Asymptotic expansions, L -values and a new Quantum Modular Form
aa r X i v : . [ m a t h . N T ] N ov ASYMPTOTIC EXPANSIONS, L -VALUES AND A NEWQUANTUM MODULAR FORM EDGAR COSTA, KORNEEL DEBAENE, AND JO ˜AO GUERREIRO
Abstract.
In 2010 Zagier introduced the notion of a quantum modular form.One of his first examples was the ”strange” function F ( q ) of Kontsevich. Herewe produce a new example of a quantum modular form by making use ofsome of Ramanujan’s mock theta functions. Using these functions and theirtransformation behaviour, we also compute asymptotic expansions similar toexpansions of F ( q ). Introduction and statement of results
In 2010, Zagier [Zag10] sketched a definition of a quantum modular form. Unlikeclassical modular forms, which are defined on the upper half plane, a quantummodular form is a function only defined on a subset of P ( Q ). This set can be seenas the border of the upper half plane under the action of the modular group. Aquantum modular form should behave well under the action of some subgroup ofthe modular group, and should comply with some analytical constraints. Recallthe action of an element γ = (cid:18) a bc d (cid:19) ∈ SL ( Z ) on a function f , f | γ ( z ) = ( cz + d ) − k f (cid:18) az + bcz + d (cid:19) . Instead of demanding that f is invariant under this action (which would give avacuous definition), Zagier demands the difference f ( z ) − f | γ ( z ) = h γ ( z )to be a function which extends to an analytical function on P ( R ).The functions that appear in this paper have an integral period, either 1 or 2.Thus it is equivalent to view them as defined on H (or a vertical half-strip of breadth1 or 2), or the open punctured unit disc. We will discern these two domains byexclusively using the variable z on H and q on the unit disc, the relationship betweenthe two being either q = e πiz or q = e πiz . The domain of quantum modular forms,namely P ( Q ) viewed as the border of the upper half plane, corresponds to theroots of unity and the origin of the unit disc. Mathematics Subject Classification.
Key words and phrases. mock theta function, quantum modular form.The first author was partially supported by FCT doctoral grant SFRH/BD/69914/2010.The second author was partially supported by the Fonds Professor Frans Wuytack.The third author was partially supported by FCT doctoral grant SFRH/BD/68772/2010.
One of the first examples studied by Zagier is the function known as Kontsevich’sstrange function [Zag01] F ( q ) def = ∞ X n =0 ( q ; q ) n , where we employ the standard abbreviation( a ; b ) n def = (cid:0) − a (cid:1)(cid:0) − ab (cid:1)(cid:0) − ab (cid:1) · · · (cid:0) − ab n − (cid:1) , n ∈ N . The function is a priori only defined when q is a root of unity, but can be extended toa function in the open unit disc which satisfies near modular properties when we put q = e πiz with z ∈ H . Zagier also proves that F has nice asymptotical expansions as q tends to roots of unity radially. Furthermore, he establishes asymptotics for F ( ζ k )as k → ∞ , where ζ k = e πik is a primitive k th root of unity. Other asymptoticalexpansions of this kind have been studied, e.g. in [CR12] and [FOR12].Our initial object of study is the function G ( q ) = ∞ X n =0 ( − n (cid:0) q ; q (cid:1) n , which is only defined at odd roots of unity. However, this function can be extendedto the open unit disc by a straightforward manipulation, G ( q ) = 1 + ∞ X n =1 ( − n (cid:0) q ; q (cid:1) n − (cid:0) − q n − (cid:1) = 1 + ∞ X n =1 ( − n (cid:0) q ; q (cid:1) n − − ∞ X n =1 q n − ( − n (cid:0) q ; q (cid:1) n − = 1 − G ( q ) + ∞ X n =0 q n +1 ( − n (cid:0) q ; q (cid:1) n = 12 ∞ X n =0 q n +1 ( − n (cid:0) q ; q (cid:1) n ! = 12 φ ( q ) , (1)where φ ( q ) is one of Ramanujan’s mock theta functions. Ramanujan defined it inthe so-called Eulerian form as(2) φ ( q ) def = ∞ X n =0 q n ( − q ; q ) n . The proof of the equality of the two forms of φ ( q ), (1) and (2), is essential forthe proof of [FOR12, Theorem 1.3], where φ ( q ) appears intimately linked to ψ ( q ),another mock theta function, ψ ( q ) def = ∞ X n =1 q n ( q ; q ) n = X n ≥ q n +1 (cid:0) − q ; q (cid:1) n . In this paper, we shall investigate how the properties of G ( q ) are akin to thoseof F ( q ), and of quantum modular forms, in general. Since the subject of quantummodular forms is still in its infancy, it is important to establish examples of suchfunctions and determine their analytic properties.In this paper, we prove the following statement. SYMPTOTIC EXPANSIONS, L -VALUES AND A NEW QUANTUM MODULAR FORM 3 Theorem 1.
The pair (cid:16) q − φ ( q ) , q − ψ ( q ) (cid:17) is a vector-valued quantum modularform. More precisely, There are asymptotic expansions for φ (cid:0) e − t ζ l k +1 (cid:1) and ψ (cid:0) e − t ζ l k (cid:1) as t → + (where k, l ∈ N satisfying ( l, k + 1) = 1 and ( l, k ) = 1 , respectively). Inparticular, φ (cid:0) e − t (cid:1) ∼ ∞ X n =0 a n n ! t n as t → + , where a n = X a +2 b + c = n n ! a !(2 b )! c ! (cid:18) (cid:19) a (cid:18) (cid:19) b E a +2 b + X a +2 b = n n ! a !(2 b )! (cid:18) (cid:19) a (cid:18) (cid:19) b E a +2 b ,E n are the Euler numbers, and the summations are taken over a, b, c ∈ N . The functions φ ( q ) and ψ ( q ) satisfy the following modular transformationequations: q − φ ( q ) = r πα q − ψ ( q ) + r π α Z ∞ e − π u α cosh πu + cosh πu cosh πu du, α = − πil k + 1 ,q − ψ ( q ) = r π α q − φ ( q ) − r π α Z ∞ e − π u α cosh πu + cosh πu cosh πu du, α = − πil k , where q = e − α , q = e − π α , and k and l as above. There are asymptotic expansions for φ ( ζ k +1 ) and ψ ( ζ k ) as N ∋ k → ∞ , ζ − k +1) φ ( ζ k +1 ) ∼ p k + 1) i ζ − k +1)96 + ∞ X n =0 ζ − k +1) b n + ζ − k +1) c n (2 k + 1) n ,ζ − k ψ ( ζ k ) ∼ √ ki ζ k − ∞ X n =0 ζ − k b n + ζ − k c n (4 k ) n , where b n = π n X a +2 b = n ( − a + b (3 i ) a a !(2 b )! E a +2 b , c n = π n X a +2 b = n ( − a + b (3 i ) a b a !(2 b )! E a +2 b ,E n are the Euler numbers, and the summations are taken over a, b ∈ N .Remark. One remark on the modularity assertion is needed. As is usual in thetheory of mock theta functions (see [GM12]), we use the notation q = e − α , where ℜ α >
0. Writing α = − πiz , the transformation q q corresponds to the modulartransformation z − z . Part 2 of the theorem then says that the vector-valuedfunction f ( z ) = (cid:16) q − φ ( q ) , q − ψ ( q ) (cid:17) has the property that the vector f ( z ) − f (cid:18) − z (cid:19) q i z q iz , is given in terms of a certain improper integral, and extends to an analytic functionon P ( R ). Thus the pair (cid:16) q − φ ( q ) , q − ψ ( q ) (cid:17) is a vector-valued quantum modular EDGAR COSTA, KORNEEL DEBAENE, AND JO˜AO GUERREIRO form with respect to the groupΓ θ = (cid:26)(cid:18) a bc d (cid:19) ∈ Γ : b ≡ c (mod 2) (cid:27) , which is generated by z − z and z z + 2, the latter being the trivial transfor-mation. Acknowledgements
We would like to thank professors Ken Ono and Rob C. Rhoades for bringingthis problem to our attention. We would also like to thank the Arizona WinterSchool for creating opportunities for research and providing an excellent platformfor collaboration. 2.
Nuts and bolts
In this section we will gather the ingredients needed in the proof of our theorem.2.1.
Transformation formulae.
The near modular transformation behaviour of φ and ψ on the open unit disc — or equivalently the upper half plane via q = e πiz , z ∈ H — was found by Ramanujan and proved by Watson [Wat36]. The followingholds: q − φ ( q ) = r πα q − ψ ( q ) + r απ W ( α ) , (3a) q − ψ ( q ) = r π α q − φ ( q ) − r α π W ( α ) , (3b)where q = e − α , ℜ α >
0, and q = e − π α . W ( α ) is defined as the following integral(4) W ( α ) def = Z ∞ e − αx cosh αx + cosh αx cosh 3 αx dx. Many more of Ramanujan’s mock theta functions satisfy similar curious trans-formation formulae, many of which were proven by Watson. A review can be foundin [GM12].One technical point is that the above expression for W ( α ) does not need toconverge for ℜ α = 0. For ℜ α > αx = πu and moving theline of integration back to the real axis gives a new expression for W ( α )(5) W ( α ) = π α Z ∞ e − π u α cosh πu + cosh πu cosh πu du, which clearly converges for all purely imaginary α as well, and extends W ( t ) con-tinuously.2.2. Mordell Integral.
One important aspect of the above transformation formu-las for φ and ψ is the Mordell-type integral W ( α ) that appears as the obstructionto modularity. Zwegers [Zwe02] established in his thesis the modular properties ofthe Mordell integral. For z ∈ C and τ ∈ H let h ( z ; τ ) def = Z R e πiτx − πzx cosh πx dx. SYMPTOTIC EXPANSIONS, L -VALUES AND A NEW QUANTUM MODULAR FORM 5 Then Zwegers proves(6) h (cid:18) zτ ; − τ (cid:19) = √− iτ e − πiz τ h ( z ; τ ) . We may express W ( α ) using this notation:(7) W ( α ) = π α (cid:18) h (cid:18) −
112 ; πi α (cid:19) + h (cid:18) −
512 ; πi α (cid:19)(cid:19) . Remark.
We could also obtain a modularity statement for W ( α ) by composingequations (3a) and (3b). The result is that W ( π α ) = (cid:0) απ (cid:1) / W ( α ). However,using Zwegers function is a more general method since the function h underlies themodularity properties of a wide class of integrals.2.3. Euler Numbers.
The Euler numbers are the numbers which appear in theTaylor series expansion of the function x . They will naturally enter the proofbelow due to the following equality (see [EMOT81], for example):(8) E n = ( − n n Z R w n cosh πw dw. These numbers also show up in [BF13], where they compute an asymptoticexpansion of the Mordell integral in connection with the so-called
Kac-Wakimotocharacters. Proof of Theorem 1.1
Proof of Theorem 1.1.
An interesting property of our mock theta functions is thatthe value at an (appropriate) root of unity exists by virtue of the fact that expression(1) degenerates into a finite sum. This is the key argument for the existence of theasymptotic expansions for the radial limits, i.e. limits of the form ζ lk e − t with t → + .We start by applying it to the simplest case φ ( e − t ) with t → + . Here φ (1)being expressed as a finite sum translates to (cid:0) e − t ; e − t (cid:1) n = (cid:0) − e − t (cid:1) (cid:0) − e − t (cid:1) . . . (cid:16) − e − (2 n +1) t (cid:17) = O ( t n ) . Hence, each k th term in the asymptotic expansion can be expressed as a finite sumby expanding its first k th terms into their Taylor series.For q = e − t ζ l k +1 we have (cid:0) q ; q (cid:1) n = O (cid:16) t ⌊ n + k k +1 ⌋ (cid:17) . Therefore, each coefficient inthe asymptotic expansion of φ (cid:0) e − t ζ l k +1 (cid:1) as t → + can also be obtained as a finitesum. The same line of reasoning works for ψ (cid:0) e − t ζ l k (cid:1) . Furthermore, we can obtain an explicit asymptotic expansion for q = e − t . Ap-plying the transformation formula (3a), which is valid for all t >
0, we get e t φ (cid:0) e − t (cid:1) = r πt e π t ψ (cid:16) e − π t (cid:17) + r tπ W ( t ) . The first term of the right hand side is O (cid:16) e − O (1) t (cid:17) and the second term has beencomputed (up to a factor of 2) in [FOR12, Proof of Theorem 4.1], which settles thefirst part of the theorem.The second part of the theorem is a question of taking limits of (3a) and (3b) toappropriate roots of units. We recall that the integral W ( α ) has been continuously EDGAR COSTA, KORNEEL DEBAENE, AND JO˜AO GUERREIRO extended to α ∈ Q i in Section 2.1. Note first how the roots of unity transformunder the modular transformation q q : q = ζ l k +1 q = ζ − k − l ,q = ζ l k q = ζ − kl ( l is odd) . This implies that the components of the transformation formulas (3a) and (3b)have well defined terms for q an odd root of unity in (3a) and for q a (4 k )th rootof unity in (3b), respectively. Moreover, each of these terms is obtained by takinga limit; the Mordell integral has been continuously extended, and the other termspossess asymptotic expansions. Consequently the validity of the equations (3a) and(3b) is kept while taking the limit to the desired roots of unity.For the third part of the theorem, we apply the transformation formulas (3a)and (3b) and plug in ζ k +1 and ζ k , that is, we set α = − πi k +1 and α = − πi k ,respectively: ζ − k +1 φ ( ζ k +1 ) = p k + 1) i ζ k +124 ψ (cid:0) ζ − k − (cid:1) + r − i k + 1 W (cid:18) − πi k + 1 (cid:19) ,ζ − k ψ ( ζ k ) = r ki ζ k φ (1) − r − i k W (cid:18) − πi k (cid:19) . We use that ψ ( ± i ) = ± i and φ (1) = 2, and get ζ − k +1) φ ( ζ k +1 ) = p k + 1) i ζ − k +1)96 + r − i k + 1 W (cid:18) − πi k + 1 (cid:19) ,ζ − k ψ ( ζ k ) = √ ki ζ k − r − i k W (cid:18) − πi k (cid:19) . The last ingredient we need is the asymptotics of W (cid:0) − πim (cid:1) . We use (7) to expressthis in terms of h ( z ; τ ): W (cid:18) − πim (cid:19) = mi (cid:18) h (cid:18) −
112 ; − m (cid:19) + h (cid:18) −
512 ; − m (cid:19)(cid:19) = r mi (cid:18) e − πi m h (cid:18) − m ; 12 m (cid:19) + e − πi m h (cid:18) − m ; 12 m (cid:19)(cid:19) , where in the last equation we applied (6) with z = − m or z = − m and τ = m .For A ∈ {− , − } , h (cid:18) Am ; 12 m (cid:19) = Z R e πi m x − π Am x cosh( πx ) dx. Expanding the term e πi m x − π Am x in its Taylor series and applying identity (8), weget h (cid:18) Am ; 12 m (cid:19) ∼ ∞ X n =0 a ( A ) n m n , where a ( A ) n = π n X a +2 b = n ( − a + b (3 i ) a A b a !(2 b )! E a +2 b . SYMPTOTIC EXPANSIONS, L -VALUES AND A NEW QUANTUM MODULAR FORM 7 Plugging in A = − A = −
5, we get the asymptotics of W (cid:0) − πim (cid:1) , for m = 2 k + 1 or m = 4 k and k → ∞ , thus finishing the proof of the theorem. (cid:3) References [BF13] Kathrin Bringmann and Amanda Folsom. On the asymptotic behavior of Kac-Wakimoto characters.
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J.London Math. Soc. , S1-11(1):55, 1936.[Zag01] Don Zagier. Vassiliev invariants and a strange identity related to the Dedekind eta-function.
Topology , 40(5):945–960, 2001.[Zag10] Don Zagier. Quantum modular forms. In
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Ph.D. Thesis (Advisor: D. Zagier), UniversiteitUtrecht , 2002. (Edgar Costa)
Courant Institute of Mathematical Sciences, New York University,251 Mercer Street, New York, N.Y. 10012-1185, U.S.A
E-mail address: [email protected] (Korneel Debaene)
Vakgroep Wiskunde, Ghent University, Krijgslaan 281, S22, 9000Gent, Belgium
E-mail address: [email protected] (Jo˜ao Guerreiro)
Department of Mathematics, Columbia University, Rm 509, MC 4406,2900 Broadway, New York, NY 10027, U.S.A