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On p-adic modular forms and the Bloch–Okounkov theorem
- Michael J. Griffin^{1},
- Marie Jameson^{2} and
- Sarah Trebat-Leder^{3}Email author
https://doi.org/10.1186/s40687-016-0055-z
© The Author(s) 2016
- Received: 23 September 2015
- Accepted: 12 January 2016
- Published: 10 May 2016
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
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.
Zagier revisited this work in [11], giving a significantly shorter proof of the Bloch–Okounkov Theorem and studying additional properties of the q-bracket.
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
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 \(\mathcal {Q}_k(\tau )\)
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
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
Proof
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.
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
Jacobi forms are invariant under the action of matrices with respect to the slash operator and elliptic transformations.
Definition 3
- 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
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 \(\mathcal {Q}_k^{(p)} \in \widetilde{M}_k(p^2)\).
Proposition 4.3
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.
Proposition 4.4
Proof
We are now ready to prove part (d) of Theorem 1.1.
Proof of Theorem1.1(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
Declarations
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.
Open AccessThis 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.
Authors’ Affiliations
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