proof of Theorem 3.1 in . These series diverge, but using an idea of Hecke (cf., § 2), who used it to derive the transformation law of the weight 2 Eisenstein series E2, one finds the non-holomorphic completion term .
It is easy to check that is indeed a harmonic weak Maaß form of weight . As a mock modular form, the function is rather peculiar since it is basically the only example of such an object which is holomorphic at the cusps of Γ0(4) (cf., Section 7).
where a = e2 π iu, b = e2 π iv, and q = e2 π i τ.
where for the second equality in (9) we refer to Lemma 1.7 in .
This function R has some nice properties, a few of which are collected in the following propositions.
(, Proposition 1.9). The function R fulfills the elliptic transformation properties
(i) R (u + 1; τ) = - R (u ; τ).
(ii) R (u ; τ) + e- 2π iu - π i τR (u + τ ; τ) = 2 e- π iu - π i τ/4.
(iii) R (- u) = R (u).
The following proposition has already been mentioned in . The proof is a straightforward computation.
which will henceforth be referred to as the completion of the Appell-Lerch sum A1.
Results and discussion
As we mentioned in the introduction, we would like to relate the two summands for each coefficient in the power series in Conjecture 1 to some sort of modular object. For that purpose, we recall the definition of Rankin-Cohen brackets as given in p. 53 of , which differs slightly (see below) from the original one in Theorem 7.1 of .
Here, the letter Γ denotes the usual gamma function.
It is well-known (cf., Theorem 7.1) that if f,g transform like modular forms of weight k and ℓ, respectively, then [ f,g] n transforms like a modular form of weight k + ℓ + 2 n and that [ f,g] 0 = f · g. The interaction of the first Rankin-Cohen bracket, which itself fulfills the Jacobi identity of Lie brackets, and the regular product of modular forms give the graded algebra of modular forms the additional structure of a Poisson algebra (cf., p. 53).
Note that our definition of the Rankin-Cohen bracket differs from the one in Theorem 7.1 of , by a factor of n!(- 2 π i) n which guarantees that if f and g have integer Fourier coefficients, so does [ f,g] n .
with λ ℓ as in (2). The coefficient of X2 k + 1is identically 0.
where g (τ) : = f(|D|τ). This yields the assertion by plugging in , , and D=1.
Since in the Rankin-Cohen brackets that we consider here, we have linear combinations of products of derivatives of a mock modular form and a regular modular form, one could call an object like this a quasi mixed mock modular form.
where again a = e2π iuand b = e2 π iv.
First we remark that the right-hand side of the identity to be shown is actually well-defined because as a function of u, A1(u,v;τ) has simple poles in (cf., Proposition 1.4) which cancel out if the sum is only taken over odd integers. Thus, the equation actually makes sense.
This is easily seen to be the same as .
transforms like a modular form of weight 2 k + 2. Because the Fourier coefficients of the holomorphic parts grow polynomially, they are holomorphic at the cusps as well.
are indeed equal up to sign and that the function in (14) is modular on Γ0(4).
This shows that we will need some specific information about the derivatives of the Jacobi theta series and the R-function evaluated at the torsion point .
A simple and straight forward calculation gives us the following result.
with Θ as in (6) and ϑ as in (5).
The identities (19) and (20) follow directly by applying the transformation properties (iii), (i), and (ii) of R in Proposition 2.
with β as in (10). Note that for convenience, we define sgn(0):=1.
we get the assertion by a straightforward calculation.
Now, we take a closer look at (16).
Again, we only show the former equation, the latter follows from the transformation laws. For simplicity, we omit the arguments of the functions considered.
Interchanging the sums gives the desired result.
holds true for alland the function in (14) is modular on Γ0(4).
does as well.
and hence the corollary.
we see that for any (not necessarily holomorphic) modular form f of even weight k on Γ0(4), the function is a modular form of the same weight on Γ0(4) as well. In particular, this applies to for all .
The second summand in (14) transforms like a modular form on Γ0(4).
the assertion follows.
We now prove Theorem 1 using Corollary 1. The proof is an induction on m. Since the base case m=0 gives an alternative proof of the class number relation (3) by Eichler, we give this as a proof of an additional theorem.
We recall that this function is a modular form of weight 2 on Γ0(4) (cf. e.g. , Proposition 1.1).
is indeed a holomorphic modular form of weight 2 on Γ0 (4) as well.
Since the space of modular forms of weight 2 on Γ0 (4) is two-dimensional, the assertion follows by comparing the first two Fourier coefficients of the function above and .
The proof of this given in  involves topological arguments about the action of Hecke operators on the Riemann surface associated to Γ0(2) on the one hand and arithmetic of quaternion orders on the other hand.
The base case of our induction is treated above, thus suppose that (23) holds true for one .
For simplicity, we omit again the argument in the occuring R derivatives.
which proves Conjecture 1.
The fact that we actually get a cusp form can be seen in the following way:
and by definition vanishes at the cusp i ∞ for all . So by the above equation, it vanishes at every cusp of Γ0(4).
The first two of the above relations were already mentioned in .
The formula (4) looks indeed very similar to the Eichler-Selberg trace formula as given in , so one might ask whether our result gives a similar trace formula for Hecke operators on the space of cusp forms of weight k on Γ0(4). Computer experiments suggest that in fact for k ≥ 1, the coefficient of X2 kin (4) equals , where denotes the n th Hecke operator on . This will be shown in an upcoming publication , since it requires different methods than the ones applied here.
The author’s research is supported by the DFG-Graduiertenkolleg 1269 ‘Global Structures in Geometry and Analysis.’ This paper is part of the author’s PhD thesis, written under the supervision of Prof. Dr. K. Bringmann at the Universität zu Köln, whom the author would like to thank for suggesting this topic as part of his PhD thesis . He also thanks his colleagues at the Universtät zu Köln, especially Dr. Ben Kane, Maryna Viazovska, and René Olivetto, for the many fruitful and helpful discussions.
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