Quantum modularity of partial theta series with periodic coefficients

Theorem 1.1.

Let f be a function with period M≥2 and support Sf⁢(k0). Let α∈Q. If f is even, then Θf⁢(α) is a quantum modular form of weight 32 on AM with respect to ΓM. If f is odd, then θf⁢(α) is a “strong” quantum modular form of weight 12 on Q with respect to ΓM and is a quantum modular form of weight 12 on BM with respect to ΓM.

Remark 1.2.

(i) The main novelty of Theorem 1.1 is that one does not require Θf⁢(z) or θf⁢(z) to be a cusp form. For example, consider θψ⁢(z), where ψ is given in Section 4.2 (cf. [20]). Otherwise, one can invoke (2.4), (2.9) and [8, Theorem 1.1].

(ii) In Theorem 1.1, Θf⁢(z) satisfies

for all γ=(abcd)∈ΓM and α∈AM, where

Here, rγ,f:ℝ→ℂ is a C∞ function which is real-analytic in ℝ∖{γ-1⁢(i⁢∞)} and χ is a multiplier given by

(iii) For τ∈ℍ-:={τ∈ℂ:Im(τ)<0}, let Θ^f⁢(τ) denote the non-holomorphic Eichler integral

In Theorem 1.1, θf⁢(α) is a “strong” quantum modular form in the following sense (see [24] or [30]):

1. θf and Θ^f “agree to infinite order” at all rational numbers (see Lemma 2.6);

2. for τ∈ℍ- and γ∈ΓM, we have

   Θ^f⁢(τ)-(Θ^f|12,χ⁢γ)⁢(τ)=rγ,f⁢(τ),

   where

   rγ,f⁢(τ)=1i⁢M⁢∫γ-1⁢(i⁢∞)i⁢∞Θf⁢(w)⁢(w-τ)-12⁢d⁢w.

Here, rγ,f⁢(τ) is a holomorphic function in ℍ-, extends as a ℂ∞ function to ℝ and is real-analytic in

Also, χ is the multiplier as in (1.6). A close inspection of the techniques in [24] reveals that one needs convergence of Θ^f⁢(τ) for τ∈ℍ- (and not necessarily at rational points) to deduce the strong quantum modularity property for θf⁢(z). To ensure this condition, Θf⁢(z) does not have to be a cusp form. For a similar approach, see [3, 17].

(iv) If f is a function with period M≥2 and support Sf⁢(k0), then θf⁢(z) is a sum of a modular form and a (strong) quantum modular form both of weight 12 and Θf⁢(z) is a sum of a modular form and a quantum modular form both of weight 32. To see this, write f as f⁢(n)=fe⁢(n)+fo⁢(n), where

Clearly, fe⁢(n) (respectively, fo⁢(n)) is an even (respectively, odd) function of period M with support contained in Sf⁢(k0). Indeed, if Sf,e⁢(k0) denotes (respectively, Sf,o⁢(k0)) the support of fe⁢(n) (respectively, fo⁢(n)), then Sf,e⁢(k0)=ℳf,e⁢(k0)∪{M-k:k∈ℳf,e⁢(k0)} (respectively, Sf,o⁢(k0)=ℳf,o⁢(k0)∪{M-k:k∈ℳf,o⁢(k0)}) for some ℳf,e⁢(k0),ℳf,o⁢(k0)⊆ℳf⁢(k0). Thus, we have

Now, apply Lemma 2.1 to θf(e)⁢(z) and Theorem 1.1 to θf(o)⁢(z). Similarly, apply Theorem 1.1 to Θf(e)⁢(z) and Lemma 2.2 to Θf(o)⁢(z).

(v) As pointed out by the referee, there is a “duality” in the proof of Theorem 1.1. If f is even, then the quantum modularity of Θf⁢(z) is driven by the modularity of θf⁢(z) and if f is odd, then the (strong) quantum modularity of θf⁢(z) is driven by the modularity of Θf⁢(z). See Lemmas 2.1 and 2.2.

Lemma 2.1.

Let f be an even function with period M≥2 and support Sf⁢(k0). For all γ=(abcd)∈ΓM, we have

Proof.

From (1.3), we have

where

Note that (2.3) follows by changing n→-n-1 in the second sum in (2.2). Since support of f is Sf⁢(k0), (2.3) yields

where θ⁢(z;k,M) is the theta series

By [28, Proposition 2.1], we see that θ⁢(z;k,M) satisfies

for all γ=(abcd)∈ΓM. Also, since γ∈ΓM, we have for some integer j that

where n has been replaced by n-2⁢j⁢k in the second sum in (2.6). Noting that

(2.1) now follows from (2.4) and (2.5)–(2.7). ∎

Lemma 2.2.

Let f be an odd function with period M≥2 and support Sf⁢(k0). For all γ=(abcd)∈ΓM, we have

Proof.

If M is even, then f⁢(M2)=0 for odd f. So we have

where

Using [28, Proposition 2.1] (with A=[M], ν=1 and P⁢(m)=m), we have

for all γ=(abcd)∈ΓM. Also, since γ∈ΓM, we have for some integer j that

where n has been replaced by n-2⁢j⁢k in the sum in (2.11). Thus, combining (2.9)–(2.11) yields (2.8). ∎

Lemma 2.3.

Let f be an even function with period M≥2 and support Sf⁢(k0). Then

where, for k∈Mf⁢(k0), k′=k2-k02⁢M∈Z≥0.

Proof.

We have

where (2.13) follows from Jacobi’s triple product identity

with z→-qk and q→qM2 in (2.14). As in the proof of Lemma 2.1, we have

Thus, (2.13) and (2.15) imply (2.12), where, for k∈ℳf⁢(k0), k′=k2-k02⁢M. Since k′∈ℤ implies k2≥k0, we conclude that k′∈ℤ≥0. ∎

Lemma 2.4.

Let f be an even function with period M≥2 and support Sf⁢(k0). Assume ∑′k∈Mf⁢(k0)f⁢(k)=0. Let α∈Q be such that α≠γ⁢(i⁢∞) for any γ∈ΓM. Then we have

Proof.

For any 1≤k<M, we obtain the following upon using [27, Chapter 5, page 76] with u=(y-i⁢α)⁢M and x=kM:

As in the proof of Lemma 2.1, we have

and so (2.16) follows from (2.17), (2.18) and ∑′k∈ℳf⁢(k0)f⁢(k)=0. ∎

Corollary 2.5.

Let f be an even function with period M≥2 and support Sf⁢(k0). Assume ∑′k∈M⁢(k0)f⁢(k)=0. Then we have

where o⁢(1)→0 as y→0+ and

Proof.

First, we note that γ⁢(i⁢∞)≠0 for all γ∈ΓM. Thus, we choose α=0 in Lemma 2.4 to get

where q1=e-πM⁢y and rM⁢(k,n)=e2⁢π⁢i⁢n⁢kM. Interchanging the sums in (2.19), we find

where

At this point, note that gM⁢(k0,n)=O⁢(1) and q1→0 as y→0+. This yields the result. ∎

Lemma 2.6.

Let f be an odd function with period M≥2 and support Sf⁢(k0). Then, as t→0+, we have for (p,q)=1 that

where Cf,k0(n):=∑k∈Mf⁢(k0)f(k)Cf(n,k) and

Thus, in the sense of Lawrence and Zagier [24, page 103], (2.20) and (2.21) imply that θf and Θ^f “agree to infinite order” at all rational numbers.

Proof.

For t>0, we have

For each k∈ℳf⁢(k0), Cf⁢(n,k) is an odd function with period Mq or 2⁢M⁢q according as 2 divides p or does not divide p. Also, Cf⁢(n,k) has mean value zero. This implies that Cf,k0⁢(n) is an odd function with period Mq or 2⁢M⁢q according as 2 divides p or does not divide p and with mean value zero. Thus, by [24, Proposition, page 98] and (2.22), we obtain

Next, we turn to Θ^f⁢(τ), where τ=x+i⁢y with y<0. First, we have, for w∈ℍ,

Thus, by the change of variable w→w+τ and contour integration, it follows that

To evaluate the integral on the right-hand side of (2.23), we let w→i⁢M⁢wπ⁢n2. This yields