# On *p*-adic modular forms and the Bloch–Okounkov theorem

- 529 Downloads

## Abstract

Bloch–Okounkov studied certain functions on partitions *f* called shifted symmetric polynomials. They showed that certain *q*-series arising from these functions (the so-called *q*-brackets \(\left<f\right>_q\)) are quasimodular forms. We revisit a family of such functions, denoted \(Q_k\), and study the *p*-adic properties of their *q*-brackets. To do this, we define regularized versions \(Q_k^{(p)}\) for primes *p*. We also use Jacobi forms to show that the \(\left<Q_k^{(p)}\right>_q\) are quasimodular and find explicit expressions for them in terms of the \(\left<Q_k\right>_q\).

### Keywords

Congruences for modular forms*p*-adic modular forms Jacobi forms

### Mathematics Subject Classification

11F33 11F50## 1 Introduction and statement of results

*p*-adic modular forms, which are

*p*-adic limits of compatible families of

*q*-expansions of classical level one modular forms. The first example of this phenomenon comes from the Eisenstein series

*k*is a positive even integer, \(\tau \in \mathbb {H},\) \(q = e^{2 \pi i \tau }\), \(B_k\) is the

*k*th Bernoulli number, \(\sigma _{k - 1}(n)\) is the sum of the \(k - 1\) powers of the divisors of

*n*, and \(\widetilde{M}_k\) is the space of weight

*k*quasimodular forms. For primes \(p \ge 5\), we regularize \(G_k\) to

*p*-adic modular form for \(k \not \equiv 0 \pmod {p - 1}\). Note that if instead \( k \equiv 0 \pmod {p - 1}\), then the constant term is not

*p*-integral, but the normalized Eisenstein series \(E_k(\tau )\) satisfies \( E_k(\tau ) \equiv 1 \pmod {p^r}\) whenever \(k \equiv 0 \pmod {\phi (p^r)}\).

Katz [7] and others have reformulated and expanded this theory to consider *p*-adic modular forms as *p*-adic analytic functions on elliptic curves. However, in this paper we will only consider *p*-adic modular forms in the sense of Serre.

In this article, we wish to examine the *p*-adic properties of certain quasimodular forms \(\langle Q_k \rangle _q\) and show that they are in many ways analogous to the Eisenstein series \(G_k\) and fit into Serre’s framework.

*q*-bracket of

*f*” to be the following formal power series obtained by “averaging”:

*q*-series \(\left<f\right>_q\) is a quasimodular form on the full modular group. The space of shifted symmetric polynomials is generated by functions \(Q_k\), which we will define explicitly in Sect. 2; the first few are given by

*q*-bracket \(\left<f\right>_q\). We give \(\mathcal {R}\) a grading by assigning to \(Q_k\) the weight

*k*. Then the Bloch–Okounkov Theorem states that for \(f \in \mathcal {R}\) homogeneous of grading

*k*, \(\left<f\right>_q\) is a quasimodular form of weight

*k*on the full modular group. In their paper, Bloch and Okounkov also defined so-called

*n*-point correlation functions which are related to \(\left<Q_k\right>_q,\) and gave a formula for them involving derivatives of a theta function. We will use a special case of this result in equation (7) of Sect. 4.

Zagier revisited this work in [11], giving a significantly shorter proof of the Bloch–Okounkov Theorem and studying additional properties of the *q*-bracket.

*q*-series.

**Theorem 1.1**

- (a)
If \(k_1, k_2 \not \equiv 0 \pmod {p - 1}\), then \(\mathcal {Q}_{k_1}^{(p)} \equiv \mathcal {Q}_{k_2}^{(p)} \pmod {p^r}\) whenever \(k_1 \equiv k_2 \pmod {\phi (p^r)}.\)

- (b)
If \(k \not \equiv 0 \pmod {p - 1}\), then \(\mathcal {Q}_{k}^{(p)}\) is a

*p*-adic modular form. - (c)
If \(p > k\), then the modulo

*p*filtration of \(\mathcal {Q}_k^{(p)}\) (and \(\mathcal {Q}_k\)) is \(k(p + 1)/2\). - (d)
The

*q*-series \(\mathcal {Q}_k^{(p)}\) is a quasimodular form of weight*k*on \(\Gamma _0(p^2)\). - (e)
We have that \(\mathcal {Q}_k^{(p)}(\tau ) = \mathcal {Q}_k(\tau ) - p^{k - 1}\mathcal {Q}_k(p^2 \tau ) - p^{k - 1}f_k^{(p)},\) where \(f_k^{(p)}\) is given explicitly in Sect. 4. In particular, \(f_k^{(p)}\) is supported on \(q^N\) with \(\genfrac(){}{}{2}{p} = \genfrac(){}{}{N}{p}\).

*Remark*

In [8], Lopez studied the functions \(\left<Q_{3}^{2n}\right>_q\) and showed that they also satisfy parts (a) and (b) of our theorem. It seems likely that other products of the \(Q_k\) yield quasimodular forms which satisfy similar *p*-adic properties.

*Example*

*q*-bracket \(\mathcal {Q}_2\) and its regularization \(\mathcal {Q}_2^{(5)}\) are

*q*:

In Sect. 2, we will explicitly define the functions \(Q_k:\mathcal {P}\rightarrow \mathbb {Q},\) as well as \(\mathcal {Q}_{k}^{(p)}\) and various other functions. In Sect. 3, we will prove parts (a)–(c) of Theorem 1.1. In Sect. 4, we will make a connection to the theory of Jacobi forms in order to prove parts (d) and (e) of Theorem 1.1.

## 2 Preliminary definitions

### 2.1 Definitions of \(P_k(\lambda ), Q_k(\lambda )\), and \(\mathcal {Q}_k(\tau )\)

*r*is the length of the longest diagonal in the Young diagram of \(\lambda \) (i.e.,

*r*is the size of the Durfee square of \(\lambda \)) and \(a_1, \ldots a_r\) (resp., \(b_1, \ldots , b_r\)) are the arm-lengths (resp., leg-lengths) of the cells on this diagonal. For example, the partition \(\lambda = (4, 3, 1)\) has Frobenius coordinates (2; 3, 1; 2, 0) as seen in the Young diagram below.

*k*).

*q*-bracket (as defined in 1) of \(Q_k\) is a quasimodular form for all non-negative integers

*k*. We will work with

*q*-series besides the constant term are integral.

### 2.2 Definitions of \(P_k^{(p)}(\lambda ), Q_k^{(p)}(\lambda ),\) and \(\mathcal {Q}_k^{(p)}(\tau )\)

*Remark*

By matching up conjugate partitions, we can see that \(\mathcal {Q}_k(\tau )\) and \(\mathcal {Q}_k^{(p)}(\tau )\) equal zero for odd *k*.

## 3 Congruences and *p*-adic modular forms

### 3.1 Congruences

Now we will show that our regularizations \(\mathcal {Q}_k^{(p)}(\tau )\) satisfy congruences analogous to those which are known for the Eisenstein series, proving parts (a) and (b) of Theorem 1.1. We focus on weights *k* that are not multiples of \(p - 1\), as that is when the constant term of \(\mathcal {Q}_k^{(p)}\) is *p*-integral. Note that this implies that \(p \ge 5\).

**Theorem**

*Proof*

### 3.2 *p*-adic modular forms

Now, we will use these congruence results to show that our regularizations \(\mathcal {Q}_k^{(p)}\) are *p*-adic modular forms. As before, we focus on weights *k* that are not multiples of \(p - 1\). First, we define a *p*-adic modular form.

**Definition 1**

We say that \(f = \sum a_n q^n \in \mathbb {Q}_p[[q]]\) is a *p*-adic modular form if there exists \(f_i \in M_{k_i}\) with rational coefficients which converge uniformly to the coefficients of *f* in \(\mathbb {Q}_p\). In this situation, we write \(f_i \rightarrow f\).

*Remark*

*X*. For \(p > 2\), we have that

*Remark*

Every level \(p^n\) modular form is a level 1 *p*-adic modular form of the same weight. This includes for instance the regularized Eisenstein series \(G_k^{(p)}\) for \(p \ge 5, k \not \equiv 0 \pmod {p - 1}\). In this case, we can see that \(G_{k + \phi (p^i)} \rightarrow G_k^{(p)}\).

Note that since \(E_2, E_4, E_6\) are *p*-adic modular forms, all quasimodular forms are too. Thus, since we will show that the \(\mathcal {Q}_k^{(p)}\)’s are quasimodular in Sect. 4, we will have that they are *p*-adic modular forms as well. However, we can also show this using the above congruences, which give them as *p*-adic limits of the \(\mathcal {Q}_k\)’s.

**Theorem**

*p*-adic modular form of weight

*k*, with

*Proof*

*p*-adically. Since the \(g_i\) are quasimodular, they are

*p*-adic modular forms, and hence the \(\mathcal {Q}_k^{(p)}\) are too. \(\square \)

*Remark*

If \(k \equiv 0 \pmod {p - 1}\), then \(B_k^{(p)}\) is not *p*-integral and hence we do not get congruences for the constant term of \(\mathcal {Q}_k^{(p)}\). However, just as with the Eisenstein series, we can renormalize so that the constant term is one. Kummer’s congruences for the Bernoulli numbers imply that the resulting functions will also converge *p*-adically. In the special case \(k=0,\) the result converges *p*-adically to 1.

### 3.3 Filtration

In addition to studying *p*-adic modular forms, we can also study modulo-*p* modular forms. One of the most important properties of modulo-*p* modular forms are their filtration. See [10] for more details.

**Definition 2**

Let \(p \ge 5\). The filtration of \(f \in \mathbb {F}_p[[q]]\) is denoted as *w*(*f*) and is defined to be the smallest integer *k* such that *f* is the modulo *p* reduction of a modular form of weight *k* and level 1 with coefficients in \(\mathbb {Q}\cap \mathbb {Z}_p\).

**Theorem**

(part (c) of Theorem 1.1) If \(p > k\) then the modulo *p* filtration of \(\mathcal {Q}_k^{(p)} (\)and \(\mathcal {Q}_k\)) is \(k(p + 1)/2\).

*Proof*

First, note that \(\mathcal {Q}_k \equiv \mathcal {Q}_k^{(p)} \pmod {p}\) since they only differ modulo higher powers of *p*. Thus they must have the same filtration.

*k*/ 2 in \(G_2,\) and Theorem 2 of [11] gives us the leading coefficient:

*p*. Also, it is well-known that \(w(G_2^i) = iw(G_2) = i(p + 1)\) for all \(i \ge 1\). Thus the filtration of \(\frac{(k - 1)!!\; 8^{k/2 - 1}}{k/2} G_2^{k/2}\) is \(k(p+1)/2,\) and the filtrations of the “lower degree terms” are strictly smaller. It follows that the filtration of \(\mathcal {Q}_k\) is \(k(p+1)/2\), as desired. \(\square \)

## 4 Jacobi forms and the quasimodularity of \(\mathcal {Q}_k^{(p)}\)

In this section, we will show that \(\mathcal {Q}_k^{(p)}(\tau )\) is quasimodular for every prime *p* and non-negative integer *k*. We define certain auxiliary functions \(F(z,\tau )\) and \(F^{(p)}(z,\tau )\) which are generating functions for the *q*-brackets \(\mathcal {Q}_k(\tau )\) and \(\mathcal {Q}_k^{(p)}(\tau )\), and we show these functions are Jacobi forms. We then make use of the theory of Jacobi forms to prove our quasimodularity result.

### 4.1 Definitions of \(F(z,\tau )\) and \(F^{(p)}(z,\tau )\)

We can describe the function \(F^{(p)}(z,\tau )\) in terms of \(F(z,\tau )\) as follows.

**Proposition 4.1**

*Proof*

### 4.2 Jacobi forms

*k*index

*m*slash operator by

Jacobi forms are invariant under the action of matrices with respect to the slash operator and elliptic transformations.

**Definition 3**

*k*and

*m*are integers, a holomorphic Jacobi form of weight

*k*and index

*m*for some subgroup \(\Gamma \) of \(\mathrm {SL}_2(\mathbb {Z})\) is a holomorphic function

- 1.for every \(\gamma \in \Gamma ,\)$$\begin{aligned} \left( \phi |_{k,m}\gamma \right) (z; t)=\phi (z;\tau ), \end{aligned}$$
- 2.for every pair of integers
*a*and*b*,$$\begin{aligned} \phi \left( z+a\tau +b;\tau \right) =e(-m(a^2\tau +2az))\phi (z;\tau ), \end{aligned}$$ - 3.and the function \(\phi (z;\tau )\) has a Fourier expansion of the form$$\begin{aligned} \phi (z;\tau )=\sum _{\begin{array}{c} n,r\in \mathbb {Z}\\ n\ge \frac{r^2}{4m} \end{array}}c(n,r)q^n\zeta ^r. \end{aligned}$$

We refer to the variable *z* in the definition above as the elliptic variable and to \(\tau \) as the modular variable.

A meromorphic Jacobi form is a function which satisfies properties (1), (2), and (3) in the definition above, but is required only to be meromorphic in the elliptic variable and weakly holomorphic in the modular variable—that is for fixed *z*, the function \(\tau \mapsto \phi (z;\tau )\) is holomorphic on \(\mathbb {H}\) and meromorphic at the cusps of \(\mathbb {H}/\Gamma \). For more details on Jacobi forms, see [6].

**Proposition 4.2**

The functions \(F(z;\tau )\) and \(F^{(p)}(z;\tau )\) are meromorphic Jacobi forms of weight 1 and index \(-2\) for \(\mathrm {SL}_2(\mathbb {Z})\) and \(\Gamma _0(p^2)\), respectively, with simple poles at the points \(z\in \frac{1}{2}\mathbb {Z}\oplus \frac{\tau }{2}\mathbb {Z}\), and \(z\in \frac{1}{2p}\mathbb {Z}\oplus \frac{\tau }{2}\mathbb {Z}\), respectively.

*Proof*

*c*, the action of the matrix simply permutes these terms, leaving only the expected automorphy factor, \((c\tau +d) \ e\left( -2c\frac{z^2}{c\tau +d}\right) .\) \(\square \)

### 4.3 Showing quasimodularity

In this section, we will prove the following.

**Theorem**

(part (d) of Theorem 1.1) Let \(p \ge 5\) be prime. Then we have that \(\mathcal {Q}_k^{(p)} \in \widetilde{M}_k(p^2)\).

*F*and \(F^{(p)}\) so that the

*q*-brackets and regularized

*q*-brackets arise from derivatives of these functions. Although derivatives of Jacobi forms are not generally Jacobi forms themselves, there is a certain differential operator \(Y_{m}\) which preserves the modularity properties, but sacrifices holomorphicity. Let \(Y_{m}\) be the Jacobi raising operator defined as follows (see [2], pg.43], or [5], Def.2.5]):

**Proposition 4.3**

*k*and index

*m*, then \(Y_{m} (\phi )\) transforms like a Jacobi form of weight \(k+1\) and index

*m*.

*Proof*

Eichler and Zagier show that the Taylor coefficients with respect to the elliptic variable of holomorphic Jacobi forms are quasimodular forms [6]. The idea is as follows: suppose \(\phi (z;\tau )\) is any function which is invariant under the slash operator \(|_{k,m}.\) If \(\phi (0,\tau )\) is defined, than the transformation laws imply this function in \(\tau \) transforms like a weight *k* modular form. Thus, if \(\Phi (z;\tau )\) is a holomorphic Jacobi form of index *m*, then \(Y^n_m(\Phi )(0;\tau )\) transforms like a modular form. It is not difficult then to see that the holomorphic component \(\left( \frac{1}{2\pi i}\frac{\partial }{\partial z}\right) ^n(\Phi )(0;\tau )\) must be quasimodular.

*F*and \(F^{(p)}\) both have poles at \(z=0.\) We can work around this problem by taking a residue at \(z=0\); however, we must be careful how we do this. The following procedure is similar to that followed by other authors, including Bringmann and Folsom [4], Section3] and Olivetto [9]. Suppose

*G*(

*z*) is any function which is real-analytic near \(z=0\) with a singularity of at most finite order at 0 (i.e., there is some positive integer

*j*such that \(|z|^{2j} G(z)\) is real-analytic in a neighborhood of 0). Then

*G*(

*z*) can be written as a Laurant series in

*z*and \(\overline{z},\) or in polar coordinates by setting \(z=r~e^{2 \pi i \theta }\) and \(\overline{z}=r~e^{-2 \pi i \theta }.\) Then we define \(\mathop {\mathrm{Res}}\limits _{z=0}\frac{1}{z}G(z)\) to be the limit of the integral

*G*(

*z*) at 0 is a negative power of \(z\overline{z}.\) In our case, the denominators will only have powers of the holomorphic variable.

**Proposition 4.4**

*z*near \(z=0\) with at most a singularity of finite order at \(z=0\) and which transforms like a weight

*k*index

*m*Jacobi form on \(\Gamma \). If the residue

*k*for \(\Gamma \).

*Proof*

We are now ready to prove part (d) of Theorem 1.1.

*Proof of Theorem1.1(d)*

*y*whose coefficients, \(f_m,\) are holomorphic

*q*-series. In fact we see that \(f_0=2^{k-1}\mathcal {Q}_k(\tau ),\) and each of the remaining \(f_m\) can be written in terms derivatives of brackets \(\mathcal {Q}_\ell (\tau )\) with \(\ell <k.\) Since \(\widetilde{\mathcal {Q}}_k(\tau )\) transforms like a modular form, it follows that \(\mathcal {Q}_k(\tau ) \in \widetilde{M}_{k}\). A nearly identical argument holds for \(\widetilde{\mathcal {Q}}_k^{(p)}(\tau );\) since \(\widetilde{\mathcal {Q}}_k^{(p)}(\tau )\) transforms like a modular form for \(\Gamma _0(p^2),\) we must have that \(\mathcal {Q}_k^{(p)}(\tau )\in \widetilde{M}_{k}(p^2)\). This completes the proof of Theorem 1.1(d).

### 4.4 Finding an explicit expression

Just as we can write \(G_k^{(p)}\) in terms of \(G_k\), we can write \(\mathcal {Q}_k^{(p)}\) in terms of \(\mathcal {Q}_k\). However, there is an extra correction term

**Theorem**

*Proof*

*F*,

*p*. When

*p*is an odd prime, we have that \(F(z;\tau )-F^{(p)}(z;\tau )\) is given by

*p*divides \(2m+1.\) If we separate the terms where

*p*divides

*n*, \(F(z;\tau )-F^{(p)}(z;\tau )\) becomes

*k*is even, we use Eq. (8) above to find have that

## Notes

### Acknowledgements

The authors began jointly discussing this work at the Spring School on Characters of Representations and Modular Forms held at the Max Planck Institute in Bonn, Germany, in March 2015 and are grateful for the good hospitality and excellent conference. The authors would also like to thank Don Zagier for his inspiring work, and Ken Ono and the referee for their helpful comments. The second and third authors thank the National Science Foundation for its support.

### References

- 1.Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partition. Bull. Amer. Math. Soc. (N.S.)
**18**(2), 167–171 (1988). doi: 10.1090/S0273-0979-1988-15637-6 MathSciNetCrossRefMATHGoogle Scholar - 2.Berndt, R., Schmidt,R.: Elements of the Representation Theory of the Jacobi Group. Progress in Mathematics, vol.163, Birkhäuser Verlag, Basel (1998). doi: 10.1007/978-3-0348-0283-3
- 3.Bloch, S., Okounkov, A.: The character of the infinite wedge representation. Adv. Math.
**149**(1), 1–60 (2000). doi: 10.1006/aima.1999.1845 MathSciNetCrossRefMATHGoogle Scholar - 4.Bringmann, K., Folsom, A.: Almost harmonic Maass forms and Kac–Wakimoto characters. J. Reine Angew. Math.
**694**, 179–202 (2014). doi: 10.1515/crelle-2012-0102 MathSciNetMATHGoogle Scholar - 5.Conley, C., Raum, M.: Harmonic Maass–Jacobi forms of degree \(1\) with higher rank indices. Preprint. arXiv:1012.2897 (2010)
- 6.Eichler, M., Zagier, D.:The Theory of Jacobi Forms. Progress in Mathematics, vol. 55. Birkhäuser Boston, Inc., Boston (1985). doi: 10.1007/978-1-4684-9162-3
- 7.Katz, N.M.: \(p\)-adic properties of modular schemes and modular forms. In: Modular Functions of One Variable, III (Proc. Internat.Summer School, Univ. Antwerp, Antwerp, 1972). Lecture Notes in Mathematics, vol. 350, pp. 69–190. Springer, Berlin (1973)Google Scholar
- 8.Lopez, A.: Kummer congruences arising from the mirror symmetry of an elliptic curve. Proc. Amer. Math. Soc. (accepted for publication)Google Scholar
- 9.Olivetto, R.: On the fourier coefficients of meromorphic jacobi forms. Int. J. Number Theory
**10**(6), 1519–1540 (2014). doi: 10.1142/S1793042114500419 - 10.Serre, J.-P.: Formes modulaires et fonctions zêta \(p\)-adiques. In: Modular Functions of One Variable, III (Proc. Internat.Summer School, Univ. Antwerp, 1972). Lecture Notes in Mathematics, vol. 350, pp. 191–268. Springer, Berlin (1973)Google Scholar
- 11.Zagier, D: Partitions, quasimodular forms, and the Bloch–Okounkov theorem. Ramanujan J. (2015). doi: 10.1007/s11139-015-9730-8
- 12.Zwegers, S.: Rank-crank type PDE’s for higher level Appell functions. Acta Arith.
**144**(3), 263–273 (2010). doi: 10.4064/aa144-3-4 MathSciNetCrossRefMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.