Zeros of modular forms of half integral weight
- Amanda Folsom^{1} and
- Paul Jenkins^{2}Email author
Received: 6 May 2016
Accepted: 19 August 2016
Published: 10 October 2016
Abstract
We study canonical bases for spaces of weakly holomorphic modular forms of level 4 and weights in \(\mathbb {Z}+\frac{1}{2}\) and show that almost all modular forms in these bases have the property that many of their zeros in a fundamental domain for \(\Gamma _0(4)\) lie on a lower boundary arc of the fundamental domain. Additionally, we show that at many places on this arc, the generating function for Hurwitz class numbers is equal to a particular mock modular Poincaré series, and show that for positive weights, a particular set of Fourier coefficients of cusp forms in this canonical basis cannot simultaneously vanish.
Mathematics Subject Classification
1 Introduction
In studying functions of a complex variable, a natural problem is to determine the locations of the zeros of the functions. There are a number of recent interesting results on the zeros of modular forms. For instance, the zeros of Hecke eigenforms of integer weight k become equidistributed in the fundamental domain as \(k \rightarrow \infty \) (see [10, 17]), yet such forms still have many zeros on the boundary and center line of the fundamental domain [7, 13].
Duke and the second author [5] studied zero locations for a canonical basis \(\{f_{k, m}(\tau )\}\) for spaces of integer weight weakly holomorphic modular forms for \({\text{ SL }}_2(\mathbb {Z})\). If the weight \(k \in 2\mathbb {Z}\) is written as \(12\ell + k'\) with \(k' \in \{0, 4, 6, 8, 10, 14\}\), then the basis elements have Fourier expansions of the form \(f_{k, m}(\tau ) = q^{-m} + O(q^{\ell +1})\), where \(q = e^{2\pi i \tau }\) as usual. If \(m \ge |\ell |-\ell \), then all of the zeros of \(f_{k, m}(\tau )\) in the standard fundamental domain for \({\text{ SL }}_2(\mathbb {Z})\) lie on the unit circle. These results were extended [6, 9] to similar canonical bases \(\{f_{k, m}^{(N)}(\tau )\}\) for the spaces \(M_k^{\sharp }(N)\) of weakly holomorphic modular forms of integer weight k and level \(N = 2, 3, 4\) with poles only at the cusp at \(\infty \), showing that many of the zeros of the basis element \(f_{k, m}^{(N)}(\tau )\) lie on an appropriate arc if m is large enough.
For spaces of modular forms with weight \(k \in \mathbb {Z}+\frac{1}{2}\), canonical bases with similar Fourier expansions exist. Zagier [20] defined such bases for spaces of weakly holomorphic modular forms of level 4 and weights \(\frac{1}{2}\) and \(\frac{3}{2}\) satisfying Kohnen’s plus space condition, and analogous bases were shown in [4] to exist in level 4 for all weights \(k \in \mathbb {Z}+\frac{1}{2}\). In this paper, we address the natural question of whether the zeros of the modular forms in such a basis lie on an arc.
Theorem 1.1
For \(k \in \mathbb {Z}+\frac{1}{2}\), assume the notation above. There are absolute, positive constants A and B such that if \(m \ge A|a| + B\), then at least \(2a + \frac{m}{2}\) of the \(5a + \frac{5m}{4} + \frac{b}{2} - C - \epsilon \) zeros of \(f_{k, m}\) in a fundamental domain for \(\Gamma _0(4)\) lie on the arc \(\mathcal {A}\). Additionally, \(A \le 9\) and \(B \le 109\).
The proof of the theorem uses contour integration on a generating function for the canonical basis elements \(f_{k, m}\) to approximate \(f_{k, m}\) by a real-valued trigonometric function on \(\mathcal {A}\). We note that for specific values of k, the bounds on A and B and the quantity \(2a+\frac{m}{2}\) can often be improved; for instance, if \(a \ge 0\) we may take \(A = 0\).
Theorem 1.2
We note that a direct application of Theorem 1.1 gives \(\frac{m}{2} -2\) zeros of the form \(f_{\frac{3}{2}, m}\); however, the proof of Theorem 1.1 in Sect. 3 shows that for the weight \(k=\frac{3}{2}\), with \(b=13\), there is at least one extra zero.
Our results also lead to an interesting corollary in the case of weight \(\frac{3}{2}\) for any positive integer \(m\equiv 0,1 \pmod {4}\) which is not a square. In this case, it turns out that Theorem 1.1 provides information about the zeros of the Poincaré series \(F_{-m,\frac{3}{2}}^+\) themselves. Previously, Rankin [16] addressed the problem of understanding the number of zeros of general Poincaré series of level 1 and even integer weight at least 4, and gave an explicit bound on the number of zeros lying on the intersection of the standard fundamental domain with the boundary of the unit disk. Our work leads to results analogous to those of Rankin in the case of weight \(\frac{3}{2}\), by virtue of the fact that for positive integers \(m\equiv 0,1\pmod {4}\) such that m is not a square, we have from [2] that \(f_{\frac{3}{2},m} = F_{-m,\frac{3}{2}}^+ \). Thus, we have the following Corollary to Theorem 1.1, giving locations for the zeros of certain weight \(\frac{3}{2}\) Poincaré series. The number \(C_m\) is equal to \(\frac{3}{2}\) or \(\frac{3}{4}\), depending on whether m is even or odd.
Corollary 1.3
Assume the notation above, and let \(m\equiv 0,1 \pmod {4}\) such that m is not a square. There is an absolute positive constant A such that if \(m \ge A\), then at least \(\frac{m}{2}\) of the \(\frac{5m}{4}+\frac{3}{2} - C_m - \epsilon \) zeros of the Poincaré series \(F_{-m,\frac{3}{2}}^+ \) in a fundamental domain for \(\Gamma _0(4)\) lie on the arc \(\mathcal {A}\). Additionally, \(A \le 111\).
We note again that the bounds appearing here, for \(k = \frac{3}{2}\), are better than appear in the general statement of Theorem 1.1. Details supporting this computation appear in Sect. 6.
As an additional application, for any weight \(k \in \mathbb {Z}+\frac{1}{2}\), Theorem 1.1 implies the following non-vanishing theorem for coefficients of modular forms in the canonical basis for \(M_{2-k}^!\).
Theorem 1.4
Assume the notation from Theorem 1.1. Let \(m \ge A\left| a\right| +B\) with \(m \equiv 0, (-1)^{s-1} \pmod {4},\) and let \(\{f_{2-k, i}(\tau )\}\) be the canonical basis for the space \(M_{2-k}^!\). For any integer \(M > 3a + \frac{3m}{4} + \frac{b}{2} - C\), it is impossible for the Fourier coefficients of \(q^m\) in each of the first M basis elements \(f_{2-k, i}(\tau )\) with \(i \not \equiv m \pmod {4}\) to simultaneously vanish.
If \(2-k\) is positive and large enough, many of the basis elements in Theorem 1.4 are actually cusp forms. An analogous result in integer weights would resemble a weaker form of Lehmer’s conjecture on the nonvanishing of the coefficients of the cusp form \(\Delta (\tau )\), since \(\Delta \) is the first canonical basis element \(f_{12, -1}\) of weight 12. See also [3] for additional results on the nonvanishing of coefficients of cusp forms of half integral weight.
In Sect. 2 of this paper, we give definitions, notation, and the proof of Theorem 1.4. The main argument in the proof of Theorem 1.1 appears in Sect. 3, with associated computations appearing in Sects. 4, 5, and 6.
2 Definitions
3 Proof of Theorem 1.1
Values of c for each b
\({\varvec{b}}\) | \({\varvec{6}}\) | \({\varvec{8}}\) | \({\varvec{9}}\) | \({\varvec{10}}\) | \({\varvec{11}}\) | \({\varvec{12}}\) | \({\varvec{13} }\) | \({\varvec{14}}\) | \({\varvec{15}}\) | \({\varvec{16}}\) | \({\varvec{17}}\) | \({\varvec{19}}\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(c \, (m \text { even})\) | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
\(c \, (m \text { odd})\) | 2 | 3 | 2 | 3 | 2 | 3 | 3 | 4 | 3 | 4 | 3 | 4 |
4 Residue sums
Theorem 4.1
Proof of Theorem 4.1
Case 2: \(z\in R_2\). We proceed as in Case 1, and begin with (8). Here, when \(z\in R_2\), the function \(B_k(z;\tau )\) has eight poles in \(W_{\varepsilon ,v}\) as described above. Note that the sets \(\mathcal M_1\) and \(\mathcal M_2\) which determine these poles in Case 1 and Case 2 (respectively) are identical, save for the matrix \(\left( {\begin{matrix} 1 &{} -1 \\ 0 &{} 1\end{matrix}}\right) \) which is replaced by \(\left( {\begin{matrix} 1 &{} 3 \\ 0 &{} 1\end{matrix}}\right) \), and the matrix \(\left( {\begin{matrix} 2 &{} 1 \\ 1 &{} 1\end{matrix}}\right) \) which is replaced by \(\left( {\begin{matrix} -2 &{} -3 \\ 1 &{} 1\end{matrix}}\right) \). It is not difficult to see that \(e\left( \frac{m}{4}\right) = e\left( \frac{-3m}{4}\right) \), and that \(\kappa _{-1,k} = \kappa _{3,k}\), which reveals that the residue contributions arising from \(\left( {\begin{matrix} 1 &{} -1 \\ 0 &{} 1\end{matrix}}\right) \) and \(\left( {\begin{matrix} 1 &{} 3 \\ 0 &{} 1\end{matrix}}\right) \) are the same. Moreover, for \(w=w_{M_r^{(3)}}(z)\) where \(r=\pm 2\), we have from Proposition 5.1 that \(A_k(z,w)(4z+1)^k = \kappa _{1,k}\). Thus, the residue contributions arising from \(\left( {\begin{matrix} 2 &{} 1 \\ 1 &{} 1\end{matrix}}\right) \) and \(\left( {\begin{matrix} -2 &{} -3 \\ 1 &{} 1\end{matrix}}\right) \) are also the same. Thus, Case 2 is identical to Case 1. This yields Theorem 4.1 in the case \(\theta \in \left( \frac{\pi }{2},\frac{7\pi }{12}\right] \).
4.1 The integral weight case
This continuity argument does not seem to work when \(k \in \mathbb {Z}+ \frac{1}{2}\). In the integral weight case, fixing a value of z results in a quotient of modular forms in the variable \(\tau \) inside the integral, with a form of weight 2 in the numerator and a form of weight 0 in the denominator, and this ratio can be written as a logarithmic derivative, allowing the computation of residues. For half integral weight, the ratio of modular forms in \(\tau \) is of weight \(2-k\) in the numerator and weight 0 in the denominator, and does not simplify to a logarithmic derivative in the same way. At the poles of \(B_k(z; \tau )\) in the region \(W_{\varepsilon , v}\), though, the ratio \(A_k(z; \tau )\) of the numerator to the derivative of \(j(4\tau )\) is constant, and we can compute its values and use the logarithmic derivative to compute residues as before. However, when \(\theta = \frac{\pi }{2}\), the derivative of \(j(4\tau )\) is 0, and the argument breaks down.
5 Evaluating the function \(A_k(z;\tau )\)Ak
Proposition 5.1
In Sect. 5.1, we first establish the Proposition in the case \(\tau = w_{M_r^{(1)}}(z)\). We then use that result to establish the Proposition in the remaining cases in Sect. 5.2.
5.1 Proof of Proposition 5.1 part 1: \(\tau = w_{M_r^{(1)}}(z)\)
In what follows, we consider the cases \(r\in \{0,1,2,3\}\) separately. We elaborate on the cases \(r=0\) and \(r=2\); the cases \(r=1\) and \(r=3\) follow similarly. Recall that elements in \(M_k^!\) transform with character \(\psi _k = \psi _{k,\gamma } := \rho ^{2k}\) under \(\gamma = \left( {\begin{matrix} a &{} b \\ c &{} d\end{matrix}}\right) \in \Gamma _0(4)\), where \(\rho =\rho _\gamma \) is defined in Sect. 2.
Case \(r=0\). By definition, we have that \(g_{b,0}(z) \in M_{26}\big (\Gamma _0(4),\psi _{b+\frac{1}{2}}\psi _{25-b+\frac{1}{2}}\big )\). However, \(\psi _{b+\frac{1}{2}}\psi _{25-b+\frac{1}{2}} = \psi _{b+\frac{1}{2},\gamma }\psi _{25-b+\frac{1}{2},\gamma } =\big (\big (\frac{c}{d}\big )\varepsilon _d^{-1}\big )^{52}\) is the trivial character, hence, \(g_{b,0}(z) \in M_{26}(\Gamma _0(4))\), and thus, \(F_k(z,z) \in M_{2}(\Gamma _0(4))\). We also have that the function \(\frac{d}{dz} j(4z) \in M_2(\Gamma _0(4))\). The Sturm bound [18] for \(M_2(\Gamma _0(4))\) is \([ SL _2(\mathbb Z) : \Gamma _0(4)] \cdot \frac{2}{12} = 1\), and thus, after directly checking Fourier expansions for each of the 12 possible choices of \(b\in S\), we may conclude that \((-4\pi i)F_k(z,z) = \frac{d}{dz}j(4z)\), and hence, Proposition 5.1 holds in the case \(r=0\), for any \(k \in \frac{1}{2} + \mathbb Z\). Since \(\kappa _{r,k} = \kappa _{r',k}\) by definition (and \(g_{b,r}(z)=g_{b,r'}(z)\) as argued above) whenever \(r\equiv r' \equiv 0 \pmod {4}\), we conclude that Proposition 5.1 holds for \(\tau = w_{M_r^{(1)}}(z)\) for any \(r\equiv 0 \pmod {4}\) and any \(k\in \frac{1}{2}+ \mathbb Z\).
5.2 Proof of Proposition 5.1 part 2
\(\tau = w_{M_r^{(d)}}(z), d\in \{2,3,4\}\) Our proof of the Proposition in this case makes uses of the Proposition in the case \(d=1\), which we established in the previous section. We divide our proof into three cases corresponding to whether d is equal to 2, 3 or 4.
6 Bounds
Lemma 6.1
- i)
If \(\theta \in \big (\frac{\pi }{3},\frac{5\pi }{12}\big ]\), then \( |\mathcal C_{m,k}(\theta )| < \sqrt{2}.\)
- ii)
If \(\theta \in \big [\frac{7\pi }{12}, \frac{2\pi }{3}\big )\), then \(|\mathcal D_{m,k}(\theta ) | < \sqrt{2}.\)
Proof
If \(\frac{5\pi }{12} \le \theta \le \frac{7\pi }{12}\), we write \(z_1 = -\frac{1}{4} + \frac{1}{4} e^{i\theta }\), and let \(v_1 = 0.2125\). If \(\frac{\pi }{3}< \theta < \frac{5\pi }{12}\) or \(\frac{7\pi }{12}< \theta < \frac{2\pi }{3}\), we instead write \(z_2 = -\frac{1}{4} + \frac{1}{4} e^{i\theta }\) and let \(v_2 = 0.1375\). We have computed the following bounds using MAPLE for these values of \(\tau \) and z. Generally, this requires explicitly bounding the tail of the Fourier expansion using crude bounds on the Fourier coefficients, and then computing bounds on the initial terms on the appropriate interval. In some cases, these bounds on the initial terms require bounding the derivative and checking values on a grid of points, similar to the computations in [6].
Upper bounds for each b
\(\varvec{b}\) | For \({\varvec{z_1, \tau _1}}\) | For \(\varvec{z_2, \tau _2}\) |
---|---|---|
6 | 706609608 | 127222365875 |
8 | 554055912 | 51930014336 |
9 | 478100088 | 32392878212 |
10 | 427574714 | 20854624833 |
11 | 408921890 | 14417163525 |
12 | 325238946 | 8284899739 |
13 | 288599577 | 5296681421 |
14 | 273853210 | 3640432156 |
15 | 220558615 | 2114574952 |
16 | 196218970 | 1353920641 |
17 | 172470466 | 860720673 |
19 | 132750791 | 344722508 |
Declarations
Acknowledgments
Both authors read and approved the final manuscript.
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.
Acknowledgements
The first author is grateful for the support of NSF Grant DMS-1449679. This work was partially supported by a grant from the Simons Foundation (#281876 to Paul Jenkins). The authors also thank the referee for helpful comments.
Competing interests
The authors declare that they have no competing interests.
Authors’ Affiliations
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