Quantum mock modular forms arising from eta-theta functions
Amanda Folsom, Sharon Garthwaite, Soon-Yi Kang, Holly Swisher, Stephanie Treneer
aa r X i v : . [ m a t h . N T ] A p r QUANTUM MOCK MODULAR FORMS ARISING FROM ETA-THETA FUNCTIONS
AMANDA FOLSOM, SHARON GARTHWAITE, SOON-YI KANG, HOLLY SWISHER, STEPHANIE TRENEER
Abstract.
In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, weunify the eta-theta functions by constructing mock modular forms from the eta-theta functions with evencharacters, such that the shadows of these mock modular forms are given by the eta-theta functions with oddcharacters. In addition, we prove that our mock modular forms are quantum modular forms. As corollaries,we establish simple finite hypergeometric expressions which may be used to evaluate Eichler integrals of theodd eta-theta functions, as well as some curious algebraic identities. Introduction and Statement of Results
One of the most well-known modular forms of weight 1 / η -function, defined for τ in theupper half-plane H := { τ ∈ C | Im( τ ) > } by η ( τ ) := q ∞ Y n =1 (1 − q n ) = X m ≥ (cid:16) m (cid:17) q m , (1)where q = e ( τ ), and (cid:0) ·· (cid:1) is the Kronecker symbol (throughout we set e ( u ) := e πiu ). More generally, eta-products , functions of the form c Y j =1 η ( a j τ ) b j , (2)where a j , b j , and c are positive integers, have been of interest not only within the classical theory of modularforms, but also in connection to the representation theory of finite groups. Conway and Norton [8] showedthat many character generating functions for the “Monster” group M , the largest of the finite sporadicsimple groups, could be realized as eta-quotients , which are of the same form as the functions in (2), butallow negative integer exponents b j . Mason [19] similarly exhibited many character generating functionsfor the Mathieu group M as multiplicative eta-products, meaning their q -series have multiplicative coef-ficients, as seen in (1) for example. This relationship to character generating functions in part motivatedDummit, Kisilevsky and McKay [9] to classify all multiplicative eta-products. Later, Martin [18] classifiedall multiplicative integer weight eta-quotients.In addition to being a simple example of a multiplicative q -series, the right-most function in (1) is alsoan example of a theta function , which is of the form θ χ ( τ ) := X n χ ( n ) n ν q n , (3)where χ is an even (resp. odd) Dirichlet character, and ν equals 0 (resp. 1). The sum in (3) is taken over n ∈ Z or n ∈ N , depending on whether or not χ is trivial. It is well-known that such functions are ordinarymodular forms of weight 1 / ν . While (1) shows an eta-quotient which is also a theta function, it is not truein general that all multiplicative eta-quotients are also theta functions. This question was studied by LemkeOliver [20], who classified all eta-quotients which are also theta functions; in particular, his classificationgives six odd eta-theta functions E m , and eighteen even eta-theta functions e n (some of which are twists bycertain principal characters). By “odd (resp. even) eta-theta function”, we mean an eta-quotient which isalso a theta function with odd (resp. even) character. See Section 2.1 for more on these functions.Modular theta functions also naturally emerge in the theory of harmonic Maass forms, which are certainnon-holomorphic functions that transform like modular forms. Harmonic Maass forms c M , as originallydefined by Bruinier and Funke [5], naturally decompose into two parts as c M = M + M − , where M is the olomorphic part of c M , and M − is the non-holomorphic part of c M . For example, when viewed as a functionof τ , we now know that the function q − X n ≥ q n ( − q ; q ) n + 2 i √ Z i ∞− τ P n ∈ Z (cid:0) n + (cid:1) e πiz (cid:0) n + 16 (cid:1) p − i ( z + τ ) dz, (4)where ( a ; q ) n := Q n − j =0 (1 − aq j ), is a harmonic Maass form due to work of Zwegers [26]. Its holomorphicpart, namely the q -hypergeometric series in (4), is one of Ramanujan’s original mock theta functions , certaincurious q -series whose exact modular properties were unknown for almost a century. Beautifully, all ofRamanujan’s original mock theta functions turn out to be examples of holomorphic parts of harmonic Maassforms [26], and we now define after Zagier [23] a mock modular form to be any holomorphic part of a harmonicMaass form. Mock modular forms come naturally equipped with a shadow , a certain modular cusp formrelated via a differential operator, which we formally define in Section 2.3. In the example given above in(4), it turns out that the shadow of Ramanujan’s q -hypergeometric mock theta function is essentially themodular theta function given in the numerator of the integral appearing there (up to a simple multiplicativefactor).As mentioned above, character generating functions for M appear as multiplicative eta-products. Wenow also know that there is a rich Moonshine phenomenon surrounding mock modular forms. Eguchi, Ooguriand Tachikawa [11], in analogy to the original Moonshine conjectures, observed that certain characters forthe Mathieu group M appeared to be related to mock modular forms. Their work was later generalizedand greatly extended by Cheng, Duncan, and Harvey [7], who developed an “umbral Moonshine” theory.Their umbral Moonshine conjectures were recently proved by Duncan, Griffin, and Ono [10].These connections serve as motivation for the first set of results in this paper. In Section 3, we unify theeta-theta functions by constructing mock modular forms which encode them in the following ways. We definefunctions V mn using the even eta-theta functions e n , and in Theorem 1.1, we prove that these functions V mn are mock modular forms, with the additional property that their shadows are given by the odd eta-thetafunctions E m .To describe these results, we introduce some notation. The functions V mn are indexed by pairs ( m, n )where m ∈ T ′ := { , , , , ′ , ′′ , , } and n ∈ N , and the admissible values for n are dependent on m .Throughout, we will call a pair ( m, n ) admissible if it is used to index one of our functions V mn . In total,there are 59 admissible pairs ( m, n ) where m ∈ T ′ . When we restrict m ∈ T := { , , , , , } , a particularsubset which we also consider, there are a total of 43 admissible pairs ( m, n ). We provide a complete listof these functions in the Appendix. The groups A mn , integers k ( mn ) γ mn , ℓ ( mn ) γ mn , r ( mn ) γ mn , and s ( mn ) γ mn , and roots ofunity ε ( m ) γ mn appearing in Theorem 1.1 below and throughout are defined in Section 3; the root of unity ψ isdefined in Lemma 2.1, and the constants c m are defined in Section 5. Theorem 1.1.
For any admissible pair ( m, n ) with m ∈ T ′ , the functions V mn are mock modular forms ofweight / with respect to the congruence subgroups A mn . Moreover, for m ∈ T , the shadow of V mn is givenby a constant multiple of the odd eta-theta function E m (cid:16) τc m (cid:17) . In particular, the functions V mn , m ∈ T ′ ,may be completed to form harmonic Maass forms b V mn of weight / on A mn , which satisfy for all γ mn = (cid:0) a mn b mn c mn d mn (cid:1) ∈ A mn , and τ ∈ H , b V mn ( γ mn τ ) = ψ ( γ mn ) − ( − k ( mn ) γmn + ℓ ( mn ) γmn + r ( mn ) γmn + s ( mn ) γmn ε ( m ) γ mn ( c mn τ + d mn ) b V mn ( τ ) . Because there are infinitely many mock modular forms with a given shadow, we are additionally motivatedto construct our functions V mn so that they are in some sense canonical. One way of doing this is by utilizingthe even eta-theta functions e n in the construction of these functions, as we have already mentioned. Further,we show in Theorem 1.2 that our mock modular forms V mn are also quantum modular forms, a propertythat is not necessarily true of all mock modular forms. A quantum modular form, as defined by Zagier[24] in 2010, is a complex function defined on an appropriate subset of the rational numbers, as opposed tothe upper half-plane, which transforms like a modular form, up to the addition of an error function thatis suitably continuous or analytic in R . (See Section 5 for more detail.) The theory of quantum modularforms is in its beginning stages; constructing explicit examples of these functions remains of interest, as does nswering the question of how quantum modular forms may arise from mock modular forms (see the recentarticles [3, 6, 13], for example).The quantum sets S mn of rational numbers and groups G mn pertaining to the forms V mn are defined inSection 4 and Section 5, and the constants ℓ m , a m , b m , c m and κ mn appearing below are defined in Section5. Here and throughout, the numbers ℓ mn are defined to equal ℓ m or 2, depending on whether or not n = 1.For N ∈ N , we define ζ N := e (1 /N ), and for r ∈ Z , we let M r := ( r ). Theorem 1.2.
For any admissible pair ( m, n ) with m ∈ T , the functions V mn are quantum modular formsof weight / on the sets S mn \ n − ℓ mn o for the groups G mn . In particular, the following are true. (i) For all x ∈ H ∪ S mn \ (cid:8) − (cid:9) , we have that V mn ( x ) + ζ ℓ m (2 x + 1) − V mn ( M x ) = − ic m Z i ∞ E m (cid:0) uc m (cid:1)p − i ( u + x ) du. (ii) For n = 1 and m ∈ { , , } , for all x ∈ H ∪ S mn \ {− } , we also have that V m ( x ) − ζ − ( x + 1) − V m ( M x ) = − ic m Z i ∞ E m (cid:0) uc m (cid:1)p − i ( u + x ) du. (5)(iii) For all x ∈ H ∪ S mn , we have that V mn ( x ) − ζ κ mn a m V mn ( x + κ mn b m ) = 0 . (6)One interesting feature of Theorem 1.2 is that it leads to simple, yet non-obvious, closed expressions forthe Eichler integrals of the odd eta-theta functions E m appearing on the right hand side of (5). Moreover,(6) leads to curious algebraic identities. To describe these results, we define the truncated q -hypergeometricseries for integers h ∈ Z , k ∈ N (gcd( h, k ) = 1) by F h,k ( z , z ) := k − X n =0 ( − ζ h k ; ζ h k ) n ζ n ( n +1) h k ( z ; ζ h k ) n +1 ( z ; ζ h k ) n +1 . (7)The additional constants d m , H m = H m ( h, k ) , and K m = K m ( h, k ) appearing in Corollary 1.3 below aredefined in Section 6. From Theorem 1.2, we have the following corollary. Corollary 1.3.
The Eichler integrals of the odd eta-theta functions E m satisfy the following identities. (i) Let m ∈ { , , , , } , and hk ∈ S m \ { − ℓ m } . Then we have that − ic m Z i ∞ ℓm E m ( zc m ) q − i (cid:0) z + hk (cid:1) dz = i ℓ m ζ d m ha m c m k F h,k ( − i ℓ m − ζ hc m k , − i − ℓ m ζ d m ha m k )(8) − ζ − ℓ m ζ d m H ℓm a m c m K ℓm (cid:0) ℓ m hk + 1 (cid:1) − F H ℓm ,K ℓm ( − i ℓ m − ζ H ℓm c m K ℓm , − i − ℓ m ζ d m H ℓm a m K ℓm ) . Moreover, we have for m ∈ { , , } that F h,k ( − i ℓ m − ζ hc m k , − i − ℓ m ζ d m ha m k ) + F h,k ( i ℓ m − ζ hc m k , i − ℓ m ζ d m ha m k ) = 0 . (9)(ii) Let hk ∈ S \ {− } . Then we have that − i Z i ∞ E ( z/ q − i (cid:0) z + hk (cid:1) dz = − ζ h k F h,k ( ζ h k , ζ h k ) − ζ h k F h,k ( ζ h k , ζ h k )(10) + ζ − (cid:0) hk + 1 (cid:1) − (cid:16) ζ H K F H ,K ( ζ H K , ζ H K ) + ζ H K F H ,K ( ζ H K , ζ H K ) (cid:17) . Moreover, we have that F h,k ( ζ h k , ζ h k ) + F h,k ( ζ h k , ζ h k ) + F h,k ( − ζ h k , − ζ h k ) + F h,k ( − ζ h k , − ζ h k ) = 0 . (11) emark. The analogous result to (9) also holds for m ∈ { , } , however, the identity for these m is trivial. We illustrate Corollary 1.3 in the following example.
Example.
Let h/k = 1 / ∈ S \ { − } . By Corollary 1.3, the Eichler integral of the eta-quotient E appearing in (8) may be evaluated exactly as − i Z i ∞ / E ( z/ dz q − i ( z + ) = ζ −
732 2 X n =0 ( − ζ ; ζ ) n ζ n ( n +1)12 ( iζ ; ζ ) n +1 ( − iζ ; ζ ) n +1 − (cid:0) (cid:1) ζ − X n =0 ( − ζ ; ζ ) n ζ n ( n +1)20 ( iζ ; ζ ) n +1 ( − iζ ; ζ ) n +1 ≈ . . i. Moreover, we have the following curious algebraic identity from (9): X n =0 ( − ζ ; ζ ) n ζ n ( n +1)12 ( iζ ; ζ ) n +1 ( − iζ ; ζ ) n +1 + X n =0 ( − ζ ; ζ ) n ζ n ( n +1)12 ( − iζ ; ζ ) n +1 ( iζ ; ζ ) n +1 = 0 . (12)While (12) may appear elementary, we point out that term by term, the two sums appearing are quitedifferent. That is, let a ( n ) := ( − ζ ; ζ ) n ζ n ( n +1)12 ( iζ ; ζ ) n +1 ( − iζ ; ζ ) n +1 , b ( n ) := ( − ζ ; ζ ) n ζ n ( n +1)12 ( − iζ ; ζ ) n +1 ( iζ ; ζ ) n +1 . The following table gives the values of each summand appearing in (12); other than the fact that the lastsummands satisfy a (2) = − b (2), term-by-term cancellation in (12) is not apparent. Other examples whichwe numerically computed behaved similarly. n a ( n ) b ( n )0 ≈ . − . i ≈ . − . i ≈ − . . i ≈ . − . i ≈ . ≈ − . V mn satisfy the stronger property that their appropriate transfor-mation properties hold on both a subset of Q and the upper half-plane H , it is natural to ask if the functions V mn also extend into the lower half-plane H − := { z ∈ C | Im( z ) < } . Indeed, in Section 2.1, we define for m ∈ T the functions e E m ( z ) for z ∈ H − . Upon making the change of variable z = − τ /c m , where τ ∈ H (andhence z ∈ H − ), we show in Proposition 1.4 that as τ → x ∈ S mn ⊆ Q from the upper half-plane, and henceas z → − x/c m ∈ Q from the lower half-plane, the functions e E m (cid:0) − x/c m (cid:1) are quantum modular formswhich transform exactly as our functions V mn ( x ) do in Theorem 1.2, up to multiplication by a constantwhich can be explicitly determined. Proposition 1.4.
For m ∈ T , the functions e E m are quantum modular forms of weight / . In particular,for any x ∈ S mn , up to multiplication by a constant, the functions e E m (cid:0) − x/c m (cid:1) satisfy the transformationlaws given in Theorem 1.2 for the functions V mn ( x ) . Series similar to the functions e E m defined in (16) which instead arise from ordinary integer weight cuspforms were studied originally by Eichler (and are also often referred to as “Eichler integrals”), and were shownto play fundamental roles within the theory of integer weight modular forms. In the present setting, themodular objects E m related to the series e E m are not of integral weight, and many aspects of Eichler’s theorybecome complicated. Nevertheless, in their fundamental work [17], Lawrence and Zagier successfully considerEichler’s theory in the half integer weight setting; moreover, their work led to some of the first examples ofquantum modular forms. The functions e E m may also be viewed as partial theta functions , which as seriesare similar to ordinary modular theta functions, but which are not modular in general [1]. Related resultson quantum modular forms similar to those given in Proposition 1.4 may be found in [4, 12, 24] among otherplaces; we follow their methods to prove Proposition 1.4.2. Preliminaries for Theorem 1.1 and Theorem 1.2
In this section we review previous work of Lemke Oliver [20], Zwegers [25], and the third author [16], andmake some preparations for our proofs of Theorems 1.1 and 1.2. .1. Work of Lemke Oliver on eta-theta functions.
We begin with the Dedkind eta-function (1), whosewell-known weight 1 / Lemma 2.1.
For all γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ) and τ ∈ H , we have that (13) η ( γτ ) = ψ ( γ ) ( cτ + d ) η ( τ ) , where ψ ( γ ) is a th root of unity, which can be given explicitly in terms of Dedekind sums [21] . In particular,we have that η (cid:18) − τ (cid:19) = √− iτ η ( τ ) . In recent work, Lemke Oliver [20] proves that there are only eighteen weight 1 / τ kτ ; we list these as e ( τ ) = η ( τ ) η (2 τ ) = X n ∈ Z (cid:18) − (cid:16) n (cid:17) (cid:19) q n = X n ∈ Z ( − n q n ,e ( τ ) = η (2 τ ) η ( τ ) η (4 τ ) = X n ∈ Z q n ,e ( τ ) = η (24 τ ) = X n ≥ (cid:18) n (cid:19) q n ,e ( τ ) = η (48 τ ) η (72 τ ) η (24 τ ) η (144 τ ) = X n ≥ (cid:16) n (cid:17) q n ,e ( τ ) = η (8 τ ) η (32 τ ) η (16 τ ) = X n ≥ (cid:18) n (cid:19) q n ,e ( τ ) = η (16 τ ) η (8 τ ) = X n ≥ (cid:16) n (cid:17) q n ,e ( τ ) = η (3 τ ) η (18 τ ) η (6 τ ) η (9 τ ) = X n ≥ (cid:18) (cid:16) n (cid:17) − (cid:16) n (cid:17) (cid:19) q n ,e ( τ ) = η (6 τ ) η (9 τ ) η (36 τ ) η (3 τ ) η (12 τ ) η (18 τ ) = X n ≥ (cid:16) n (cid:17) q n ,e ( τ ) = η (48 τ ) η (24 τ ) η (96 τ ) = X n ≥ (cid:18) n (cid:19) q n ,e ( τ ) = η (24 τ ) η (96 τ ) η (144 τ ) η (48 τ ) η (72 τ ) η (288 τ ) = X n ≥ (cid:18) n (cid:19) q n ,e ( τ ) = η ( τ ) η (4 τ ) η (6 τ ) η (2 τ ) η (3 τ ) η (12 τ ) = X n ∈ Z (cid:18) − (cid:16) n (cid:17) (cid:19) q n ,e ( τ ) = η (2 τ ) η (3 τ ) η ( τ ) η (6 τ ) = X n ∈ Z (cid:18) − (cid:16) n (cid:17) − (cid:16) n (cid:17) + 3 (cid:16) n (cid:17) (cid:19) q n ,e ( τ ) = η (8 τ ) η (48 τ ) η (16 τ ) η (24 τ ) = X n ≥ (cid:18) (cid:16) n (cid:17) − (cid:16) n (cid:17) (cid:19) q n . (14)Lemke Oliver establishes a similar list for eta-quotients of weight 3 / Our ordering here differs from Lemke Oliver’s. E ( τ ) = η (8 τ ) = X n ≥ (cid:18) − n (cid:19) nq n , E ( τ ) = η (48 τ ) η (24 τ ) η (96 τ ) = X n ≥ (cid:18) − n (cid:19) nq n , E ( τ ) = η (16 τ ) η (8 τ ) η (32 τ ) = X n ≥ (cid:18) − n (cid:19) nq n , E ( τ ) = η (24 τ ) η (48 τ ) = X n ≥ (cid:16) n (cid:17) nq n , E ( τ ) = η (3 τ ) η (12 τ ) η (6 τ ) = X n ≥ (cid:16) n (cid:17) nq n , E ( τ ) = η (6 τ ) η (3 τ ) = X n ≥ (cid:16) (cid:16) n (cid:17) − (cid:16) n (cid:17)(cid:17) nq n .We also define the following functions for z ∈ H − , e E ( z ) = X n ≥ (cid:18) − n (cid:19) e − πizn , e E ( z ) = X n ≥ (cid:18) − n (cid:19) e − πizn , e E ( z ) = X n ≥ (cid:18) − n (cid:19) e − πizn , e E ( z ) = X n ≥ (cid:16) n (cid:17) e − πizn , e E ( z ) = X n ≥ (cid:16) n (cid:17) e − πizn , e E ( z ) = X n ≥ (cid:16) (cid:16) n (cid:17) − (cid:16) n (cid:17)(cid:17) e − πizn .(16)Although these functions are not modular forms, as series, their relationship to the modular eta-thetafunctions E m is apparent. As discussed in Section 1, these functions may be viewed as formal Eichlerintegrals of the modular eta-theta functions E m , or, as partial theta functions. Connections between thesetypes of functions and mock modular and quantum modular forms have been explored in a number of works,including [4, 12, 13, 15, 22, 24].2.2. Work of Zwegers on mock theta functions related to unary theta functions.
Zwegers [25]provides a mechanism for constructing mock theta functions with shadow related to a given unary thetafunction of weight 3 /
2. These mock theta functions (which we discuss further in this context in Section 2.3)feature the weight 1/2 theta functions ϑ ( v ; τ ) := X n ∈ Z e πi ( n + )( v + ) q ( n + ) . It is well-known that these theta functions may be written as(17) ϑ ( v ; τ ) = − iq e − πiv Y n ≥ (1 − q n )(1 − e πiv q n − )(1 − e − πiv q n ) . We note that ϑ ( v ; τ ) also satisfies the explicit modularity properties described in the following lemma. Lemma 2.2 ([21], (80.31) and (80.8)) . For λ, µ ∈ Z , γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ) , z ∈ C , and τ ∈ H , we have that ϑ ( z + λτ + µ ; τ ) = ( − λ + µ q − λ e − πiλz ϑ ( z ; τ ) , (18) ϑ (cid:18) zcτ + d ; γτ (cid:19) = ψ ( γ ) ( cτ + d ) e πicz cτ + d ϑ ( z ; τ ) . (19) In particular, we have that ϑ (cid:18) zτ ; − τ (cid:19) = − i √− iτ e πiz τ ϑ ( z ; τ ) . Now for τ ∈ H and u, v ∈ C \ ( Z τ + Z ), Zwegers defines(20) µ ( u, v ; τ ) := e πiu ϑ ( v ; τ ) X n ∈ Z ( − n e πinv q n ( n +1)2 − e πiu q n . Zwegers also defines for u ∈ C and τ ∈ H the Mordell integral h by(21) h ( u ) = h ( u ; τ ) := Z R e πiτx − πux cosh πx dx. We will make use of the following properties of µ . emma 2.3 (Zwegers, Prop. 1.4 and 1.5 of [25]) . Let µ ( u, v ) := µ ( u, v ; τ ) and h ( u ; τ ) be defined as in (20)and (21). Then we have (1) µ ( u + 1 , v ) = − µ ( u, v ) , (2) µ ( u, v + 1) = − µ ( u, v ) , (3) µ ( − u, − v ) = µ ( u, v ) , (4) µ ( u + z, v + z ) − µ ( u, v ) = πi ϑ ′ (0) ϑ ( u + v + z ) ϑ ( z ) ϑ ( u ) ϑ ( v ) ϑ ( u + z ) ϑ ( v + z ) , for u, v, u + z, v + z / ∈ Z τ + Z ,and the modular transformation properties, (5) µ ( u, v ; τ + 1) = e − πi µ ( u, v ; τ ) , (6) √− iτ e πi ( u − v ) /τ µ (cid:0) uτ , vτ ; − τ (cid:1) + µ ( u, v ; τ ) = i h ( u − v ; τ ) . Additionally, we will use the following theorem of the third author [16], relating a certain specializationof µ ( u, v ; τ ) to a universal mock theta function. Theorem 2.4 (Kang [16]) . If α ∈ C such that α Z τ + Z , then µ (cid:16) α, τ τ (cid:17) = iq g ( e ( α ); q ) − e ( − α ) q η ( τ ) η ( τ ) ϑ (2 α ; τ ) , where g is the universal mock theta function defined by g ( z ; q ) := ∞ X n =0 ( − q ) n q n ( n +1) / ( z ; q ) n +1 ( z − q ; q ) n +1 . The function µ is completed by defining the real-analytic function R ( u ; τ ) := X ν ∈ + Z sgn( ν ) − Z ( ν + a ) √ y e − πt dt ! ( − ν − e − πiν τ − πiνu , with y = Im( τ ) and a = Im(u)Im( τ ) . For τ ∈ H and u, v ∈ C \ ( Z τ + Z ), Zwegers defines(22) b µ ( u, v ; τ ) := µ ( u, v ; τ ) + i R ( u − v ; τ ) . The following explicit transformation properties show that b µ transforms like a two-variable (non-holomorphic)Jacobi form of weight 1 / Lemma 2.5 (Zwegers, Prop. 1.11(1,2) of [25]) . Let b µ ( u, v ; τ ) be defined as in (22). Then (1) b µ ( u + kτ + l, v + mτ + n ; τ ) = ( − k + l + m + n e πi ( k − m ) τ +2 πi ( k − m )( u − v ) b µ ( u, v ; τ ) , for k, l, m, n ∈ Z , and (2) b µ (cid:16) ucτ + d , vcτ + d ; aτ + bcτ + d (cid:17) = v ( γ ) − ( cτ + d ) e − πic ( u − v ) / ( cτ + d ) b µ ( u, v ; τ ) , for γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ) , with v ( γ ) defined as in (13). As we shall see in Theorem 2.7, these completions are related to the unary theta function defined for a, b ∈ R and τ ∈ H by(23) g a,b ( τ ) := X n ∈ Z ( n + a ) e πib ( n + a ) q ( n + a )22 . The following transformation properties show, in particular, that g a,b is a modular form of weight 3 / a and b are rational. Lemma 2.6 (Zwegers, Prop. 1.15 of [25]) . The function g a,b satisfies the following: (1) g a +1 ,b ( τ ) = g a,b ( τ ) , (2) g a,b +1 ( τ ) = e πia g a,b ( τ ) , (3) g − a, − b ( τ ) = − g a,b ( τ ) , (4) g a,b ( τ + 1) = e − πia ( a +1) g a,a + b + ( τ ) , (5) g a,b ( − τ ) = ie πiab ( − iτ ) / g b, − a ( τ ) . We have rewritten this formula to account for our definition of ϑ and µ , which differs from [16]. In particular, writing ϑ K and µ K to indicate the notation in [16], we have that ϑ = − iϑ K and µ = iµ K . he unary theta function g a,b is related to both R and h by the following theorem. Theorem 2.7 (Zwegers, Thm. 1.16 of [25]) . For τ ∈ H , we have the following two results.When a ∈ ( − , ) and b ∈ R , (24) Z i ∞− τ g a + ,b + ( z ) p − i ( z + τ ) dz = − e πia ( b + ) q − a R ( aτ − b ; τ ) . Also, when a, b ∈ ( − , ) , (25) Z i ∞ g a + ,b + ( z ) p − i ( z + τ ) dz = − e πia ( b + ) q − a h ( aτ − b ; τ ) . We extend Theorem 2.7 in the following result, which we will use in our proof of Theorem 1.2.
Lemma 2.8.
Let τ ∈ H . i) For b ∈ R \ Z , Z i ∞− τ g ,b + ( z ) p − i ( z + τ ) dz = − ie (cid:18) − τ b (cid:19) R (cid:16) τ − b ; τ (cid:17) + i. ii) For b ∈ ( − , ) \ { } , Z i ∞ g ,b + ( z ) p − i ( z + τ ) dz = − ie (cid:18) − τ b (cid:19) h (cid:16) τ − b ; τ (cid:17) + i. iii) For a ∈ ( − , ) \ { } , Z i ∞ g a +1 / , ( z ) p − i ( z + τ ) dz = − e (cid:18) − a τ + a (cid:19) h (cid:18) aτ −
12 ; τ (cid:19) + e ( a ) √− iτ . Proof of Lemma 2.8. If b ∈ R \ Z , we have that g ,b + ( z ) = O (cid:0) e − π Im( z ) (cid:1) . If a ∈ ( − / , / \ { } , we havethat g a + , ( z ) = O (cid:16) e − πv Im( z ) (cid:17) for some v > z ) → ∞ . These facts justify the convergence of theintegrals in Lemma 2.8.The proof of (24) in [25] yields that the integral on the left hand side of i) in Lemma 2.8 equals − ie (cid:18) − τ b (cid:19) X v ∈ + Z sgn (cid:18) v + 12 (cid:19) − Z ( v + ) √ τ )0 e − πu du ! ( − v − e (cid:18) − v τ − v (cid:16) τ − b (cid:17)(cid:19) . (26)Now for all v ∈ ( + Z ) \ {− } , we have that sgn (cid:0) v + (cid:1) = sgn( v ). For v = − , we have that 0 =sgn (cid:0) − + (cid:1) = sgn (cid:0) − (cid:1) + 1. Making these substitutions into (26) and simplifying proves part i) of Lemma2.8.Part ii) and part iii) of Lemma 2.8 now follow as argued in Remark 1.20 in [25] by using part i) of Lemma2.8 above (rather than (24)) where necessary. (cid:3) In the following section, we review the connection between b µ and the theory of harmonic Maass forms.2.3. Harmonic Maass forms of weight / , and period and Mordell integrals. Following Bruinierand Funke [5], a harmonic Maass form b f : H → C is a non-holomorphic extension of a classical modularform. It is a smooth function such that for a weight κ ∈ Z , if Γ ⊆ SL ( Z ) and χ is a Dirichlet charactermodulo N , then for all γ = (cid:0) a bc d (cid:1) ∈ Γ and τ ∈ H we have b f ( γτ ) = χ ( d )( cτ + d ) κ b f ( τ ). Moreover, b f mustvanish under the weight κ Laplacian operator defined, for τ = x + iy , by∆ κ := − y (cid:18) ∂ ∂x + ∂ ∂y (cid:19) + iκy (cid:18) ∂∂x + i ∂∂y (cid:19) . Additionally, b f must have at most linear exponential growth at all cusps.The Fourier series of a harmonic Maass form b f of weight κ naturally decomposes as the sum of a holo-morphic and a non-holomorphic part. We refer to the holomorphic part f as a mock modular form of weight after Zagier [23]. In the special case κ ∈ { / , / } , we refer to f as a mock theta function . Moreover, aharmonic Maass form b f of weight κ is mapped to a classical modular form of weight 2 − κ by the differentialoperator ξ κ := 2 iy κ · ∂∂τ . The image of b f under ξ κ is called the shadow of f . We next show that certain specializations of the function µ are essentially mock theta functions with shadows related to g a,b . Similar results are known, however inthis paper we require and thus establish the precise statement given in Proposition 2.8. To state it, we definefor a function g : H → C its complement g c ( τ ) := g ( − τ ) . For τ ∈ H , we define for a, b ∈ R and u, v ∈ C \ ( Z τ + Z ) the function c M a,b ( τ ) := −√ e πia ( b + ) q − a b µ ( u, v ; τ ) . (27)We denote the holomorphic part of c M a,b by M a,b , that is, M a,b ( τ ) := −√ e πia ( b + ) q − a µ ( u, v ; τ ) . Proposition 2.9.
Let τ ∈ H , and u, v ∈ C \ ( Z τ + Z ) . If u − v = aτ − b for some a, b ∈ R , then the function c M a,b ( τ ) satisfies (i) ξ (cid:16) c M a,b ( τ ) (cid:17) = g ca + ,b + ( τ ) , (ii) ∆ ( c M a,b ( τ )) = 0 . Remark.
Part (ii) of Proposition 2.9 together with the transformation laws established in Lemma 2.5 showthat c M a,b is essentially a harmonic Maass form of weight / for suitable v, a, and b ; we illustrate this moreprecisely in the proof of Theorem 1.1.Proof. Here and throughout, we write τ = x + iy . We begin by establishing part (i). We have that ξ (cid:16) c M a,b ( τ ) (cid:17) = ξ (cid:18) −√ e πia ( b + ) q − a µ ( u, v ; τ ) − √ e πia ( b + ) q − a · i R ( aτ − b ; τ ) (cid:19) = 2 iy ∂∂τ (cid:18) −√ e πia ( b + ) q − a · i R ( aτ − b ; τ ) (cid:19) = −√ y e − πia ( b + ) q − a ∂∂τ R ( aτ − b ; τ ) . (28)It is shown in [25, (1.5)] that ∂∂τ R ( aτ − b ; τ ) = − i √ y e − πa y X n ∈ Z ( − n (cid:0) n + a + (cid:1) e − πi ( n + ) x e − π ( n + ) y e − πi ( n + ) ( ax − b ) e − π ( n + ) ay . (29)Taking the conjugate of (29) and q − a / , we find that (28) becomes= − ie − πia ( b + ) q a X n ∈ Z ( − n (cid:0) n + a + (cid:1) q ( n + ) q ( n + ) a e − πi ( n + ) b = X n ∈ Z (cid:0) n + a + (cid:1) q ( n + a + ) e − πi ( n + a + )( b + ) = g a + , − b − ( τ ) . Using the definition of g a + , − b − ( τ ), it is not difficult to show that g a + , − b − ( τ ) = g a + ,b + ( − τ ) = g ca + ,b + ( τ ) . (30)This proves part (i).To prove part (ii), we use the fact that the weight 1 / = − ξ ξ . Theresult follows by applying − ξ to the expression given in part (i) of the Proposition, using (30). (cid:3) .4. Converting setting of Lemke Oliver to notation of Zwegers.
We now express the eta-thetafunctions from (14) and (15) in terms of the theta functions ϑ ( v ; τ ) and g a,b ( τ ). We first observe that wecan convert the sums over positive integers in the definitions of the functions E m into sums over all integers,and then write the E m in terms of the functions g a,b . For example, we see from the definition of g a,b ( τ ) in(23) that 4 g , (32 τ ) = 4 X n ∈ Z (cid:18) n + 14 (cid:19) q ( n + ) = X n ∈ Z (4 n + 1) q (4 n +1) = E ( τ ) . By similar methods we find the following result.
Lemma 2.10.
For τ ∈ H , we have that E ( τ ) = 4 g , (32 τ ) = X n ∈ Z (4 n + 1) q (4 n +1) , (31) E ( τ ) = 4 e − πi g , (32 τ ) = X n ∈ Z ( − n (4 n + 1) q (4 n +1) ,E ( τ ) = 3 g , (18 τ ) = X n ∈ Z (3 n + 1) q (3 n +1) ,E ( τ ) = 12 e − πi g , (288 τ ) + 12 e − πi g , (288 τ )= X n ∈ Z ( − n (12 n + 1) q (12 n +1) + X n ∈ Z ( − n (12 n + 5) q (12 n +5) ,E ( τ ) = 6 g , (72 τ ) = X n ∈ Z (6 n + 1) q (6 n +1) ,E ( τ ) = 3 e − πi g , (18 τ ) = X n ∈ Z ( − n (3 n + 1) q (3 n +1) . From Proposition 2.9 and Lemma 2.10, we see that to construct forms c M a,b whose images under the ξ -operator are equal to a constant multiple of E m ( τ /k m ) for some suitable constants k m we are only restrictedby u − v for u, v ∈ C \ ( Z τ + Z ), not by u and v individually. Since the theta function ϑ ( v ; τ ) appears as aprominent factor in the definition of b µ from (22), we again use the classification in [20] to restrict to those ϑ ( v ; τ ) which are eta-quotients of weight 1 / Lemma 2.11.
For τ ∈ H , we have that ϑ (cid:16) τ τ (cid:17) = − iq − e (cid:16) τ (cid:17) , ϑ (cid:16) τ τ (cid:17) = − iq − e (cid:16) τ (cid:17) , (32) ϑ (cid:18) τ −
12 ; τ (cid:19) = q − e (cid:16) τ (cid:17) , ϑ (cid:18) τ −
12 ; τ (cid:19) = q − e (cid:16) τ (cid:17) ,ϑ (cid:16) τ τ (cid:17) = − iq − e (cid:16) τ (cid:17) , ϑ (cid:16) τ τ (cid:17) = − iq − e (cid:16) τ (cid:17) ,ϑ (cid:18) τ −
12 ; τ (cid:19) = q − e (cid:16) τ (cid:17) , ϑ (cid:18) τ −
12 ; τ (cid:19) = q − e (cid:16) τ (cid:17) . Proof.
Using (17), we have that ϑ (cid:18) τ −
12 ; 2 τ (cid:19) = − iq e − πi ( τ − ) Y n ≥ (1 − q n )(1 − e πi ( τ − ) q n − )(1 − e − πi ( τ − ) q n )= q − Y n ≥ (1 − q n )(1 + q n − ) = q − Y n ≥ (1 − q n )(1 + q n ) (1 + q n ) · (1 − q n ) (1 − q n ) (1 − q n ) (1 − q n ) = q − Y n ≥ (1 − q n ) (1 − q n ) (1 − q n ) = q − η (2 τ ) η ( τ ) η (4 τ ) = q − e ( τ ) , which is the first identity above with τ → τ /
2. The rest follow from similar arguments. (cid:3) ote that ϑ (cid:0) τ ; τ − (cid:1) = e ( − ) q − e (cid:0) τ (cid:1) and ϑ (cid:0) τ − / τ − (cid:1) = e ( ) q − e (cid:0) τ (cid:1) , which are not ofthe form ϑ ( v ; τ ). Similarly, e , e , and e cannot be written in the form ϑ ( v ; τ ). Hence, we restrict ourconstructions to the first eight e n functions.3. Eta-theta functions and mock modular forms
We are now ready to construct our families of mock theta functions. For each weight 3 / E m we construct eight corresponding functions V mn , one for each weight 1 / e n . However,in some cases the V mn are degenerate due the presence of poles. Here, we will focus on the construction of V as the other constructions follow similarly. Our goal for V is to construct a function that has shadowassociated to E , and the factor e in its series representation.First, we observe from Lemma 2.10 that E ( τ ) = 4 g , (32 τ ). Thus, we make the change τ τ andconsider a function of the form µ ( u, v ; 32 τ ) as in (20). We choose v so that the theta function ϑ ( v ; 32 τ )appearing in (20) is in terms of e . By Lemma 2.11 we see that we should choose v = 16 τ so that ϑ (16 τ ; 32 τ ) = − iq − e (16 τ ). By Proposition 2.9 the corresponding function M − , − (32 τ ) has shadowrelated to g , (32 τ ), so long as u − τ = u − v = −
14 (32 τ ) − (cid:0) − (cid:1) = − τ + . Thus we choose u = 8 τ + 1 /
2, and calculate the series form of − q − µ (8 τ + 1 / , τ ; 32 τ ) using (20). Ourfinal step is to renormalize with τ τ /
32. We obtain V ( τ ) = q − / e ( τ / X n ∈ Z ( − n q ( n +1) / q n +1 / = w q t µ (cid:16) u (11) τ , v (11) τ ; τ (cid:17) := − q − / µ (cid:18) τ , τ τ (cid:19) . We repeat this process for each of the remaining e n , to find V n as above with shadow related to E andwith the theta function e n as a factor in its series representation. We find that the construction of V leadsto the choice u = 0 ∈ C \ (32 Z τ + Z ), and so this fails to produce a mock modular form due to the existenceof poles. Each of the V n are fully listed in the Appendix.We repeat this entire process for each E m with m ∈ { , , , } . The case E requires some additionalcare as E ( τ ) = 12 e − πi g , (288 τ ) + 12 e − πi g , (288 τ ). In this case we build two different forms V ′ n and V ′′ n , one for each g a,b function, using the process described above. We then add these forms to create ourdesired mock theta function. For example, V ( τ ) = V ′ ( τ ) + V ′′ ( τ ) = − q − / e ( τ / X n ∈ Z ( − n q ( n +1) / − q n +1 / + − q − / e ( τ / X n ∈ Z ( − n q ( n +1) / − q n +5 / , = iq − / µ (cid:16) τ , τ τ (cid:17) + iq − / µ (cid:18) τ , τ τ (cid:19) . All of the V mn , including the V n , are listed in the Appendix. Remark.
Using Lemma 2.3, we see V ( τ ) = − q − / µ (cid:18) − τ , τ τ (cid:19) = − q − / µ (cid:18) τ , − τ τ (cid:19) = − q − / µ (cid:18) − τ , τ −
12 ; τ (cid:19) = V ( τ ) . A comparison of coefficients reveals that all other series are unique.
Remark.
A coefficient search on the On-Line Encyclopedia of Integer Sequences suggests the followingequalities: − q / V (12 τ ) = ψ ( q ) , q / V (3 τ ) = χ ( q ) , q − / V (6 τ ) = ρ ( q ) , − q / V (4 τ ) = A ( q ) , − q / V (4 τ ) = U ( q ) , q / V (4 τ ) = U ( q ) , where ψ ( q ) , χ ( q ) , and ρ ( q ) are Ramanujan’s third order mock thetas, A ( q ) is Ramanujan’s second order mocktheta, and U ( q ) and U ( q ) are Gordon and McIntosh’s eighth order mock thetas. These series are definedin [2] and [14] . The latter three equalities follow from the definitions in [14] and Lemma 2.3. .1. Proof of Theorem 1.1.
First, we wish to establish the mock modularity of the 51 functions V mn foradmissible ( m, n ) when m ∈ T ′ \{ } . We will make use of Proposition 2.9, but must also establish the modulartransformation properties of these functions. For such pairs ( m, n ), the functions V mn , as summarized in theAppendix, may be expressed in terms of the µ -function, and parameters w m , t m , u ( mn ) τ , v ( mn ) τ as V mn ( τ ) = w m q t m µ ( u ( mn ) τ , v ( mn ) τ ; τ ) . (33)We denote their completions by b V mn ( τ ) := w m q t m b µ ( u ( mn ) τ , v ( mn ) τ ; τ ) . (34)When m = 4, for any admissible n , we consider V n ( τ ) = V ′ n ( τ ) + V ′′ n ( τ ) , and the completions b V n ( τ ) := b V ′ n ( τ ) + b V ′′ n ( τ ) . Toward establishing their modular properties, we establish the following preliminary lemmas, the first ofwhich follows directly from Lemma 2.5. Throughout, for x τ ∈ C \ ( Z τ + Z ), we define for any γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ), e x γ,τ := x γτ · ( cτ + d ) . Lemma 3.1.
Let γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ) , τ ∈ H , and u τ , v τ ∈ C \ ( Z τ + Z ) . Suppose e u γ,τ = u τ + k γ · τ + ℓ γ , and e v γ,τ = v τ + r γ · τ + s γ , for some integers k γ , ℓ γ , r γ , s γ . Then we have that b µ ( u γτ , v γτ ; γτ ) = b µ (cid:18) e u γ,τ cτ + d , e v γ,τ cτ + d ; γτ (cid:19) = ψ ( γ ) − ( − k γ + ℓ γ + r γ + s γ ( cτ + d ) q ( kγ − rγ )22 e (cid:18) − c ( e u γ,τ − e v γ,τ ) cτ + d ) + ( k γ − r γ )( u τ − v τ ) (cid:19) b µ ( u τ , v τ ; τ ) . (cid:3) We next establish two technical lemmas, Lemma 3.2 and Lemma 3.3, which will allow us to efficientlyestablish the mock modularity of our functions V mn , when combined with Lemma 3.1 and Proposition 2.9above. Lemma 3.2.
Let γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ) , τ ∈ H , j ∈ { , } , and u ( j ) τ , v ( j ) τ ∈ C \ ( Z τ + Z ) . Suppose there existconstants k ( j ) γ , ℓ ( j ) γ , r ( j ) γ , s ( j ) γ ∈ R satisfying e u ( j ) γ,τ = u ( j ) τ + k ( j ) γ · τ + ℓ ( j ) γ , and e v ( j ) γ,τ = v ( j ) τ + r ( j ) γ · τ + s ( j ) γ . Further,define the difference functions d ( j ) τ := u ( j ) τ − v ( j ) τ , e d ( j ) γ,τ := e u ( j ) γ,τ − e v ( j ) γ,τ ,δ ( j ) γ := k ( j ) γ − r ( j ) γ , ρ ( j ) γ := ℓ ( j ) γ − s ( j ) γ . Suppose that d (1) τ = d (2) τ . Then we have that e d (1) γ,τ = e d (2) γ,τ , δ (1) γ = δ (2) γ , and ρ (1) γ = ρ (2) γ . Proof of Lemma 3.2.
The first assertion follows from the fact that d (1) τ = d (2) τ , that cτ + d = 0, and thedefinitions of e u ( j ) γ,τ and e v ( j ) γ,τ . To prove the second and third assertions, we have that e u ( j ) γ,τ = u ( j ) τ + k ( j ) γ · τ + ℓ ( j ) γ and e v ( j ) γ,τ = v ( j ) τ + r ( j ) γ · τ + s ( j ) γ . Subtracting the second of these equalities from the first, we have that e d ( j ) γ,τ = d ( j ) τ + δ ( j ) γ · τ + ρ ( j ) γ . But d (1) τ = d (2) τ and e d (1) γ,τ = e d (2) γ,τ , which implies that δ (1) γ · τ + ρ (1) γ = δ (2) γ · τ + ρ (2) γ .The second and third assertions now follow, using the fact that δ ( j ) γ and ρ ( j ) γ are constants in R , and τ ∈ H . (cid:3) In order to utilize Lemma 3.1 to determine the modular transformation properties for the functions V mn ,we need to know for which γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ) we have that e u ( mn ) γ,τ − u ( mn ) τ ∈ Z τ + Z , and e v ( mn ) γ,τ − v ( mn ) τ ∈ Z τ + Z . We note the following lemma, which follows directly by using the definition of e x γ,τ . emma 3.3. Let x τ ∈ C \ ( Z τ + Z ) be of the form x τ = ατ + βN , where N ∈ N , and ≤ α, β ≤ N − . For fixed γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ) , we have that e x γ,τ − x τ ∈ Z τ + Z if andonly if the following congruences hold αa + βc ≡ α (mod N ) αb + βd ≡ β (mod N ) . (cid:3) The following corollary follows directly from Lemma 3.3.
Corollary 3.4.
In the context of the above lemma, when α = 0 , and β is relatively prime to N , then e x γ,τ − x τ ∈ Z τ + Z if and only if γ ∈ Γ ( N ) . Similarly, if β = 0 , and α is relatively prime to N , then e x γ,τ − x τ ∈ Z τ + Z if and only if γ ∈ Γ ( N ) . (cid:3) In Table 1, we list the congruence subgroups A mn for each mock modular form V mn in Theorem 1.1.These are computed using Lemma 3.3 and Corollary 3.4, and are used in the proof of Theorem 1.1 below. Table 1. congruence subgroups A mn for each mock modular form V mn n \ m , ′ , ′′ (4) ∩ Γ (2) Γ (4) Γ (6) ∩ Γ (2) Γ (12) Γ (6) ∩ Γ (2) Γ (6)2 Γ (4) ∩ Γ (2) Γ (4) ∩ Γ (2) Γ (6) ∩ Γ (2) Γ (12) ∩ Γ (2) Γ (6) ∩ Γ (2) Γ (6) ∩ Γ (2)3 Γ (12) ∩ Γ (2) Γ (12) Γ (6) ∩ Γ (2) Γ (12) Γ (3) ∩ Γ (2) Γ (6)4 Γ (12) ∩ Γ (2) Γ (12) ∩ Γ (2) Γ (6) ∩ Γ (2) Γ (12) ∩ Γ (2) −− Γ (6) ∩ Γ (2)5 Γ (4) ∩ Γ (2) −− Γ (12) ∩ Γ (2) Γ (12) Γ (12) ∩ Γ (2) Γ (12)6 −− Γ (4) ∩ Γ (2) Γ (12) ∩ Γ (2) Γ (12) ∩ Γ (2) Γ (12) ∩ Γ (2) Γ (12) ∩ Γ (2)7 Γ (12) ∩ Γ (2) Γ (12) Γ (6) ∩ Γ (2) Γ (12) Γ (6) ∩ Γ (2) −− (12) ∩ Γ (2) Γ (12) ∩ Γ (2) −− Γ (12) ∩ Γ (2) Γ (6) ∩ Γ (2) Γ (6) ∩ Γ (2) Proof of Theorem 1.1.
We first consider the functions V mn where m ∈ T ′ \{ } . After doing so, we will addressthe more delicate case of m = 4. We begin by considering (for m ∈ T ′ \ { } ) the defining parameters u ( mn ) τ and v ( mn ) τ from the Appendix, as well as their associated values e u ( mn ) γ,τ and e v ( mn ) γ,τ , where γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ).For γ ∈ A mn as defined in Table 1, we may write e u ( mn ) γ,τ = u ( mn ) τ + k ( mn ) γ · τ + ℓ ( mn ) γ , e v ( mn ) γ,τ = v ( mn ) τ + r ( mn ) γ · τ + s ( mn ) γ , where k ( mn ) γ , ℓ ( mn ) γ , r ( mn ) γ , s ( mn ) γ ∈ Z . For example, when ( m, n ) = (2 , , we have that u (22) τ := τ − = τ − and v (22) τ := τ − = τ − . By Lemma 3.3, we see that e u (22) γ,τ − u (22) τ ∈ Z τ + Z if and only if a + 2 c ≡ b + 2 d ≡ , whereas e v (22) γ,τ − v (22) τ ∈ Z τ + Z if and only if a + c ≡ b + d ≡ . Recalling that ad − bc = 1, a straightforward calculation shows that these congruences are simultaneouslysatisfied if and only if γ ∈ Γ (4) ∩ Γ (2), which is A in Table 1. hus, for general m ∈ T ′ \ { } we may apply Lemma 3.1, which reveals that b V mn ( γ mn τ ) = ψ ( γ mn ) − ( − k ( mn ) γmn + ℓ ( mn ) γmn + r ( mn ) γmn + s ( mn ) γmn ( c mn τ + d mn ) φ ( m ) n,γ mn ,τ b V mn ( τ ) , where for γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ), the functions φ ( m ) n,γ,τ are defined by φ ( m ) n,γ,τ := e ( t m γτ ) e (cid:18) − c cτ + d ) (˜ u ( mn ) γ,τ − ˜ v ( mn ) γ,τ ) (cid:19) q ( k ( mn ) γ − r ( mn ) γ ) e (cid:16) ( u ( mn ) τ − v ( mn ) τ )( k ( mn ) γ − r ( mn ) γ ) (cid:17) q − t m . (35)Next, we define the difference functions d ( mn ) τ := u ( mn ) τ − v ( mn ) τ , e d ( mn ) γ,τ := e u ( mn ) γ,τ − e v ( mn ) γ,τ , δ ( mn ) γ := k ( mn ) γ − r ( mn ) γ . By hypotheses, we have that d ( mn ) τ = D ( m ) τ , for some function D ( m ) τ , which is independent of n . Thus, byLemma 3.2, we have for any n such that ( m, n ) is an admissible pair that e d ( mn ) γ,τ = e D ( m ) γ,τ and δ ( mn ) γ = ∆ ( m ) γ ,for some functions e D ( m ) γ,τ and ∆ ( m ) γ which are independent of n . For example, when m = 2, we have that D (2) τ = − τ , e D (2) γ,τ = −
14 ( aτ + b ) , ∆ (2) γ = 1 − a . Thus, the functions φ ( m ) n,γ,τ defined in (35) are in fact independent of n ; that is, φ ( m ) n,γ,τ = Φ ( m ) γ,τ , whereΦ ( m ) γ,τ := e ( t m γτ ) e (cid:18) − c cτ + d ) ( e D ( m ) γ,τ ) (cid:19) q (∆ ( m ) γ ) e (cid:16) D ( m ) τ ∆ ( m ) γ (cid:17) q − t m . After some simplification, using the fact that det γ = 1 for any γ ∈ SL ( Z ), we find that Φ ( m ) γ,τ = ε ( m ) γ , where ε ( m ) γ := e ( abt m ) , m = 2 , ′ , ′′ , ,e (cid:0) − a − ab +4 c (cid:1) , m = 1 ,e (cid:0) − a − ab +18 c − cd (cid:1) , m = 3 ,e (cid:0) − a − ab +18 c − cd (cid:1) , m = 5 , and in particular, is a root of unity, and thus also independent of τ . Thus, we have shown for m ∈ T ′ \ { } that b V mn ( γ mn τ ) = ψ ( γ mn ) − ( − k ( mn ) γmn + ℓ ( mn ) γmn + r ( mn ) γmn + s ( mn ) γmn ε ( m ) γ mn ( c mn τ + d mn ) b V mn ( τ ) , (36)as desired.When m = 4, some additional care is required, as the function V n is formed by adding V ′ n and V ′′ n .While the groups A ′ n and A ′′ n are equal, a priori, it is not clear for a matrix γ n = (cid:0) a bc d (cid:1) ∈ A ′ n = A ′′ n = A n that the two multipliers( − k (4 ′ n ) γn + ℓ (4 ′ n ) γn + r (4 ′ n ) γn + s (4 ′ n ) γn ε (4 ′ ) γ n and ( − k (4 ′′ n ) γn + ℓ (4 ′′ n ) γn + r (4 ′′ n ) γn + s (4 ′′ n ) γn ε (4 ′′ ) γ n are equal, which we desire in order to give a transformation property for the functions b V n by adding thetransformations in (36) when m = 4 ′ and m = 4 ′′ . By parity considerations, a direct calculation reveals thatindeed, ( − k (4 ′ n ) γn + ℓ (4 ′ n ) γn + r (4 ′ n ) γn + s (4 ′ n ) γn = ( − k (4 ′′ n ) γn + ℓ (4 ′′ n ) γn + r (4 ′′ n ) γn + s (4 ′′ n ) γn . For the remaining roots of unity weuse fact that a = 1 + 12 a ′ and b = 12 b ′ for some integers a ′ and b ′ , and hence, ε (4 ′′ ) γ n = ζ − (1+12 a ′ )12 b ′ = ζ − b ′ ( − − a ′ b ′ = ζ − b ′ ( − − a ′ b ′ = ζ − a ′ )12 b ′ = ε (4 ′ ) γ n . Thus, when m = 4, (36) holds as well, with the multiplier( − k (4 n ) γ n + ℓ (4 n ) γ n + r (4 n ) γ n + s (4 n ) γ n ε (4) γ n := ( − k (4 ′ n ) γ n + ℓ (4 ′ n ) γ n + r (4 ′ n ) γ n + s (4 ′ n ) γ n ε (4 ′ ) γ n = ( − k (4 ′′ n ) γ n + ℓ (4 ′′ n ) γ n + r (4 ′′ n ) γ n + s (4 ′′ n ) γ n ε (4 ′′ ) γ n . To show that the functions b V mn are harmonic Maass forms, we must additionally show that they are annihi-lated by the operator ∆ . As summarized in the Appendix, we have for any admissible pair ( m, n ) that the unction b V mn ( τ ) may be expressed, up to multiplication by an easily determined constant α mn , as follows: b V n ( τ ) = α n c M − , − ( τ ) , b V ′ n ( τ ) = α ′ n c M − , ( τ ) , b V n ( τ ) = α n c M − , − ( τ ) , b V n ( τ ) = α n c M − , ( τ ) , b V ′′ n ( τ ) = α ′′ n c M − , ( τ ) , b V n ( τ ) = α n c M − , ( τ ) , b V n ( τ ) = α n c M − , − ( τ ) , b V n ( τ ) = α n (cid:16) c M − , ( τ ) + c M − , ( τ ) (cid:17) , (37)where the functions c M a,b ( τ ) are defined in (27). We then apply Proposition 2.9 to see that the functions b V mn are annihilated by the operator ∆ . That these forms satisfy adequate growth conditions follows fromtheir definitions. Clearly, the functions V mn are the holomorphic parts of the forms b V mn , hence, are mockmodular.Finally, we prove that for m ∈ T , the functions V mn have, up to a constant multiple, shadows givenby the weight 3 / E m (cid:16) τc m (cid:17) . To show this, we use (37), Proposition 2.9, and Lemma2.10. In the case of m = 1, combining (37) with Proposition 2.9 part (i) shows that up to a constant,the mock modular forms V n have shadows given by g c , ( τ ) . It is not difficult to show by definition that g c , ( τ ) = g , ( τ ) . We previously established in Lemma 2.10 that E ( τ /
32) = 4 g , ( τ ) , hence, we haveproved that the functions V n ( τ ) have shadows given by a (computable) constant multiple of the eta-thetafunction E ( τ / V mn for the other values of m followby a similar argument. (cid:3) Quantum sets
In order to establish quantum modularity of the functions V mn , we must first determine viable sets ofrationals. We call a subset S ⊆ Q a quantum set for a function F with respect to the group G ⊆ SL ( Z ) ifboth F ( x ) and F ( M x ) exist (are non-singular) for all x ∈ S and M ∈ G .4.1. Utilizing Theorem 2.4.
By examining our catalogue of V mn in the Appendix, we see that preciselywhen n = 1 we have a µ -function in the form given in Theorem 2.4 of the third author. Using Theorem 2.4in these cases, and the notation from the Appendix, we directly obtain the following lemma. Lemma 4.1.
For m ∈ T ′ \{ } , we have that V m ( τ ) = iw m q t m + · g e u ( m τ ! ; q ! − w m q t m + e − u ( m τ ! · η ( τ ) η ( τ ) ϑ ( u ( m τ ; τ ) . Furthermore, when n = 1 we are still able to utilize Theorem 2.4. Let m ∈ T ′ \{ } , and n admissible.Then by Lemma 2.3 (4), and using the fact that ϑ ′ (0; τ ) = − πη ( τ ) , we see that(38) V mn ( τ ) − V m ( τ ) = iw m q t m η ( τ ) ϑ (cid:16) τ + u ( mn ) τ ; τ (cid:17) ϑ ( u ( mn ) τ − u ( m τ ; τ ) ϑ ( u ( m τ ; τ ) ϑ (cid:0) τ ; τ (cid:1) ϑ ( u ( mn ) τ ; τ ) ϑ ( v ( mn ) τ ; τ ) =: F mn ( τ ) . We will explicitly show in Lemma 5.1 that these functions F mn ( τ ) transform like weakly holomorphic modularforms of weight 1 /
2. Since we can write(39) V mn ( τ ) = V m ( τ ) + F mn ( τ )for m ∈ T ′ \{ } , and for m = 4,(40) V n ( τ ) = V ( τ ) + F ′ n ( τ ) + F ′′ n ( τ ) , once we establish quantum sets for the V m , which we will do in Section 4.2, we can use (39) and (40) tofind subsets that are quantum sets for the more general V mn , for each m ∈ T . We do so in Section 4.3. .2. Determining quantum sets for V m . Observe from Lemma 4.1 that for m ∈ T ′ \{ } , V m ( τ ) is asum of two of terms, each with a coefficient that is a constant times a power of q . The first term is of theform(41) g ( a m q bm ; q ) = ∞ X n =0 ( − q ; q ) n q n ( n +1) / ( a m q bm ; q ) n +1 ( a − m q bm − bm ; q ) n +1 , where a m = 1 when m is even, a m = i when m is odd, and ( b , b , b , b ′ , b ′′ , b , b ) = (4 , , , , / , , η ( τ ) η ( τ ) ϑ ( u ( m τ ; τ ) , which by (17) we see is equal to ie ( u ( m τ ) multiplied by the infinite product f m ( τ ) := ( q ; q ) ∞ ( − q ; q ) ∞ ( a m q bm ; q ) ∞ ( a − m q bm − bm ; q ) ∞ . (42)Note that for any τ ∈ Q , constant and q -power multiples of these terms will not affect whether V m ( τ ) and V m ( M τ ) exist. Thus, we may determine quantum sets for each V m ( τ ) by examining the sum and productappearing in equations (41) and (42). We seek rational numbers h/k ∈ Q such that for sufficiently large n ,0 = (cid:18) − e (cid:18) h k (cid:19) ; e (cid:18) h k (cid:19)(cid:19) n = n Y j =1 (cid:18) e (cid:18) jh k (cid:19)(cid:19) , (43)and hence, the infinite sum defining the function g in (41) terminates, and can be explicitly evaluated. Theidentity in (43) holds if and only if jh/k is an odd integer for some 1 ≤ j ≤ n . This can never happen when h is even, and when h is odd, then j = k causes the series to terminate at n = k . Thus, the largest possibleset for which (43) can hold is(44) S := { h/k ∈ Q | h ∈ Z , k ∈ N , gcd( h, k ) = 1 , h ≡ } . We also set S ′ := { h/k ∈ S | h ≡ ± } (45) S ev := { h/k ∈ S | k ≡ } S od := { h/k ∈ S | k ≡ } , and define the subsets S m ⊆ S by S , S , S := SS , S := S ′ S := S ′ ∪ S ev . We will prove the following theorem.
Theorem 4.2.
For m ∈ T , the set S m is a quantum set for V m with respect to the group G m , where G := h ( ) , ( ) i ⊂ Γ (2) ∩ Γ (4) ,G := h ( ) , ( ) i ⊂ Γ (4) ,G , G := h ( ) , ( ) i ⊂ Γ (2) ∩ Γ (6) ,G := h ( ) , ( ) i ⊂ Γ (12) ,G := h ( ) , ( ) i ⊂ Γ (6) . Before proving Theorem 4.2 we prove two lemmas which analyze the behavior of V m on S m . Lemma 4.3.
For each m ∈ T ′ \{ } we have that g ( a m q bm ; q ) is well-defined for τ ∈ S m . roof. We have seen above that for any τ ∈ S , the series in (41) terminates at n = k . We further requirethat ( a m e ( h b m k ); e ( h k )) n +1 , and ( a − m e ( ( b m − h b m k ); e ( h k )) n +1 do not vanish before the termination of the series.First, we note that (cid:18) a m e (cid:18) h b m k (cid:19) ; e (cid:18) h k (cid:19)(cid:19) n +1 = n Y j =0 (cid:18) − a m e (cid:18) ( b m j + 1) h b m k (cid:19)(cid:19)(cid:18) a − m e (cid:18) ( b m − h b m k (cid:19) ; e (cid:18) h k (cid:19)(cid:19) n +1 = n Y j =0 (cid:18) − a − m e (cid:18) ( b m ( j + 1) − h b m k (cid:19)(cid:19) . When m is even, we have a m = a − m = 1, and when m is odd, we have a m = i and a − m = − i . Thus for m even we need to avoid the existence of an r ∈ Z and 0 ≤ j ≤ n such at least one of the following hold, h ( b m j + 1) = 2 b m kr, (46) h ( b m ( j + 1) −
1) = 2 b m kr. (47)This can never occur for the cases m = 2 , ′ because b = 4 and b ′ = 12 are even while h is odd.When m = 4 ′′ , multiplying the equations through by 5 gives a similar contradiction since 5 b ′′ = 12 is evenwhile 5, h are odd. Thus when m = 2 , ′ , ′′ , S is the largest set of rationals over which the sum defining g ( a m q bm ; q ) terminates.For m = 6, we have b = 3, so we see that one of h (3 j + 1) = 6 kr or h (3 j + 2) = 6 kr can occur when h ≡ k ≡ ± k ≡ j = k − , thenwe have that 0 < j < k is an integer and so is r = h (3 j +1)6 k . Similarly, if k ≡ j = k − , thenwe have that 0 < j < k is an integer and so is r = h (3( j +1) − k . However, when h ≡ ± m = 6, S is thelargest set of rationals over which the sum defining g ( a m q bm ; q ) terminates.Similarly, when m is odd we wish to avoid the existence of an r ∈ Z and 0 ≤ j ≤ n such that at least oneof the following hold, 2 h ( b m j + 1) = b m k (4 r − , (48) 2 h ( b m ( j + 1) −
1) = b m k (4 r + 1) . (49)This can never occur for the case m = 1 because here b = 4 while h is odd, meaning the left hand sideof neither equation is divisible by 4. Thus when m = 1, S is the largest set of rationals over which the sumdefining g ( a m q bm ; q ) terminates.For m = 3, we have b = 3 so equations (48), (49) become2 h (3 j + 1) = 3 k (4 r − , h (3 j + 2) = 3 k (4 r + 1) . When h ≡ ± h ≡ k is odd, we see that neither (48) nor (49) can be satisfied.However if h ≡ k is even, then one of 2 h (3 j + 1) = 3 k (4 r −
1) or 2 h (3 j −
2) = 3 k (4 r + 1)can occur for some 0 ≤ j < k . This is because then k ≡ ± h ≡ k ≡ j = k − ∈ N , and r = ( h (3 j +2)3 k − ∈ Z .(2) Let h ≡ k ≡ j = k − ∈ N , and r = ( h (3 j +2)3 k − ∈ Z .(3) Let h ≡ k ≡ j = k − ∈ N , and r = ( h (3 j +1)3 k + 1) ∈ Z .(4) Let h ≡ k ≡ j = k − ∈ N , and r = ( h (3 j +1)3 k + 1) ∈ Z .In each of these cases observe that 0 ≤ j < k . Thus we see that when m = 3, actually S ′ ∪ S od is the largestset of rationals over which the sum defining g ( a m q bm ; q ) terminates. However, we will see in the nextlemma that we must eventually restrict to S .For m = 5 we have b = 6 so equations (48), (49) become h (6 j + 1) = 3 k (4 r − ,h (6 j + 5) = 3 k (4 r + 1) . hen h ≡ ± h ≡ k is even, we see that neither (48) nor (49) can be satisfied.However if h ≡ k is odd, then one of h (6 j + 1) = 3 k (4 r −
1) or h (6 j + 5) = 3 k (4 r + 1) canoccur for some 0 ≤ j < k . This is because then k ≡ ± h ≡ k ≡ j = k − ∈ N , and r = ( h (6 j +5)3 k − ∈ Z .(2) Let h ≡ k ≡ j = k − ∈ N , and r = ( h (6 j +5)3 k − ∈ Z .(3) Let h ≡ k ≡ j = k − ∈ N , and r = ( h (6 j +1)3 k + 1) ∈ Z .(4) Let h ≡ k ≡ j = k − ∈ N , and r = ( h (6 j +1)3 k + 1) ∈ Z .In each of these cases observe that 0 ≤ j < k . Thus we see that when m = 5, S is the largest set ofrationals over which the sum defining g ( a m q bm ; q ) terminates. (cid:3) We next analyze the second term from Lemma 4.1, f m ( τ ), when τ ∈ S m . The following result is used toprove the transformation formulas in the next section. Lemma 4.4.
For each m ∈ T ′ \{ } , f m ( τ ) vanishes for each τ ∈ S m .Proof. We observe from (42) that the product ( − q ; q ) ∞ = Q n ≥ (1 + q n ) appears in the numerator of f m ( τ ). Thus, as in our analysis of the g term, we see that for τ = hk ∈ S the n = k term of this productwill be 0. Similarly, the n = k term of ( q ; q ) ∞ will also vanish. Thus to show that f m ( τ ) = 0 for τ ∈ S m it remains to show that the products in the denominators of f m ( τ ) are finite and nonzero on S m for terms1 ≤ n ≤ k (when expressed as products indexed by n ). We see that the terms appearing in ( a m q bm ; q ) ∞ and ( a − m q bm − bm ; q ) ∞ , are the squares of terms appearing in the denominators of g ( a m q bm ; q ). We analyzethem similarly as in Lemma 4.3. For τ = hk , we have (cid:18) ( − m e (cid:18) hb m k (cid:19) ; e (cid:18) hk (cid:19)(cid:19) ∞ = Y n ≥ (cid:18) − ( − m e (cid:18) ( b m n + 1) hb m k (cid:19)(cid:19)(cid:18) ( − m e (cid:18) ( b m − hb m k (cid:19) ; e (cid:18) hk (cid:19)(cid:19) ∞ = Y n ≥ (cid:18) − ( − m e (cid:18) ( b m ( n + 1) − hb m k (cid:19)(cid:19) . Thus for m even we wish to avoid the existence of an r ∈ Z and 0 ≤ n ≤ k such at least one of the followinghold, h ( b m n + 1) = b m kr,h ( b m ( n + 1) −
1) = b m kr. For m = 2 , ′ we have that b m is even and h is odd so this cannot occur. When m = 4 ′′ , multiplying theequations through by 5 gives a similar contradiction since 5 b ′′ = 12 is even while 5, h are odd. When m = 6,we have b = 3. But for hk ∈ S , we have h ≡ ± m is odd, we must show there is no r ∈ Z such that h ( b m n + 1) = b m k (2 r + 1) , (50) h ( b m ( n + 1) −
1) = b m k (2 r + 1) . (51)When m = 1 , b m is even and h is odd so this cannot occur. When m = 3, we have b = 3.But for hk ∈ S , we have that h ≡ ± S from S ′ . If h ≡ k is odd, then either k ≡ n = k − and r = h − in the first equation, or k ≡ n = k − and r = h − in the second equation. Both instances result in a zero in thedenominator before termination. (cid:3) Remark.
Lemmas 4.3 and 4.4 imply that for each m ∈ T , S m is our largest possible quantum set for V m . We now prove Theorem 4.2. roof. (Proof of Theorem 4.2) Let m ∈ T . By Lemmas 4.3 and 4.4, we see that each V m is well-defined for τ ∈ S m , but it remains to be seen that V m is well-defined for each M τ , where M ∈ G m . We conclude byproving that each set S m is closed under transformations by the matrices in G m . Observe that each G m has two generators, one of the form ( A ) and the other of the form ( B ) for positive integers A m , B m . For h/k ∈ S we have T m, ( h/k ) := (cid:18) A m (cid:19) hk = hk + A m h , T m, ( h/k ) := (cid:18) B m (cid:19) hk = h + B m kk . Since gcd( h, k ) = 1, we have gcd( h, A m k + h ) = gcd( h + B m k, h ) = 1. Moreover, we note that B m is evenfor each m , so when h is odd we have that h + B m k is odd, and thus T m ( h/k ) , T ′ m ( h/k ) ∈ S for all τ ∈ S .Thus for m = 1 , , ′ , ′′ , T m ( h/k ) , T ′ m ( h/k ) ∈ S m for all τ ∈ S m . When m = 3 , , B m = 6 so that h + B m k ≡ h (mod 6). Thus for m = 3 , ,
6, we see that T ′ m ( h/k ) ∈ S m for all τ ∈ S m .To see also that T m ( h/k ) ∈ S m for all τ ∈ S m , we only need to observe that in the case m = 5, when k iseven, then k + 2 h is also even.Now we need to also consider the inverses T − m, ( h/k ) := (cid:18) − A m (cid:19) hk = hk − A m h , T − m, ( h/k ) := (cid:18) − B m (cid:19) hk = h − B m kk . When k − A m h is positive, the same arguments as above go through. When k − A m h is negative, we observethat T − m ( h/k ) := (cid:18) − A m (cid:19) hk = − hA m h − k , has a positive denominator, and so again we use the arguments above. (cid:3) Determining quantum sets for general V mn . In Section 4.2, we determined the quantum sets S m for the function V m . In this section, we will use (39) and (40) to determine the more general quantum sets S mn for the functions V mn with n = 1. Observe that our previous discussion shows that we must require S mn ⊆ S m for each m ∈ T . We define the sets S mn for any m ∈ T and admissible n below; for completeness,we also include the sets S m previously determined. In Lemma 4.5, we establish that these sets are indeedappropriate, by showing that the auxiliary functions F mn appearing in (39) and (40) vanish at any rationalpoint in S mn .We define the 43 subsets S mn ⊆ S m by S , S , S , S , S , S , S := SS , S , S , S , S , S := S ev S , S , S , S , S , S , S , S , S , S , S , S , S := S ′ S , S , S , S := S od S , S , S := S ′ ∩ S ev S , S := S ′ ∩ S od S := S ′ ∪ S od S , S , S , S , S , S , S := S ′ ∪ S ev . Lemma 4.5.
For m ∈ T \{ } and hk ∈ S mn , or for m ∈ { ′ , ′′ } and hk ∈ S n , we have that F mn (cid:18) hk (cid:19) = 0 . Proof of Lemma 4.5.
Note that by applying the triple product formula (17) to each of the ϑ -functions ap-pearing in the definition of F mn in (38), we can cancel the four copies of ( q ; q ) ∞ appearing in the denominatorwith four of the five copies appearing in the numerator (three of which arise from the function η ). Thus wemay write F mn ( τ ) as a constant multiple of q multiplied by( q ; q ) ∞ ( q ; q ) ∞ · ( e ( τ + u ( mn ) τ ); q ) ∞ ( e ( − τ − u ( mn ) τ ); q ) ∞ ( e ( u ( m τ ); q ) ∞ ( e ( − u ( m τ ) q ; q ) ∞ ( e ( u ( mn ) τ ); q ) ∞ ( e ( − u ( mn ) τ ) q ; q ) ∞ ( e ( v ( mn ) τ ); q ) ∞ ( e ( − v ( mn ) τ ) q ; q ) ∞ . (52) bserve that for any τ = h/k ∈ S , we have that ( q ; q ) ∞ vanishes at the k th term when expanded,and ( q ; q ) ∞ never vanishes. Moreover, we have already demonstrated in the proof of Lemma 4.4 that( e ( u ( m τ ); q ) ∞ ( e ( − u ( m τ ) q ; q ) ∞ does not vanish for τ = h/k ∈ S m , as this term appears in the denominatorof f m . Thus, it suffices to show that when τ = h/k ∈ S mn each of the products in(53) ( e ( u ( mn ) τ ); q ) ∞ ( e ( − u ( mn ) τ ) q ; q ) ∞ ( e ( v ( mn ) τ ); q ) ∞ ( e ( − v ( mn ) τ ) q ; q ) ∞ is non-vanishing for terms 1 ≤ s ≤ k , when expanded as products indexed by s . Next, we observe that v ( mn ) τ depends only on n . In particular, v ( mn ) τ = τc n when n is odd, and v ( mn ) τ = τc n − when n is even, where( c , c , c , c , c , c , c , c ) = (2 , , , , , , , m, n ),( e ( v ( mn ) τ ); q ) ∞ ( e ( − v ( mn ) τ ) q ; q ) ∞ = ( ( q cn ; q ) ∞ ( q cn − cn ; q ) ∞ when n odd( − q cn ; q ) ∞ ( − q cn − cn ; q ) ∞ when n even . When n is odd, we have that for τ = h/k ,( q cn ; q ) ∞ ( q cn − cn ; q ) ∞ = Y j ≥ (cid:18) − e (cid:18) ( c n j + 1) hc n k (cid:19)(cid:19) (cid:18) − e (cid:18) ( c n ( j + 1) − hc n k (cid:19)(cid:19) . Thus for n odd we wish to avoid the existence of an r ∈ Z and 0 ≤ j ≤ k such that at least one of thefollowing hold: h ( c n j + 1) = c n kr,h ( c n ( j + 1) −
1) = c n kr. For n = 1 , , c n is even and h is odd so this cannot occur. When n = 3, we have c = 3. Butfor hk ∈ S , we have h ≡ ± n is even, we have that for τ = h/k ,( − q cn ; q ) ∞ ( − q cn − cn ; q ) ∞ = Y j ≥ (cid:18) − e (cid:18) ( c n j + 1) hc n k − (cid:19)(cid:19) (cid:18) − e (cid:18) ( c n ( j + 1) − hc n k − (cid:19)(cid:19) , so in this case we need to avoid the existence of an r ∈ Z and 0 ≤ j ≤ k such that at least one of thefollowing hold, 2 h ( c n j + 1) = c n k (2 r + 1) , h ( c n ( j + 1) −
1) = c n k (2 r + 1) . When n = 6 we have c n = 4 and since h is odd so this cannot occur for any element in S . When n = 2, bothequations reduce to the equation h (2 j + 1) = k (2 j + 1). In the definitions of the sets S m , we see that ineach case k is even, and so this equation can never be satisfied for an element of S m . When n = 4, we havethe equations 2 h (3 j + 1) = 3 k (2 r + 1), and 2 h (3 j + 2) = 3 k (2 r + 1). We see that these can not be satisfiedwhen h h ≡ k odd. Thus, for elements of S m they cannot be satisfied.Similarly, when n = 8, we have the equations 2 h (6 j + 1) = 6 k (2 r + 1), and 2 h (6 j + 5) = 3 k (6 r + 1), whichalso can’t be satisfied when h h ≡ k even. The definitions of S m shows that we are always in one of these cases.Thus, we have reduced the problem to showing that when τ = h/k ∈ S mn , the products(54) ( e ( u ( mn ) τ ); q ) ∞ ( e ( − u ( mn ) τ ) q ; q ) ∞ are non-vanishing in their first k terms when expanded. Although at first glance it would seem that we havemany cases to consider, in fact we have already done most of the work, we just need to compare each case tothe defined set S mn . Comparing the values of u ( mn ) τ when m > v ( mn ) τ that we have alreadyconsidered, and using that e ( ) = e ( − ), we see that there are only about a dozen left to consider. Moreover,the cases that are merely a negative multiple can be reduced fairly easily to the original case. Thus the only u ( mn ) τ we will consider here are u (13) τ = τ /
12 + 1 / u (14) τ = τ / u (15) τ = 1 /
2, and u (4 ′′ τ = 5 τ / − / or u (13) τ = τ /
12 + 1 /
2, (54) becomes( − q ; q ) ∞ ( − q ; q ) ∞ = Y j ≥ (cid:18) − e (cid:18) (12 j + 1) h k − (cid:19)(cid:19) (cid:18) − e (cid:18) (12( j + 1) − h k − (cid:19)(cid:19) , and so we wish to avoid the existence of an r ∈ Z and 0 ≤ j ≤ k such that at least one of the following hold, h (12 j + 1) = 6 k (2 r + 1) ,h (12( j + 1) −
1) = 6 k (2 r + 1) . But since h is odd this can never occur.For u (14) τ = τ /
12, (54) becomes( q ; q ) ∞ ( q ; q ) ∞ = Y j ≥ (cid:18) − e (cid:18) (12 j + 1) h k (cid:19)(cid:19) (cid:18) − e (cid:18) (12( j + 1) − h k (cid:19)(cid:19) , and so we wish to avoid the existence of an r ∈ Z and 0 ≤ j ≤ k such that at least one of the following hold, h (12 j + 1) = 12 kr,h (12( j + 1) −
1) = 12 kr, which again can never occur since h is odd.For u (15) τ = 1 /
2, (54) becomes( − q ) ∞ ( − q ; q ) ∞ = 2 Y j ≥ (cid:18) − e (cid:18) jhk − (cid:19)(cid:19) , so we wish to avoid the existence of an r ∈ Z and 0 ≤ j ≤ k such that 2 hj = k (2 r + 1) holds, which can’toccur in S since k is odd.Lastly, when u (4 ′′ τ = 5 τ / − /
2, (54) becomes( − q ; q ) ∞ ( − q ; q ) ∞ = Y j ≥ (cid:18) − e (cid:18) (12 j + 5) h k − (cid:19)(cid:19) (cid:18) − e (cid:18) (12( j + 1) − h k − (cid:19)(cid:19) , and so we wish to avoid the existence of an r ∈ Z and 0 ≤ j ≤ k such that at least one of the following hold, h (12 j + 5) = 6 k (2 r + 1) ,h (12( j + 1) −
5) = 6 k (2 r + 1) , which can never occur since h is odd. (cid:3) Quantum modularity of the V mn We now make more precise the notion of a quantum modular form. For k ∈ Z , a quantum modular formof weight k on the set S for the group G is a complex-valued function f such that S is a quantum set for f with respect to the group G ⊆ SL ( Z ). Further, for all γ = (cid:0) a bc d (cid:1) ∈ G , and for all x ∈ S ( x = − dc ), thefunctions h f,γ ( x ) := f ( x ) − ǫ ( γ )( cx + d ) − k f (cid:18) ax + bcx + d (cid:19) are suitably continuous or analytic in R , as defined by Zagier in [24]. In this paper, we will consider realanalytic functions h f,γ . The ǫ ( γ ) are appropriate complex numbers, such as those that arise naturally in thetheory of half-integer weight modular forms.In this section, we prove Theorem 1.2 and Proposition 1.4, the first of which in particular establishes thequantum modularity of the functions V mn . We begin by defining for m ∈ T the numbers ℓ m := ( , m = 1 , , , , m = 2 , , , a m := , m = 1 , , , m = 3 , , , m = 4 , , m = 5 , b m := ( a m , m = 1 , , , , a m , m = 3 , , c m := ( a m , m = 1 , , , , a m , m = 3 , , nd let ℓ ′ = ℓ ′′ := 1 , b ′ = b ′′ := 12 , and a ′ = a ′′ = 24. We define the following groups G , G , G , G := h ( ) , ( ) i ⊂ Γ (2) ∩ Γ (4) ,G , G , G , G , G , G , G G , G , G , G n , G , G , G , G := h ( ) , ( ) i ⊂ Γ (2) ∩ Γ (12) ,G , G , G , G , G , G G , G , G , G , G , G := h ( ) , ( ) i ⊂ Γ (2) ∩ Γ (6) . The sets G m are as defined in Theorem 4.2, and the sets G n above are defined for any admissible n when m equals 4. We also define the constants κ n := ( , n ∈ { , } , , n ∈ { , , , } , κ n := ( , n ∈ { , } , , n ∈ { , , , } , κ n := ( , n ∈ { , , , } , , n ∈ { , } ,κ n := ( , n ∈ { , , , } , , n ∈ { , } , κ n := ( , n ∈ { , , , } , , n ∈ { , } , as well as κ m = κ n = κ ′ n = κ ′′ n := 1 for any admissible pair ( m, , n ), (4 ′ , n ) or (4 ′′ , n ). We recallthat for r ∈ Z , we let M r := ( r ).In Section 5.1, we first sketch the general proof of Theorem 1.2 when m ∈ T and n = 1, and then providedetails for the case when ( m, n ) = (1 , m, n ), in Section 5.2,we deduce the result for all remaining pairs ( m, n ). In Section 5.3, we prove Proposition 1.4.5.1. Proof of Theorem 1.2 for ( m, n ) = ( m, . General Proof of Theorem 1.2 when m ∈ T, n = 1 . For r ∈ N we have M r = ST − r S − , where S = (cid:0) −
11 0 (cid:1) and T = ( ), and we define τ r := T − r S − τ = − /τ − r . Using the fact that M r τ = Sτ r , we find bystraightforward but lengthy calculations using the expressions for V m given in (33) (and the Appendix)combined with Lemma 2.3 that V m ( M ℓ m τ ) = ζ − ℓ m ( ℓ m τ + 1) V m ( τ ) + I m ( τ ) + J m ( τ ) . (55)The functions I m and J m are defined by Mordell integrals h ( z ; τ ), which we then simplify to Eichler integralsof weight 3 / g a,b , using either Theorem 2.7 (for m = 2 , ,
6) or Lemma 2.8 (for = 1 , , I ( τ ) := − ζ i e (cid:0) τ (cid:1) √− iτ h (cid:0) τ + ; τ (cid:1) = i √ τ + 1 Z g , ( u ) p − i ( u + τ ) du + i √− iτ , J ( τ ) := 12 i q − √ τ + 1 h (cid:0) τ − ; τ (cid:1) = i √ τ + 1 Z i ∞ g , ( u ) p − i ( u + τ ) du − i √− iτ , I ( τ ) := 12 √− iτ h (cid:0) ; τ (cid:1) = i √ τ + 1 Z g , ( u ) p − i ( u + τ ) du, J ( τ ) := − ζ q − √ τ + 1 h (cid:0) τ ; τ (cid:1) = i √ τ + 1 Z i ∞ g , ( u ) p − i ( u + τ ) du, I ( τ ) := − ζ i e (cid:0) τ (cid:1) √− iτ h (cid:0) τ + ; τ (cid:1) = i √ τ + 1 Z g , ( u ) p − i ( u + τ ) du + i √− iτ , J ( τ ) := 12 i q − √ τ + 1 h (cid:0) τ − ; τ (cid:1) = i √ τ + 1 Z i ∞ g , ( u ) p − i ( u + τ ) du − i √− iτ , I ( τ ) := 12 √− iτ (cid:0) h (cid:0) ; τ (cid:1) + h (cid:0) ; τ (cid:1)(cid:1) = iζ √ τ + 1 Z ζ − g , ( u ) + ζ − g , ( u ) p − i ( u + τ ) du, J ( τ ) := − ζ √ τ + 1 (cid:16) q − h (cid:0) τ ; τ (cid:1) + q − h (cid:0) τ ; τ (cid:1)(cid:17) = iζ √ τ + 1 Z i ∞ ζ − g , ( u ) + ζ − g , ( u ) p − i ( u + τ ) du, I ( τ ) := − ζ − e (cid:0) τ (cid:1) √− iτ h (cid:0) τ + ; τ (cid:1) = i √ τ + 1 Z g , ( u ) p − i ( u + τ ) du + i √− iτ , J ( τ ) := 12 i q − √ τ + 1 h (cid:0) τ − ; τ (cid:1) = i √ τ + 1 Z i ∞ g , ( u ) p − i ( u + τ ) du − i √− iτ , I ( τ ) := 12 √− iτ h (cid:0) ; τ (cid:1) = iζ − √ τ + 1 Z g , ( u ) p − i ( u + τ ) du, J ( τ ) := 12 i ζ − q − √ τ + 1 h (cid:0) τ ; τ (cid:1) = iζ − √ τ + 1 Z i ∞ g , ( u ) p − i ( u + τ ) du, Table 2.
Mordell and Eichler integrals I m and J m For τ ∈ H , the transformation law in part (i) of Theorem 1.2 when m ∈ { , , } and n = 1, and thetransformation law in part (ii) of Theorem 1.2 when m ∈ { , , } and n = 1 (both of which pertain to V m ( M ℓ m τ ) , m ∈ T ) now follow from (55), Table 2, and Lemma 2.10. The transformation law in part (i)of Theorem 1.2 for τ ∈ H when m ∈ { , , } and n = 1 follows after a short calculation by iterating thetransformation law given in part (ii), applying Lemma 2.6, and simplifying.The transformation law (under τ → τ + b m ) in part (iii) of Theorem 1.2 follows for τ ∈ H by a directcalculation using Lemma 2.3.Having established parts (i), (ii), and (iii) of Theorem 1.2 for τ ∈ H for n = 1, we have continuation to τ = x ∈ S m \ {− } in part (i), to τ = x ∈ S m \ {− } in part (ii), and to x ∈ S m in part (iii), by Theorem4.2 and the argument given in Section 4. As argued in [4, 6, 13, 22, 24], for example, the integrals appearingin parts (i) and (ii) of Theorem 1.2 are real analytic functions, except at − / − (cid:3) Detailed Proof of Theorem 1.2 for ( m, n ) = (1 , . As summarized in the Appendix or (33), we may write V ( τ ) = − q − µ ( τ / / , τ / τ ) . (56)Thus, we have V ( M τ ) = − e (cid:18) − M τ (cid:19) µ (cid:18) Sτ , Sτ Sτ (cid:19) = − e (cid:18) − M τ (cid:19) e − τ (cid:18)
14 + τ (cid:19) ! √− iτ (cid:18) − µ (cid:18) −
14 + τ , −
12 ; τ (cid:19) + 12 i h (cid:18)
14 + τ τ (cid:19)(cid:19) = ζ e (cid:18) τ (cid:19) √− iτ µ (cid:18) −
14 + τ , −
12 ; τ (cid:19) + I ( τ ) , (57)where I ( τ ) := − i ζ e (cid:18) τ (cid:19) √− iτ h (cid:18) τ τ (cid:19) . Here, we have used Lemma 2.3(6). Next, recalling that τ = S − τ −
2, we (repeatedly) apply Lemma2.3(5) followed by a second application of Lemma 2.3(6), as well as Lemma 2.3(1, 3), and find after somesimplification that (57) equals (2 τ + 1) V ( τ ) + I ( τ ) + J ( τ ) , (58)where J ( τ ) := 12 i e (cid:16) − τ (cid:17) (2 τ + 1) h (cid:18) τ −
12 ; τ (cid:19) . We re-write I ( τ ) using part (ii) of Lemma 2.8, and obtain after some simplification that I ( τ ) equals − √− iτ Z g , (cid:0) − u (cid:1)p i ( u − + τ − ) duu + i √− iτ = − √− iτ Z g , (cid:0) − u (cid:1)p i ( u − + τ − ) duu + i √− iτ = − √− iτ ( − i ) Z g , ( u ) p i ( u − + τ − ) duu + i √− iτ = − √ iτ τ ( − i ) Z g , ( u ) p − i ( u + τ ) du + i √− iτ = 12 √ τ + 1 Z g , ( u ) p − i ( u + τ ) du + i √− iτ (59)where we have used Lemma 2.6. We also re-write J ( τ ) using part (iii) of Lemma 2.8, and obtain after somesimplification that J ( τ ) equals 12 √ τ + 1 Z i ∞ g , ( u ) p − i ( u + τ ) du − i √− iτ . (60)Next, we re-write g / , ( z ) = ig / , ( τ ) = i E ( z/ τ → τ / (2 τ + 1)) given inpart (i) of Theorem 1.2 for τ ∈ H .For the second transformation law (under τ → τ + 4) in part (iii) of Theorem 1.2, we again use Lemma2.3. From (56), we have V ( τ + 4) = − ζ − q µ (cid:18) τ , τ , τ + 4 (cid:19) = − ζ − q µ (cid:18) τ , τ , τ (cid:19) = ζ − V ( τ ) , as desired. We have continuation to τ = x ∈ S \ {− } by Theorem 4.2 and the argument in Section 4 . (cid:3) .2. Proof of Theorem 1.2 for n = 1 . To prove the theorem in the remaining cases, we establish Lemma5.1 below, which shows that the auxiliary functions F mn , defined in (38), are weakly holomorphic modularforms, and provides explicit transformation properties. Lemma 5.1.
The functions F mn are weakly holomorphic modular forms of weight / . In particular, for τ ∈ H , the following are true. (i) For m ∈ T ′ \ { } , for each admissible n , we have that F mn ( τ ) + i ℓ m (2 τ + 1) − F mn (cid:18) τ τ + 1 (cid:19) = 0 . (ii) For m ∈ T ′ \ { } , for each admissible n , we have that F mn ( τ ) − ζ κ mn a m F mn ( τ + κ mn b m ) = 0 . We postpone the proof of Lemma 5.1 until the end of this section, and first prove Theorem 1.2 for theremaining functions V mn (i.e. n = 1). Proof of Theorem 1.2 for n = 1 . We begin by re-writing the functions V mn using (39) and (40), which wepreviously established. Note that F m ( τ ) is identically equal to zero for each m ∈ T ′ \ { } . We next usethe fact that for fixed m and each admissible n , we have that S mn ⊆ S m and G mn ⊆ G m . Previously,in Section 4, we showed that if x ∈ S m , then M x ∈ S m for any M ∈ G m . A nearly identical argumentshows that for fixed m and each admissible n , that if x ∈ S mn , then M x ∈ S mn for any M ∈ G mn . Thus,for fixed m and each admissible n ( n = 1), the quantum modular transformation properties given in parts(i) and (iii) of Theorem 1.2 for the functions V mn with n = 1 now follow from the transformation propertiesestablished in Section 5.1 for the functions V m in Theorem 1.2 restricted to the subsets S mn ⊆ S m and thesubgroups G mn ⊆ G m , combined with Lemma 5.1 and Lemma 4.5. This concludes the proof of Theorem1.2 in the remaining cases ( n = 1). (cid:3) Proof of Lemma 5.1.Proof of part (i).
The proof of part (i) of Lemma 5.1 makes use of Lemma 2.1 and Lemma 2.2. We divide ourproof into six cases, corresponding to six possible values of m . For m = 1, we give an explicit proof for eachadmissible n . For the remaining cases ( m ∈ { , , ′ , ′′ , , } ), we provide a sketch of proof for brevity’s sake,as the proofs in these cases are nearly identical to the case m = 1. To begin, we list some transformationproperties of certain specialized Jacobi ϑ -functions under M := ( ) which we will make use of: ϑ (cid:18) αM τ + 12 ; M τ (cid:19) = − iq − − α (2 τ + 1) e (cid:0) ( α + 1) τ + (cid:1) τ + 1 ! ϑ (cid:18) ατ + 12 ; τ (cid:19) , (61) ϑ ( αM τ ; M τ ) = − i (2 τ + 1) e ( ατ ) τ + 1 ! ϑ ( ατ ; τ ) , (62)where α ∈ C . To establish (61) and (62), we have used Lemma 2.2, and the fact that ψ ( M ) = − i . Case m = 1 .We have by definition that F n ( τ ) := − iq − η ( τ ) ϑ (cid:16) τ + u (1 n ) τ ; τ (cid:17) ϑ (cid:16) u (1 n ) τ − τ − ; τ (cid:17) ϑ (cid:0) τ + ; τ (cid:1) ϑ (cid:0) τ ; τ (cid:1) ϑ ( u (1 n ) τ ; τ ) ϑ ( v (1 n ) τ ; τ ) . Using transformation properties from (13), (18), (61), and (62), we find after some straightforward calcula-tions that F n ( M τ ) = i (2 τ + 1) ρ n ( M ) F n ( τ ) , (63)where ρ ( M ) := q + e τ + 1 − τ
32 + (cid:18) τ (cid:19) + (cid:18) τ + 12 (cid:19) − (cid:18) τ (cid:19) − (cid:18) τ (cid:19) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) !! , ( M ) := q + e τ + 1 − τ
32 + (cid:18) τ
12 + 12 (cid:19) + (cid:16) τ (cid:17) − (cid:18) τ (cid:19) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) − (cid:18) τ
12 + 12 (cid:19) !! ,ρ ( M ) := q + e τ + 1 − τ
32 + (cid:18) τ (cid:19) + (cid:18) τ (cid:19) − (cid:18) τ (cid:19) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) − (cid:18) τ (cid:19) !! ,ρ ( M ) := q + e τ + 1 − τ
32 + (cid:18) τ (cid:19) − (cid:18) τ (cid:19) − (cid:16) τ (cid:17) − (cid:18) τ + 12 (cid:19) !! ,ρ ( M ) := q + e τ + 1 − τ
32 + (cid:18) τ
12 + 12 (cid:19) + (cid:16) τ (cid:17) − (cid:18) τ (cid:19) − (cid:16) τ (cid:17) − (cid:18) τ
12 + 12 (cid:19) − (cid:16) τ (cid:17) !! ,ρ ( M ) := q + e τ + 1 − τ
32 + (cid:18) τ (cid:19) + (cid:18) τ (cid:19) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) − (cid:18) τ (cid:19) − (cid:18) τ (cid:19) !! . After simplifying, one finds that ρ n ( M ) = − i for each admissible n . Using this fact, Lemma 5.1 followsfrom (63) for each F n . Case m ∈ { , , ′ , ′′ , , } .We proceed as above in the case m = 1. Using transformation properties from (61), (62), and (18), we findafter some straightforward calculations that F mn ( M τ ) = i (2 τ + 1) ρ mn ( M ) F mn ( τ ) , (64)where for any admissible n , ρ mn ( M ) = ( , m ∈ { , ′ , ′′ , } , − i, m ∈ { , } . For example, for ( m, n ) ∈ { (2 , , (2 , , (3 , , (4 ′ , , (4 ′′ , , (5 , , (6 , } , we have that ρ ( M ) := q e τ + 1 − τ
32 + (cid:18) τ (cid:19) + (cid:18) τ + 12 (cid:19) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) − (cid:18) τ (cid:19) − (cid:18) τ (cid:19) !! = 1 ,ρ ( M ) := q − e τ + 1 − τ
32 + (cid:18) τ (cid:19) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) − (cid:18) τ + 12 (cid:19) !! = 1 ,ρ ( M ) := q +1 e τ + 1 − τ
72 + (cid:18) τ (cid:19) + (cid:18) τ (cid:19) − (cid:18) τ (cid:19) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) !! = − i,ρ ′ ( M ) := q e τ + 1 − τ
288 + (cid:18) τ
12 + 12 (cid:19) + (cid:18) τ + 12 (cid:19) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) − (cid:18) τ
12 + 12 (cid:19) − (cid:18) τ (cid:19) !! = 1 ,ρ ′′ ( M ) := q e τ + 1 − τ
288 + (cid:18) τ (cid:19) + (cid:16) τ (cid:17) − (cid:18) τ (cid:19) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) !! = 1 ,ρ ( M ) := q + e τ + 1 − τ
18 + (cid:18) τ (cid:19) − (cid:18) τ (cid:19) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) !! = − i,ρ ( M ) := q e τ + 1 − τ
72 + (cid:18) τ (cid:19) + (cid:18) τ + 12 (cid:19) − (cid:16) τ (cid:17) − (cid:16) τ (cid:17) − (cid:18) τ (cid:19) − (cid:18) τ (cid:19) !! = 1 . Proof of part (ii).
The proof in this case also follows by direct calculations using the definition of thefunctions F mn , as well the transformations η ( τ + b ) = ζ b η ( τ ) , ϑ ( z + a ; τ + b ) = ( − a ζ b ϑ ( z ; τ ) , (65)which hold for any a, b ∈ Z , and follow from (13), (18), and (19). We provide details in the cases m ∈{ , , , } and leave the remaining cases m ∈ { ′ , ′′ , } to the reader for brevity, as the proofs follow in a imilar manner. Case m ∈ { , } . In this case, b m = 4 , a m = 8, and t m = − /
32. Using (65) and a direct calculation, wefind that the portion of F mn independent of ϑ -functions satisfies iw m η ( τ + κ mn b m ) e πit m ( τ + κ mn b m ) = − ζ − κ mn a m · iw m η ( τ ) q t m . (66)Thus, it suffices to show that under τ τ + κ mn b m , the functions F mn ( τ ) / ( iw m η ( τ ) q t m ), which arequotients of ϑ -functions, map to −F mn / ( iw m η ( τ ) q t m ) . We compute using the definitions of u ( mn ) τ and v ( mn ) τ that u (1 n ) τ + κ n b = ( u (1 n ) τ ± , n ∈ { , , , , } ,u (1 n ) τ , n = 5 , v (1 n ) τ + κ n b = v (1 n ) τ + 4 , n ∈ { , } ,v (1 n ) τ + 2 , n ∈ { , , } ,v (1 n ) τ + 1 , n = 5 ,u (2 n ) τ + κ n b = ( u (2 n ) τ ± , n ∈ { , , , , } ,u (2 n ) τ , n = 6 , v (2 n ) τ + κ n b = v (2 n ) τ + 4 , n ∈ { , } ,v (2 n ) τ + 2 , n ∈ { , , } ,v (2 n ) τ + 1 , n = 6 ,u (11) τ + κ n b = ( u (11) τ + 3 , n ∈ { , , , } ,u (11) τ + 1 , n ∈ { , } , u (21) τ + κ n b = ( u (21) τ + 3 , n ∈ { , , , } ,u (21) τ + 1 , n ∈ { , } . The claim now follows after combining the above with the transformation for the ϑ -function given in (65). Case m ∈ { , } . In this case, b m = 6 , a m = 3 , and t m = − /
72. In this case, analogous to (66), we obtain iw m η ( τ + κ mn b m ) e πit m ( τ + κ mn b m ) = ζ − κ mn a m · iw m η ( τ ) q t m . Thus, it suffices to show that under the functions F mn ( τ ) / ( iw m η ( τ ) q t m ), which are quotients of ϑ -functions,remain invariant under τ τ + κ mn b m . Using the definititions of u ( mn ) τ and v ( mn ) τ , we find that u (3 n ) τ + κ n b = u (3 n ) τ + 2 , n = 2 ,u (3 n ) τ + 1 , n ∈ { , , , } ,u (3 n ) τ , n = 7 , v (3 n ) τ + κ n b = v (3 n ) τ + 3 , n ∈ { , , } ,v (3 n ) τ + 2 , n ∈ { , } ,v (3 n ) τ + 1 , n = 7 ,u (6 n ) τ + κ n b = u (6 n ) τ + 2 , n = 2 ,u (6 n ) τ + 1 , n ∈ { , , , } ,u (6 n ) τ , n = 8 , v (6 n ) τ + κ n b = v (6 n ) τ + 3 , n ∈ { , , } ,v (6 n ) τ + 2 , n ∈ { , } ,v (6 n ) τ + 1 , n = 8 ,u (31) τ + κ n b = ( u (31) τ + 2 , n ∈ { , , , } ,u (31) τ + 4 , n ∈ { , } , u (61) τ + κ n b = ( u (61) τ + 2 , n ∈ { , , , } ,u (61) τ + 4 , n ∈ { , } . The claim now follows after combining the above with the transformation for the ϑ -function given in (65). (cid:3) Proof of Proposition 1.4.
We follow a method of proof and argument originally due to Zagier in [24],which was later generalized in [4], and used also in [12], for example; we refer the reader to these sources formore explicit details, and provide a detailed sketch of proof here. The functions E m are modular forms ofweight 3 /
2, and satisfy, for all τ ∈ H and γ = (cid:0) a bc d (cid:1) ∈ G m ⊆ SL ( Z ), the transformation E m ( γτ ) = ν m ( γ )( cτ + d ) E m ( τ ) . (67)Here, ν m and G m are suitable multipliers and subgroups (respectively), and can be explicitly determinedusing the definitions of the functions E m . We define the function E ∗ m ( − τ ) ( τ ∈ H ) by E ∗ m ( − τ ) := Z i ∞− τ E m ( u ) √ u + τ du. Using (67), it is not difficult to show for all τ ∈ H and γ ∈ G m that E ∗ m ( − τ ) − ( − cτ + d ) − ν − m ( γ ) E ∗ m ( γ ( − τ )) = Z i ∞− dc E m ( u ) √ u + τ du. (68) nder a change of variable in the integrand, with an appropriate choice of matrix γ , up to multiplicationby a constant (which can be explicitly determined), we find that the transformations given in (68) for thefunctions E ∗ m (cid:0) − τ/c m (cid:1) are identical to the transformations given for the functions V mn ( x ) in Theorem1.2 for x ∈ S mn ⊆ Q , as τ = x + iy → x from the upper half-plane (as y → + ), or equivalently, as z = − τ /c m → − x/c m from the lower half-plane.On the other hand, we also have that the asymptotic expansions of E ∗ m ( − τ ) and e E m ( − τ ) agree atrational numbers r/s , that is, with τ = r/s + iy ∈ H , as y → + ; this fact is established more generally in[4, Proposition 2.1]. Thus, the functions e E m inherit the transformation properties satisfied by the functions E ∗ m at appropriate rationals, and hence, transform (up to the aforementioned change of variable, up to aconstant multiple) like the functions V mn in Theorem 1.2, as claimed.6. Corollaries
In this section, we prove Corollary 1.3, in which we evaluate Eichler integrals of eta-theta functions E m appearing in Theorem 1.2 as finite q -hypergeometric sums, and establish related curious algebraic identities.We define for m ∈ { , , , , } the numbers d m by d m := , m = 1 , , , m = 3 , , , m = 5 . For h/k ∈ Q with gcd( h, k ) = 1, we define for positive integers m the numbers H m = H m ( h, k ) := ( h, mh + k > , − h, mh + k < , (69) K m = K m ( h, k ) := | mh + k | . Proof of Corollary 1.3.
We first establish (8) and (10). To do so, we begin with parts (i) and (ii) of Theorem1.2 in the case n = 1. We then use Lemma 4.1 to re-write the functions V m in terms of the functions f m and g . By Lemma 4.4, we have that the functions f m vanish at rationals in S m . From Lemma 4.3, we alsohave that the remaining functions in Lemma 4.1, defined using the function g , are defined at rationals in S m . Moreover, the proof of Lemma 4.3 more specifically reveals that the functions defined using the infinitesums g in Lemma 4.1 in fact truncate, and become finite sums. Identities (8) and (10) of Corollary 1.3 thenfollow by a direct calculation using the definition of the functions F h,k given in (7), and the numbers H m and K m in (69). The claimed identities in (9) and (11) follow similarly. We begin with part (iii) of Theorem1.2 in the case n = 1, then apply Lemma 4.1, Lemma 4.4, and Lemma 4.3. (cid:3) Acknowledgements
This research was supported by the American Institute of Mathematics through their SQuaREs (Struc-tured Quartet Research Ensembles) program. The authors are deeply grateful to AIM, and in particularto Brian Conrey and Estelle Basor, for the generous support and consistent encouragement throughout thisproject. The first author is additionally grateful for the support of NSF CAREER grant DMS-1449679, andfor the hospitality provided by the Institute for Advanced Study, Princeton, under NSF grant DMS-128155.
Appendix
Here we tabulate all of our mock modular forms V mn , for any admissible pair ( m, n ), as originally definedas quotients of Lambert-type series and the eta-theta functions e n , and also in terms of Zwegers’ µ -function.These functions have normalized shadow E m ( τ ), meaning their shadows are equal to a constant multiple ofthe function E m (2 τ /c m ), where c m is defined in Section 5. We note that embedded in these tables are thedefinitions of the constants w m , t m , u ( mn ) τ , and v ( mn ) τ . able E1. Mock theta functions with normalized shadow E ( τ ), where u (1 n ) τ − v (1 n ) τ = − τ + . V n ( τ ) Series w q t µ (cid:16) u (1 n ) τ , v (1 n ) τ ; τ (cid:17) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +1) / q n +1 / − q − / µ ( τ + , τ ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z q ( n +1) / − q n +1 / − q − / µ ( τ , τ − ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +5 / / q n +1 / − q − / µ ( τ + , τ ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z q ( n +5 / / − q n +1 / − q − / µ ( τ , τ − ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +3 / / q n − q − / µ ( , τ ; τ ) V ( τ ) — − q − / µ (0 , τ − ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +2 / / q n − / − q − / µ ( − τ + , τ ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z q ( n +2 / / − q n − / − q − / µ ( − τ , τ − ; τ ) Table E2.
Mock theta functions with normalized shadow E ( τ ), where u (2 n ) τ − v (2 n ) τ = − τ . V n ( τ ) Series w q t µ (cid:16) u (2 n ) τ , v (2 n ) τ ; τ (cid:17) V ( τ ) − q − / e ( τ / X n ∈ Z ( − n q ( n +1) / − q n +1 / iq − / µ ( τ , τ ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z q ( n +1) / q n +1 / iq − / µ ( τ − , τ − ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z ( − n q ( n +5 / / − q n +1 / iq − / µ ( τ , τ ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z q ( n +5 / / q n +1 / iq − / µ ( τ − , τ − ; τ ) V ( τ ) — iq − / µ (0 , τ ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z q ( n +3 / / q n iq − / µ ( − , τ − ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z ( − n q ( n +2 / / − q n − / iq − / µ ( − τ , τ ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z q ( n +2 / / q n − / iq − / µ ( − τ − , τ − ; τ ) able E3. Mock theta functions with normalized shadow E ( τ ), where u (3 n ) τ − v (3 n ) τ = − τ + . V n ( τ ) Series w q t µ (cid:16) u (3 n ) τ , v (3 n ) τ ; τ (cid:17) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +1) / q n +1 / − q − / µ ( τ + , τ ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z q ( n +1) / − q n +1 / − q − / µ ( τ , τ − ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +5 / / q n +1 / − q − / µ ( τ + , τ ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z q ( n +5 / / − q n +1 / − q − / µ ( τ , τ − ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +3 / / q n +1 / − q − / µ ( τ + , τ ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z q ( n +3 / / − q n +1 / − q − / µ ( τ , τ − ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +2 / / q n − q − / µ ( , τ ; τ ) V ( τ ) — − q − / µ (0 , τ − ; τ ) able E4. Mock theta functions with normalized shadow E ( τ ), where u (4 ′ n ) τ − v (4 n ) τ = − τ and u (4 ′′ n ) τ − v (4 n ) τ = − τ . V n ( τ ) = Series w q t ′ µ (cid:16) u (4 ′ n ) τ , v (4 n ) τ ; τ (cid:17) V ′ n ( τ ) + V ′′ n ( τ ) + w q t ′′ µ (cid:16) u (4 ′′ n ) τ , v (4 n ) τ ; τ (cid:17) V ( τ ) = − q − / e ( τ / X n ∈ Z ( − n q ( n +1) / − q n +1 / iq − / µ ( τ , τ ; τ ) V ′ ( τ ) + V ′′ τ ) + − q − / e ( τ / X n ∈ Z ( − n q ( n +1) / − q n +5 / + iq − / µ ( τ , τ ; τ ) V ( τ ) = q − / e ( τ / X n ∈ Z q ( n +1) / q n +1 / iq − / µ ( τ − , τ − ; τ ) V ′ ( τ ) + V ′′ ( τ ) + q − / e ( τ / X n ∈ Z q ( n +1) / q n +5 / + iq − / µ ( τ − , τ − ; τ ) V ( τ ) = − q − / e ( τ / X n ∈ Z ( − n q ( n +5 / / − q n − / iq − / µ ( − τ , τ ; τ ) V ′ ( τ ) + V ′′ ( τ ) + − q − / e ( τ / X n ∈ Z ( − n q ( n +5 / / − q n +1 / + iq − / µ ( τ , τ ; τ ) V ( τ ) = − q − / e ( τ / X n ∈ Z q ( n +5 / / q n − / iq − / µ ( − τ − , τ − ; τ ) V ′ ( τ ) + V ′′ ( τ ) + − q − / e ( τ / X n ∈ Z q ( n +5 / / q n +1 / + iq − / µ ( τ − , τ − ; τ ) V ( τ ) = − q − / e ( τ / X n ∈ Z ( − n q ( n +3 / / − q n − / iq − / µ ( − τ , τ ; τ ) V ′ ( τ ) + V ′′ ( τ ) + − q − / e ( τ / X n ∈ Z ( − n q ( n +3 / / − q n +1 / + iq − / µ ( τ , τ ; τ ) V ( τ ) = q − / e ( τ / X n ∈ Z q ( n +3 / / q n − / iq − / µ ( − τ − , τ − ; τ ) V ′ ( τ ) + V ′′ ( τ ) + q − / e ( τ / X n ∈ Z q ( n +3 / / q n +1 / + iq − / µ ( τ − , τ − ; τ ) V ( τ ) = − q − / e ( τ / X n ∈ Z ( − n q ( n +2 / / − q n − / iq − / µ ( − τ , τ ; τ ) V ′ ( τ ) + V ′′ ( τ ) + − q − / e ( τ / X n ∈ Z ( − n q ( n +2 / / − q n +1 / + iq − / µ ( τ , τ ; τ ) V ( τ ) = q − / e ( τ / X n ∈ Z q ( n +2 / / q n − / iq − / µ ( − τ − , τ − ; τ ) V ′ ( τ ) + V ′′ ( τ ) + q − / e ( τ / X n ∈ Z q ( n +2 / / q n +1 / + iq − / µ ( τ − , τ − ; τ ) able E5. Mock theta functions with normalized shadow E ( τ ), where u (5 n ) τ − v (5 n ) τ = − τ + . V n ( τ ) Series w q t µ (cid:16) u (5 n ) τ , v (5 n ) τ ; τ (cid:17) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +1) / q n +1 / − q − / µ ( τ + , τ ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z q ( n +1) / − q n +1 / − q − / µ ( τ , τ − ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +5 / / q n − q − / µ ( , τ ; τ ) V ( τ ) — − q − / µ (0 , τ − ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +3 / / q n − / − q − / µ ( − τ + , τ ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z q ( n +3 / / − q n − / − q − / µ ( − τ , τ − ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z ( − n q ( n +2 / / q n − / − q − / µ ( − τ + , τ ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z q ( n +2 / / − q n − / − q − / µ ( − τ , τ − ; τ ) Table E6.
Mock theta functions with normalized shadow E ( τ ), where u (6 n ) τ − v (6 n ) τ = − τ . V n ( τ ) Series w q t µ (cid:16) u (6 n ) τ , v (6 n ) τ ; τ (cid:17) V ( τ ) − q − / e ( τ / X n ∈ Z ( − n q ( n +1) / − q n +1 / iq − / µ ( τ , τ ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z q ( n +1) / q n +1 / iq − / µ ( τ − , τ − ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z ( − n q ( n +5 / / − q n +1 / iq − / µ ( τ , τ ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z q ( n +5 / / q n +1 / iq − / µ ( τ − , τ − ; τ ) V ( τ ) − q − / e ( τ / X n ∈ Z ( − n q ( n +3 / / − q n +1 / iq − / µ ( τ , τ ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z q ( n +3 / / q n +1 / iq − / µ ( τ − , τ − ; τ ) V ( τ ) — iq − / µ (0 , τ ; τ ) V ( τ ) q − / e ( τ / X n ∈ Z q ( n +2 / / q n iq − / µ ( − , τ − ; τ ) eferences
1. George E. Andrews,
Ramanujan’s “lost” notebook. I. Partial θ -functions , Adv. in Math. (1981), no. 2, 137–172.2. George E. Andrews and Bruce C. Berndt, Ramanujan’s lost notebook. Part I , Springer, New York, 2005.3. Kathrin Bringmann, Amanda Folsom, and Robert C. Rhoades,
Unimodal sequences and “strange” functions: a family ofquantum modular forms , Pacific J. Math. (2015), no. 1, 1–25.4. Kathrin Bringmann and Larry Rolen,
Half-integral weight Eichler integrals and quantum modular forms , J. Number Theory (2016), 240–254.5. Jan Hendrik Bruinier and Jens Funke,
On two geometric theta lifts , Duke Math. J. (2004), no. 1, 45–90.6. Jennifer Bryson, Ken Ono, Sarah Pitman, and Robert C. Rhoades,
Unimodal sequences and quantum and mock modularforms , Proc. Natl. Acad. Sci. USA (2012), no. 40, 16063–16067.7. Miranda C. N. Cheng, John F. R. Duncan, and Jeffrey A. Harvey,
Umbral moonshine , Commun. Number Theory Phys. (2014), no. 2, 101–242.8. J. H. Conway and S. P. Norton, Monstrous moonshine , Bull. London Math. Soc. (1979), no. 3, 308–339.9. D. Dummit, H. Kisilevsky, and J. McKay, Multiplicative products of η -functions , Finite groups—coming of age (Montreal,Que., 1982), Contemp. Math., vol. 45, Amer. Math. Soc., Providence, RI, 1985, pp. 89–98.10. John F. R. Duncan, Michael J. Griffin, and Ken Ono, Moonshine , Res. Math. Sci. (2015), Art. 11, 57.11. Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa, Notes on the K surface and the Mathieu group M , Exp. Math. (2011), no. 1, 91–96.12. Amanda Folsom, Caleb Ki, Yen Nhi Truong Vu, and Bowen Yang, Strange combinatorial quantum modular forms , submit-ted.13. Amanda Folsom, Ken Ono, and Robert C Rhoades,
Mock theta functions and quantum modular forms , Forum of Mathe-matics, Pi, vol. 1, Cambridge Univ Press, 2013, p. e2.14. Basil Gordon and Richard J. McIntosh,
Some eighth order mock theta functions , J. London Math. Soc. (2) (2000), no. 2,321–335.15. Kazuhiro Hikami, Quantum invariant for torus link and modular forms , Comm. Math. Phys. (2004), no. 2, 403–426.16. Soon-Yi Kang,
Mock jacobi forms in basic hypergeometric series , Compositio Mathematica (2009), no. 03, 553–565.17. Ruth Lawrence and Don Zagier,
Modular forms and quantum invariants of -manifolds , Asian J. Math. (1999), no. 1,93–107, Sir Michael Atiyah: a great mathematician of the twentieth century.18. Yves Martin, Multiplicative η -quotients , Trans. Amer. Math. Soc. (1996), no. 12, 4825–4856.19. Geoffrey Mason, M and certain automorphic forms , Finite groups—coming of age (Montreal, Que., 1982), Contemp.Math., vol. 45, Amer. Math. Soc., Providence, RI, 1985, pp. 223–244.20. Robert J Lemke Oliver, Eta-quotients and theta functions , Advances in Mathematics (2013), 1–17.21. Hans Rademacher,
Topics in analytic number theory , Springer-Verlag, New York-Heidelberg, 1973, Edited by E. Grosswald,J. Lehner and M. Newman, Die Grundlehren der mathematischen Wissenschaften, Band 169.22. Don Zagier,
Vassiliev invariants and a strange identity related to the Dedekind eta-function , Topology (2001), no. 5,945–960.23. , Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann) , Ast´erisque (2009),no. 326, Exp. No. 986, vii–viii, 143–164 (2010), S´eminaire Bourbaki. Vol. 2007/2008.24. ,
Quantum modular forms , Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010,pp. 659–675.25. S. P. Zwegers,
Mock theta functions , Ph.D. thesis.26. ,
Mock θ -functions and real analytic modular forms , q -series with applications to combinatorics, number theory, andphysics (Urbana, IL, 2000), Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 269–277.-series with applications to combinatorics, number theory, andphysics (Urbana, IL, 2000), Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 269–277.