1 Correction to: Ramanujan J (2015) 36:103–116 https://doi.org/10.1007/s11139-014-9621-4
This note corrects the proof of Theorem 1.1 of [1], and extends the statement of the result to odd m and also furnishes the missed statement with regard to the funding obtained from the European Research council and that provided to A. Folsom in the article note.
2 Introduction and statement of results
Let for \(m\in \mathbb {N}\)
where \((q:=e^{2\pi i\tau }, \zeta :=e^{2\pi iz}\) with \(\tau \in {\mathbb {H}}, z\in \mathbb {C}\))
is the Jacobi theta function. Note that in contrast to [1], we write \(\varphi _m\) in order to highlight the dependence on m. Denote the coefficients of the Fourier expansion (in z) by \(\chi _r,\) so that
Define the Nebentypus character \(\psi _m\) for matrices \(\gamma = ({\begin{matrix}a&{}b\\ c&{}d\end{matrix}}) \in \Gamma _0(2)\) by
Moreover, we require the well-known Eisenstein series \(E_{2j}(\tau )\). For \(j\ge 2\), they are holomorphic modular forms, while \(E_2(\tau )\) is a quasimodular form. The Bernoulli numbers \(B_{\ell }\) are defined for non-negative integers \(\ell \) by the generating function
Theorem 1.1
For \(r \in \mathbb {Z}\) and \(m \in \mathbb {N}\), we have
where for each \(0\le j \le m\) such that \(j \equiv m \pmod {2}\), the function \(D_j\) is a modular form of weight \(-j\) on \(\Gamma _0(2)\) with Nebentypus character \(\psi _m\), as defined in (1.2).
Remark
Theorem 1.1 was given for even m in [1]; above, we have extended the statement to hold for odd m. Moreover, the proof in [1] had a mistake: the second displayed formula in the proof of Proposition 3.3 was incorrect. We thank Sander Zwegers for pointing out the mistake and for fruitful discussion.
3 Proof of Theorem 1.1
Using that, for \(\lambda , \mu \in \mathbb {Z}\), we have
we obtain that
Let for \(z_0\in \mathbb {C},\ \tau \in \mathbb {H}\)
Then, with \(z_0\) such that no pole of \(\varphi _m\) lies at the boundary of \(P_{z_0}\), we compute
Using (2.1) gives
Thus (2.2) becomes
Inserting the Fourier expansion of \(\varphi _m\) yields
So (assuming \(r\ne 0\) if m is even)
We now compute (2.2) in another way, picking \(z_0=-\frac{1}{2} -\frac{\tau }{2}\). Then the only pole of \(\varphi _m\) in \(P_{z_0}\) is at \(z=0\). So, using the Residue Theorem, (2.2) equals
Write (noting that \(\varphi _m\) is even or odd, depending on the parity of m)
Inserting the series expansion of \(e^{-2\pi irz}\), (2.4) becomes
Thus, for \(r\in \mathbb {Z}\) (with the restriction that \(r\ne 0\) if m is even) we obtain by comparing with (2.3),
This gives the first equation in Theorem 1.1.
To determine \(\chi _0\) (for m even), we plug in to (1.1), which implies
We now insert the Laurent expansions around \(z=0\) on both sides. We write the sum on r as
It is not hard to see that both sums converge absolutely for \(-v<y<0\), where \(v := {\text {Im}}(\tau ), y := {\text {Im}}(z)\). We write the second summand in (2.7) as
The first summand equals
The second summand combines with the first summand in (2.7) as using that \(\varphi _m\) is an even function of z,
Thus the right hand side in (2.6) becomes
Picking off the constant term on both sides of (2.5) then gives
as claimed.
The proof of the modularity follows from the fact that for \(\gamma =({\begin{matrix}a&{}b\\ c&{}d\end{matrix}}) \in \Gamma _0(2)\), we have that
Reference
Bringmann, K., Folsom, A., Mahlburg, K.: Quasimodular forms and \(s\ell (m\vert m)\hat{}\) characters. Ramanujan J. 36, 103–116 (2015)
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The research of K. Bringmann is supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation and the research leading to these results receives funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement No. 335220 - AQSER. A. Folsom is grateful for the support of NSF Grant DMS-1449679.
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Bringmann, K., Folsom, A. & Mahlburg, K. Correction to: Quasimodular forms and \(s\ell (m\vert m)^{\wedge }\) characters. Ramanujan J 47, 237–241 (2018). https://doi.org/10.1007/s11139-018-0069-9
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DOI: https://doi.org/10.1007/s11139-018-0069-9