Ramanujan J 2012 27:1 13 DOI 101007/11139-011-9299-9 q-expanion of vetor-valued modular form of negative weight Joe Gimenez Wiam Raji Reeived: 19 July 2010 / Aepted: 2 February 2011 / Publihed online: 15 May 2011 Springer Siene+Buine Media, LLC 2011 Abtrat In thi paper, we determine the q-expanion of vetor-valued modular form Knopp and Maon in Ill J Math 48:1345 1366, 2004; Ata Arith 1102: 117 124, 2003 of large negative weight on the full modular group where we allow pole in the upper half plane and at infinity Keyword Vetor-valued modular form q-expanion Cirle method Mathemati Subjet Claifiation 2000 11B57 11F03 11F30 1 Introdution and tatement of reult In [5], the author determine the Fourier oeffiient of a laial modular form of negative weight where the form onidered ha pole only at the infinite up In [6], the Fourier oeffiient of generalized modular form of large negative weight on ubgroup of finite index in the full modular group were determined but till with the ondition that pole were not allowed in H but only at infinity In [7], the author allow pole in the upper half plane and then determine the expanion of the form of negative weight at the up In thi paper, we determine the q-expanion of vetorvalued modular form of large negative weight where we allow pole in the upper half plane Notie that the proof would work if we onider a ubgroup of finite index J Gimenez Department of Mathemati, Temple Univerity, Philadelphia, PA, USA e-mail: gimenez@templeedu W Raji Department of Mathemati, Amerian Univerity of Beirut, Beirut, Lebanon e-mail: wr07@aubedulb
2 J Gimenez, W Raji in the full modular group but we have to retrit the allowane of the pole in the upper half plane and at the infinite up Let Fτ = F 1 τ,, F p τ t be a p-tuple of meromorphi funtion in the omplex upper half-plane H and ρ : Ɣ GLp, C a p-dimenional omplex repreentationf, ρ, orimplyf, i a vetor-valued form of real negative weight k on the modular group Ɣ = SL2, Z if 1 For all V = ab d Ɣ we have F1 τ,,f p τ t k Vτ= ρa,b,,d F 1 τ,,f p τ t, 1 where the lah operator r V i defined by: F k Vτ= F ε k Vτ= εa,b,,d 1 τ + d k FVτ 2 and ɛv = 1 for all V Note that throughout the paper, we might repreent ɛv and ρv by ɛa,b,,d and ρa,b,,d, repetively, when needed 2 Eah omponent funtion F j τ ha a onvergent x-expanion meromorphi at infinity: F j τ = x m j a ν jx ν 3 ν κ j with 0 m j < 1 a poitive rational number, κ j an integer, and x = e 2πiτ We hall aume that π arg ω<π for ω C Alo ine Fτ ha only a finite number of pole in the fundamental region, we an find a number A uh that the only ingularity with Iτ A i at τ = i To bound ρv, we introdue the Eihler length of V [1] with repet to the generator S = 0 1 1 0, T = 11 01 of Ɣ Namely, we write V a a produt V =±V1 V L, where eah V L i equal to either S or T n j, for ome integer n j, no two oneutive V j are both equal to S or a power of T, and where L i minimal Eihler proved that LV n 1 log μv + n 2, where μv = a 2 + b 2 + 2 + d 2, and n 1, n 2 are ontant independent of V Alo Knopp and Maon [3] howed that ρ jm V p LV 1 K LV 1, 1 m, j p, where K 1 i a ontant that atifie ρ jm S K 1 for all 1 m, j p Therefore, if we ue the uual norm ρv = 1 m,j p ρjm V 2,
q-expanion of vetor-valued modular form of negative weight 3 we ee that ρv K 2 μv α, α= n 1 log pk 1, 4 where K 2 i independent of V Note that ine the length of V 1 i the ame a the length of V, we an ue the ame bound for ρv 1 = ρv 1 Alo given a vetor-valued modular form F, ρ, we an define f j x = x m j F j τ = a ν jx ν, ν κ j 1 j p, 5 whih are analyti in the unit irle and have a pole at x = 0 of order at wort κ j, provided that κ j i negative We define a well f 1 y fy= f p y We ee that in the unit irle, f j x ha a pole of order at wort κ j at x = 0, pole at eah point orreponding to the pole of Fτ and no other ingularitie Later we have to determine a urve C N whih lie within the unit irle and doe not pa through any of the pole of f j x Notie that in [5], the author onidered C N to be the irle of equation x = e 2π/N2 Thu if y i a point within C N and not a pole of fx,wehave f j y = 1 f j x 2πi C N x y R j N, 6 where R j N i the um of the reidue of the funtion f j x/x y at the pole of f j x inide C N We now tate the main theorem Theorem 1 If Fτ i a vetor-valued modular form of negative weight k where k >2α for ome α to be determined α i given in 4, of multiplier ytem ɛ and having ingularitie in the fundamental region, at mot a finite number of pole and a pole at infinity, then we have where fy= 2π A,ν n = =1 0 a< a,=1 min κ j 0 j p ν=1 a ν 1 0 1 0 a ν p A,ν B,ν y n R, 7 n=0 ɛ a, 1 + aa,, a ρ a, 1 + aa,, a
4 J Gimenez, W Raji 0 0 e 2πin+m j a +ν m p a e 2πin+m j a +ν m 1 a B,ν i a olumn vetor whoe jth omponent i B,ν j ν k+1/2 mj = I k+1 n + m j 4π ν m j ν + m j where aa 1 mod Alo, I k+1 i the Beel funtion of the firt kind with purely imaginary argument and R = lim N RN whih i a olumn vetor whoe jth omponent i given by R j N whih i the um of the reidue of the funtion f j x/x y at the pole of f j x inide C N 2 The proof To determine the urve C N, we firt have to determine a path w n in the τ plane and then take it image in the x plane To do o, onider the Nth Farey equene and onider the tranformation 0 1, 1 N,,a 1 1, a,, N 1 N, 1 1, a a T = 1, 1 where we define a 0 to be 1 and 0 to be N Notie that under T, the firt quadrant of the τ plane i mapped into the area bounded by a emiirle and the real axi Now if Fτ ha pole on Rτ = 0 we follow exatly the tep of [7] a we detour around them with mall emiirle extending into the right half-plane and deform the emiirle through a / and a 1 / 1 a well Thi will give a ontinuou path from τ = 1/N to τ = 1 We alo detour around the point a / along the line Iτ = B from eah emiirle to it neighbor We need to how a in [7] that the endpoint of w N differ by 1 and hene the image of C N of w N i a loed urve For w N we hooe the part of thi path between the point 1/N + i and i + N 1/N + i whih are the image of the point τ = i under the firt and lat T Notie that 1 a a 1 1 2 1 2N and thu we have for any τ on w N, 0 < Iτ < 1 2N
q-expanion of vetor-valued modular form of negative weight 5 We an now write the following equation f j y = f j e 2πiτ e 2πiτ y 1 e 2πiτ dτ R j N w N = e 2πi1 m j τ F j τ e 2πiτ y 1 dτ Rj N 8 w N Let and e 2πi1 m 1τ 0 Eτ = 0 e 2πi1 m pτ R 1 N RN = R p N Thu we an write the following olumn vetor fy= EτFτ e 2πiτ y 1 dτ RN 9 w N To evaluate the integral above, we have to break the path w N into part Similarly a in [7], we let Ω to be the part of w N that join a 1 / 1 + ib to a / + ib for all exept the firt and the lat Thu we an write EτFτ e 2πiτ y 1 dτ = I = EτFτ e 2πiτ y 1 dτ 10 w N Ω We now make the hange of variable τ = a σ + a 1 σ + 1 11 Notie that with the hange of variable done in 11, the path of integration will hange exatly a deribed in [7, p 131] The new path of integration will be denoted by Ω Hene we get I = E Ω = ɛ ρ F σ + 1 a σ + a 1 a σ + a 1 Ω E σ + 1 a σ + a 1 e 2πi a σ +a 1 σ + 1 y 1 σ + 1 2 dσ σ + 1 Fσ e 2πi a σ +a 1 σ + 1 y 1 σ + 1 k 2 dσ, where ɛ and ρ are the abbreviation of ɛa,a 1,, 1 and ρa,a 1,, 1, repetively
6 J Gimenez, W Raji Now in order to bound I, we need to ue the Eihler etimate diued in 4, and therefore we have ρ K a 2 + a 1 2 + 2 + α 2 1 K1 2α K 2 N 2α, 12 where α = n 1 logpk, and K 1, K 2 are a ontant independent of V Note that 12 hold ine in our ae we have 0 a,a 1 < for all T aording to propertie of Farey equene Sine we will let N tend to infinity, we an uppoe that C N an be hoen to ontain the irle x = 1+ y 2 Thu we have x y 1 y 13 2 for all x on the path C N For the part of Ω in the trip 1/A Iσ A, wehaverσ 0 and we have the bound a in [7, p 132], e 2πi1 ατ < 1, 14 σ + 1 2 N 2 2A 2 15 From 14, we an eaily ee that E a σ + a 1 < p 16 σ + 1 A a reult, uing 12, 13, 14, 15, 16, and the fat that Fσ i bounded on a path of finite length that i independent of N and that ontain no pole of F, we ee that N k+2α 2 I = O 17 1 y Now for the remaining part of Ω, we follow the tep of [7, p 133] by alling the upper remaining part Ω and the lower part Ω and then tranforming the later by μ = σ 1 into Ω and then determining it path, we get a σ + a 1 I = ɛ ρ E Fσ e 2πi a σ +a 1 σ + 1 y 1 σ + 1 k 2 dσ Ω σ + 1 a 1 μ a ɛ ρ E F 1 e 2πi a 1 μ a 1 μ+ y 1 Ω 1 μ + μ k 2 μ + dμ N k+2α 2 1 μ 2 d + O 1 y N = H + H k+2α 2 + O 18 1 y
q-expanion of vetor-valued modular form of negative weight 7 Thu we get H = e πik/2 ɛ ɛ 0 ρ 0 ρ Ω 1 μ k 2 dμ, a 1 μ a E 1 μ + Fμ e 2πi a 1 μ a 1 μ+ y 1 where ɛ 0 = ɛ0, 1, 1, 0 and ρ 0 = ρ0, 1, 1, 0 Notie that on Ω we have that Iz A and thu we have Fourier expanion A a reult, we write where I = e πik/2 ɛ ɛ 0 ρ 0 ρ and H = I + K, a 1 μ a E Ω 1 μ + e 2πim 1μ κ 1 ν=1 a ν1e 2πiνμ e 2πim pμ κ p ν=1 a νpe 2πiνμ a 1 μ a E Ω 1 μ + e 2πim 1μ n=1 a n 1e 2πinμ e 2πim pμ n=1 a n pe 2πinμ K = e πik/2 ɛ ɛ 0 ρ 0 ρ If μ i on Ω then τ = a 1μ a 1 μ+ K = ρ 0 ρ ω e 2πi a 1 μ a 1 μ+ y 1 1 μ k 2 dμ 19 dμ e 2πi a 1 μ a 1 μ+ i on w N and ɛ =1, thu we get y 1 1 μ k 2 e 2πim 1μ n=1 a n 1e 2πinμ Eτx y 1 1 μ k e 2πim pμ dτ, n=1 a n pe 2πinμ where ω i the part of w N that τ run over a μ run over Ω Nowfrom[7, p 134], we ee that 1 μ 2 N 2 2 20 Notie that e 2πim 1μ n=1 a n 1e 2πinμ e 2πim pμ n=1 a n pe 2πinμ
8 J Gimenez, W Raji = e 2πm 1A an 1 2 e 2πnA + + n=1 e 2πm pa an p 2 1/2 e 2πnA Alo, we have Iμ A, thu uing 13 forx, and the fat that 0 < Iτ < 1/2N Thu we have Similarly, we find 1 K = O 1 y N k+2α ω n=1 N k+2α dτ = O 1 y H = I + K, ω dτ 21 where I = ɛ ρ Ω a σ + a 1 e 2πi a σ +a 1 E σ + 1 y 1 σ + 1 k 2 σ + 1 e 2πim 1σ κ 1 ν=1 a ν1e 2πiνσ e 2πim pσ κ p ν=1 a νpe 2πiνσ dσ 22 and Now ω and 23 that N K k+2α = O dτ 23 1 y ω and ω are both part of Ω and they do not overlap Thu we ee from 21 N K k+2α + K = O dτ 24 1 y Ω We now onider I and I of 22 and 19 We make the uitable hange of variable in both integral a in [7, Set 4] to derive an equation for the um of the two integral that i imilar to the one in [7] and we get I + I = ɛ ρ 1 max κ j 1 j p ν=1 n=0 a ν 1e 2πi n+m1a / +ν m1 1/ 0 yn 2πi n+mpa / +ν mp 1/ 0 a ν pe z k 2 e 2π/ ν m 1 z+n+m 1 1/z e 2π/ ν m p z+n+m p 1/z dz 25
q-expanion of vetor-valued modular form of negative weight 9 Notiein25 that the integrand ha no ingularitie exept at z = 0 and z =, o we an deform the path of integration We form the ame path of integration a in [7, p 137] A areful heking of the etimate of the five integral along the path how no hange for our etimate exept for taking the etimate of the repreentation into onideration Thu if n + m j >0for1 j p, wehave I + I = ν m 1 n+m 1 k+1/2 I k+1 4π ν m1 ν + m 1 + ON k+2α 1 e 4AπN 2 n+m 1 ν m p n+m p k+1/2 I k+1 4π ν mp ν + m p + ON k+2α 1 e 4AπN 2 n+m p where I k i the Beel funtion of the firt kind with purely imaginary argument Alo for min 1 j p {n+m j }=0, we an ue the reult from [2] whih etimate the ame integral for thi ae Now uing 9, 17, 10, 18, 24, 25, and 26, we get max κ j fy= 2π a 1 j p ν 1 0 1 ν=1 0 a ν p ɛ ρ e 2πi n+m 1a / +ν m 1 1 / 0 0 e 2πi n+m pa / +ν m p 1 / n=0 ν m 1 n+m 1 k+1/2 I k+1 4π ν m1 ν + m 1 + ON k+2α 1 e 4AπN 2 n+m 1 ν mp n+m p k+1/2 I k+1 4π ν mp ν + m p + ON k+2α 1 e 4AπN 2 n+m p 26 yn N k+2α N k 2+2α + O dτ + O RN 27 1 y Ω 1 y Theerrortermin27 redue to { 1 O = O N k+2α 1 e 8AπN 2 n+m 1 + +e 8AπN 2 n+m p 1/2 y n n=0 } + N k+2α dτ + 1 y Ω N k+2α 2 1 y N k+2α e 8AπN 2 m 1 + +e 8AπN 2 m p 1/2
10 J Gimenez, W Raji 1 e 4AπN 2 y 1 N 1 1 N k+2α + O dτ + 1 y w N N 2 28 Notie that w N ha an upper bound independent of N and notie alo that N 2 N 1 1 N 1 N N 1 1 = N 2 = 1 k=1 a=0 =1 and thu, by applying the inequality to 28, then 27 beome max κ j fy= 2π 1 j p a ν 1 0 1 ɛ ρ ν=1 0 a ν p n=0 e 2πi n+m 1a / +ν m 1 1 / 0 0 e 2πi n+m pa / +ν m p 1 / ν m 1 n+m 1 k+1/2 I k+1 4π ν m1 ν + m 1 ν m p n+m p k+1/2 I k+1 4π yn ν mp ν + m p RN + O N k+2α e 8AπN 2 m 1 + +e 8AπN 2 m p 1/2 1 e 4AπN 2 y 1 29 Now if we let N go to infinity, we get the deired reult, but firt we have to make 29 independent of the Farey erie of order N Leta = a and = Wehavetofind an equivalent expreion to 29uinga and but not a 1 and 1 Let u define a to be the olution of the ongruene and thu we get aa 1 mod 30 a + 1 0 mod 31 Thu we have aτ + a 1 F = ɛ ρ τ + 1 k Fτ 32 τ + 1
q-expanion of vetor-valued modular form of negative weight 11 Uing the tranformation law of vetor-valued modular form along with the fat that a 1 / 1 and a/ are oneutive Farey fration, we get aτ + a 1 aτ + 1 + a / 1 + aa / F = F τ + 1 τ + 1 + a / a = ɛ a, 1 + aa,, a ρ a, 1 + aa,, a τ + 1 k e 2πim 1 +a 1 0 Fτ 33 0 e 2πim p 1 +a Comparing 32 and 33, we get ɛ ρ = ɛ a, 1 + aa,, a ρ a, 1 + aa,, a e 2πim 1 +a 1 0 0 e 2πim p 1 +a Hene, ine ν i an integer and a + 1, e 2πiν m 1 1 0 ɛ ρ 0 e 2πiν m p 1 a, 1 + aa = ɛ a, 1 + aa Subtituting into 29, we get fy= 2π =1 e 2πiν m 1 a,, a ρ,, a 0 0 e 2πiν m p a min κ j 0 j p a N ν 1 0 1 A,ν B,ν y n ν=1 0 a ν p n=0 RN + O N k+2α e 8AπN 2 m 1 + +e 8AπN 2 m p 1/2 1 e 4AπN 2 y 1, 34
12 J Gimenez, W Raji where A,ν n = 0 a< a,=1 ɛ a, 1 + aa,, a ρ a, 1 + aa,, a 0, 0 e 2πin+m j a +ν m p a e 2πin+m j a +ν m 1 a with and B,ν = ν m 1 n+m 1 k+1/2 I k+1 4π ν m1 ν + m 1 ν m p n+m p k+1/2 I k+1 4π ν mp ν + m p aa 1 mod Notie now that if N,theumin34 beome infinite and it i eaily proved that it i onvergent Alo RN will be replaed by R Notie alo that for k >2α we an ee that the error term will go to zero a N goe to infinity, and the reult follow Following exatly the ame tep a in [7], we obtain a erie repreentation for R in 7 The author in [7] aumethatfτ ha a ingle pole in the interior of the fundamental region and no other ingularitie in the region exept a poible polar ingularity at i Moreover, the value of R in other ae an be obtained in imilar method a in [7, Set 7] Suppoe now that Fτ ha a imple pole at τ = σ of reidue R, then we ee that the R i where R i i the ith omponent of the olumn vetor R i given by κ i R i = a n y n + 2πiR i e 2πi1 m iσ e 2πiσ y 1 n=1 2πiR i p=1 q= ρ q,p,p, pp + 1 q ɛ q,p,p, pp + 1 1 q 1 pσ + q k 2 e 2πi1 m iσ e 2πiσ y 1, 35 where the um are taken uh that p, q = 1, p i a olution to the ongruene pp 1 mod q,
q-expanion of vetor-valued modular form of negative weight 13 and σ = pp + 1/qσ + p pσ + q If Fτ ha everal imple pole, the value of R will be the um of expreion imilar to 35 Aknowledgement We would like to thank the referee for the patiene in arefully reheking the alulation and alo for the preie omment Referene 1 Eihler, M: Grenzkreigruppen und kettenbruhartige Algorithmen Ata Arith 11, 169 180 1965 2 Gimenez, J: Fourier oeffiient of vetor-valued modular form of negative weight and Eihler Cohomology PhD Thei, Temple Univerity, April 2007 3 Knopp, M, Maon, G: On vetor-valued modular form and their Fourier oeffiient Ata Arith 1102, 117 124 2003 4 Knopp, M, Maon, G: Vetor-valued modular form and Poinare erie Ill J Math 48, 1345 1366 2004 5 Rademaher, H, Zukerman, H: On the Fourier oeffiient of ertain modular form of poitive dimenion Ann Math 39, 433 462 1938 6 Raji, W: Fourier oeffiient of generalized modular form of negative weight Int J Number Theory 51, 153 160 2009 7 Zukerman, H: On the expanion of ertain modular form of poitive dimenion Am J Math 621/4, 127 152 1940