Hecke operators and Euler products

Transcription

1 Hecke operators and Euler products van Lint, J.H. Published: 28/10/1957 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA): van Lint, J. H. (1957). Hecke operators and Euler products Utrecht: Utrecht University General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal. Take down policy If you believe that this document breaches copyright please contact us (openaccess@tue.nl) providing details. We will immediately remove access to the work pending the investigation of your claim. Download date: 27. Jan. 2019

2 ,) НЕСКЕ OPERATORS AND EULER PRODUCTS PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE WIS- EN NATUURKUNDE AAN DE R!JKSUNIVERSITEIT ТЕ UTRECHT, ОР GEZAG VAN DE RECTOR MAGNIFICUS Dr. L. SEEKLES. HOOGLERAAR IN DE FACUL TEIT DER DIERGENEES KUNDE. VOLGENS BESLUIT VAN DE SENAAT DER UNIVERSITEIT, TEGEN DE BEDENКINGEN VAN DE FACULTEIT DER WIS- EN NATUURKUNDE ТЕ VERDEDIGEN ОР MAANDAG 28 OKTOBER 1957 ТЕ 4 UUR PRECIES DOOR JACOBUS HENDRICUS V AN LINT GEBOREN ТЕ BANDOENG ~о-~ ~i г~~:!1\..,: 01...~ I -'-' f ~. r:...п '.. ~>! 7 h.r~ р ~о 1 " 1 ' ~-- '- ' -~~-~--_:_-~-- -c----,.-j 1 Т.Н. Eii'HJ!:ovr::N \ \ DRUKKERIJ "LUCTOR ЕТ EMERGO" - LEIDEN

3 PROMOTOR: PROF. DR. F. VAN DER BLIJ

4 О. Introduction. The Fourier coefficients of modular f9rms have been studied Ьу many mathematicians. Although many interesting results have Ьееn found, still enough proьlems remain to keep us interested in this subject. One of the powerful methods used in studying the coefficients is the theory of Hecke~s operators Тп (Hecke [1]). Hecke considered modular forms of integral dimension for cong~uence groups Г (N). The operators Тп are linear comhinations of transformed functions f (ат. d + Ь), where (а Ь) О d runs through а complete system of in regard 1 to Г (N) nonequivalent transformation matrices of order п. For instance, for modular functions of dimension - l' and step 1: f 1 Тп = п'- 1 2: t(ат d Ь) d-'. ad=n bmodd,d>o The operators шар the set of integral modular forms of dimension - l' and step 1 into itse]f. These operators were studied thoroughly (Hecke [lj, Petersson [2]). It was proved that under certain conditions modular forms could Ье written as linear com Ьinations of eigenfunctions of these operators. со If f(т) = 2: а (n) е lг is eigenfunction of all Тп, the associated п=о 00 Dirichlet series ер (s) = 2: а (п) п-s can Ье written as an Euler n=l product П (1 - a(p)p-s + s(p) pk-i-2s)-1. р То find analogous results for modu1ar forms of non-integral dimension, operators of the type of Тп would have to Ье defined. Hecke [2] considered the fшictions J1 8 (т) 5-Ь(т) where а+ Ь is an odd integer, and proved that operators ТР could generally not Ье defined.

5 6 А general theory of "Hecke-operators" was given Ьу \Vohlfahrt [1]. The operators he defined, and some theorems concerning these operators, will Ье given in l. Now that we have these operators, we are of course interested in the results that can Ье found Ьу applying them. W ohlfahrt considered applications to the functions {} (т, 9, stl, Pk), defined Ьу Schoeneberg [1]. This gave formulae concerning the number of representations of an integer Ьу quadratic forms in an odd number of variaьles. For many years the coefficients of the powers of ;i (т) have been studied, and special attention has been given to the findidg of coefficients which are _ zero. The results concerning these coefficients, that have been found Ьу Newman [1, 2, 3, 4] and Rademacher [1] suggest that they can Ье proved and generalized Ьу applying Hecke-operators to the functions ;i 1 (т). This is done in 4 of this thesis after the necessary operators have been defined in 2 and 3. Wohlfahrt proved that all Hecke-operators could Ье built up out of the operators he defined excep_t if the transformed functions occuring in these operators were linearly dependent; in which case more operators could Ье defined. Sometimes the operators can on]y Ье defined if there is linear dependence, as is the case for the functions ;ia (т).э-ъ(т), studied Ьу Hecke. In 5 we shab prove some theorems about this linear dependence, and find all integral modu!ar forms for the groups Г (1_) and Г {} for which this linear dependence occurs, under the condition that 2r is an integer, if r represents the dimension of the modular forms. 1. Definitions and notation. 1.1 Г (1) is the modular group, i.e. the group of unimodular two-rowed matrices (; ~) with rational integral elements. _ Г (N) is the subgroup of matrices for which а= d - 1 (mod N) and Ь =с= О (mod N). Г 0 (N) is the subgroup of matrices for which с = О (mod N). r 0 (N) is the subgroup of matrices for which Ь =О (mod N). Га. is the subgroup generated Ьу (~ ~) and (~ -~).

6 7 We shall write (~ ~) =!, (~ ~) = U and (~ -~) = Т. Let Г Ье а subgroup of finite index of Г (1), r а real number. We shall call а function f(т), defined for lm т >О, а modular form of dimension -r and with multiplier system v for the group Г if it satisfies the following conditions (cf. Petersson [1 ]) :. 1. ln every closed region in the halfplane Im т >О, f(т) has as only singularities, at most а finite number of poles. 2. For L = (; ~) е Г and lm т > О, the following relation ho1ds: ат+ ь) f(lт) = f ( ст+ d = v(l) (ст+ d)' f(т), (1.01) where v(l) depends only on L and J v(l) J = 1. (For v(l) we shall also write v (; ~)). Here (ст+ d)' is defined as follows: For (с, d) =/=(О, О) and lm т >О, (ст+ d)' = e' 10 g(c +d), where that branch of log(cт + d) is taken, for which log(cт + d) is real if ст + d > О, с f=. О. And for с = О we define:.. sgnd-1 (ст + d)' = d' ~ 1d1' е -mr If М = (:~ ::) and S. (; ~) are rational unimodular matrices and MS = М' =(то: тз:). the number r;(m, 5) = r;(г)(m,s) т1 т2 is defined in the following way: '(т 'т + т ')' (m1 Sт + т 2 )' = r;(m, S). (ст+ d);. Then r; (М. S) = е 2 п;"', where 11 is an integer which does not depend on т. For the multipliers the following relation holds: (1.02) 3. ln the closvre of а fundamental region of Г, f(т) has as only possiьle singularities, poles in the various uniformizing variaьles. These poles do not cluster about r~al parabolic limit points of the fundamental region. So we have, if UN е Г and we write v ( UN) ~ e'lnix :

7 8 1: 2niт:n 2niN:-t со N f(т) = е 2: а(п) е ; (1.03) n=-k!г, -r, v! will denote the class of modular forn:is for the group Г, with dimension -r and multiplier system v. We call the class trivial if f(т) = О is the only modular form in the class. If f(т) is а modular form of dimension - r apd S = (с а db) а rational matrix with 1S1 > О, We then have: we shall write: f(т) 1 S = JSlr (ст+ d)-r f(sт). (1.04) (1.05) If f(т) е!г, -r, vj. and Q is а transformation matrix, we have: fj Q е jг J Q. -r, v J Qj. (1.06) where ГI Q= Г(I) 11 Q- 1 гq and v/ Q is the multiplier system determined Ьу : v(l) J Q = rr(~~g~i)q) v(qlq- 1 ) for L е Г J Q. (1.07) For further theorems on modu!ar forms of non~integral dimension, see Petersson [1]. We shall call а matrix Q primitive if the greatest common divisor of its elements is 1. If 1 Q 1 = п; we say Q is of order п. Some modular forms we shal1 consider are the following: ni'rn.2 ni-i 12 = = (12) - а. 11(т) = е П (1 - e 2 "' 1 n ) = 2: - е. 12 (Im т>о). (1.08) n=l n=i п ii(т) is а modular form for Г(l), of dimension -t. There are several expressions for the multiplier system (cf. Rademacher [2]). In the interior of the upper т~halfplane ii'(т) is free from poles and zeros, and ii (т) vanishes at т =О and т = ico. ь. 2(~) = '1 2 З-(т) = L e"'i n2 =. n=l ii(т+ 1) З-(т) is а modular form of dimension --t for the group ГО. ( 1.09)

8 9 In' the fundamental region of Г {} {} (т) has no poles. The Ье~ haviour in т = -1 can Ье found from (1.09) to Ье as follows: ( + 2;n;i ( 7: ) ( 7: ) т..;т=.е,:::...,спе п= о 1)1. С.( ) В ;+'! ~ 2ni ~ п (1.1 О) с. The Eisenstein series: 1 Gk(т) = 2,(k) m~:2 (т1т + m2)-k = ( 27Гi)k =. - 1 _L "\" rr. (,,.,\,,.2;n;i<n (1 11) - '1 ((k).(k--,-l)jn~l-к-,-1\ 1- ',."., where 11', (п) = L: d', and k is an even integer?: 4, are d/n, d> О modular forms е jг(l), -k, Ч Every modular form е jг(l),-k, 1J can Ье writteri uniquely as L: спдn(т)gk- 12 k,;(т), (O::s;п::s;U, п k - 12п-=/::. 2), where G 0 = 1, and д(т) = (т) (cf. Hecke [1]). 1.2 W е shall now give some definitions and theorems from Wohlfahrt's general theory of Hecke~operators (Wohlfahrt [1]). lf Г2 is а subgroup of finite index of Г1, we write jг 1 : Г2! for а complete system of representatives of а decomposition оfг 1 into left cosets of Г2 Theorem 1. If К= jг; -r, v! and Л = j&, -r, v*j асе two classes о{ modular forms, Q is. а trcj.nsformation matrix witb v \ Q = v* оп В п Г \ Q, and & п Г 1 Q is о{ finite index in Г(f), the operator T~(Q) defined Ьу fl T~(Q) = L: v*(v)- 1 fl QI V (1.12) v е ie : е Г\ г 1 Q! maps К into Л. Remark: If Q* = п Q- 1 where п = 1 Q 1. and Т~ ( Q) is de~ fined, then ~ (Q*) сап also Ье defined. We refer to this as ( 1.13)

9 10 Theorem 2. We define а Hecke-operator to Ье ап operator which trans{orms tlie modular forms { о{ а class jг, -r, vi into forms о{ j&, -r, v*i the operator being а linear comьination {with coetfi.cients independent о{ {) о{ transformed {uпctions f J Q with primitive Q о{ а {ixed order. Тhеп all Hecke-operators defined {or the whole class jг, -r, vi сап Ье built ар out о{ the operators defined in theorem 1 with the following exception : Let Q rип through а complete system о{ primitive transformatioп matrices of fixed order, which are left-inequivaleпt with respect to Г. lf for all f of jг, ~r, v! the transformed functions fl Q are liпearlg dependeпt, other Hecke-operators thaп those meпtioned above сап Ье defiпed. Remark: It is also of interest to consider those functions of jг, -r, v! for which linear relations for the fl Q exist. For these functions other Hecke-operators can Ье defined. Of course these operators are not necessarily Hecke-operators for the other functions of the class. Functions for which this holds are the functions 11 (т), 11 3 (т),.э-(т) and 11 4 (т),э--j (т), studied Ьij Hecke [2]. We shall give more examples in 5. Sometimes the operators defined in theorem l can Ье written in а form which is easier to handle. Consider К= jг, -r, v!, л = j&. -r, v*i Let. г (N) ~ г о & and let т~ ( Q) Ье defined and (п, N) = 1, where п = [ Q[. Then the system: LR = 92~ = jr: R = (~ b1;).ad = п, а> О, Ь mod d, (а, Ь, d) = Ч (1.14) is а complete system of in regard to Г(l) left-inequivalent, primitive transformation matrices of order n. Theorem 3. lf 92 ~ Г Q& апd have fl T~(Q) = L л(r)- 1 fl R. Re LR [& () Г(l) Q: & Г\ Г [ Q] = 1 we ( 1.15) Here л(r) = 15(М, Q) 15(MQ, V) v(m) v*(v) i{ R = MQV with М е Г, V е &. We call this the normal form о{ Т~ (Q).

10 11 /:с. ln defining the operators т~ ( Q) the main difficulty is proviпg that v* = v 1 Q. In fact if К is given and а is а given group it is possiьle that for а no multiplier system v* exists so that v* = v 1 Q and je, -r, v*! is non~trivial. For certain groups we shall study the multiplier systems in 2, апd then define some special operators of the type т~ ( Q) in Some properties of multiplier systems. We shail consider multiplier systems for the groups Г ( i), го (2) and Г.9" estaьlishing relations between v(;p ~) and v (: ь~). First we remark that for all groups Г with -,/ е Г, we have: l' (-/) == e-:л;ir (2.01) 2.1 The case Г = Г{l). Г(l) is generated Ьу U and Т for which the following relations hold: Т 2 = -I and (UT) 3 = -I. From these we find for Л = v(u) and µ, = v(t) the relations: л = ~6 е 6 and µ, = л -з = ± е 2.(2.02) Here ~6 is а б~th root of unity. We see that for Г(l) only 6 different multiplier systems are possiьle (for every r). In the following we shall use the fact that for every М е Г(l) there is an integer w (М), independent of r, so that: v(m) = лw(м). (2.03) Proof: Expressing М in the generators U and Т, we find for v(m) а product of factors rт(м;, Mj) and powers of Л and µ,. All rт(м;, Mj) are powers of е 2 "';', therefore powers of л. Using,и.,-:--- л-з we find the result (2.03). If f(т) е!г{l), -r, v!, application of (1.06) gives: {(рт) r-р(т) е jг о (р), r (р - 1), v*!, (2.04)

11 12 For f(т) = '11(т) and р > 3 а prime, v* (:р ~) = в(d) = ( :), where (:) is the quadratic restsymbol (Hecke [2]). This gives us the relation : eтj!w(;ь~)-p.w{;p~}! - (:) (2.05) Using f(т) = '11 3 (т) we find: 1,.1 \ \;). (2.06) We now prove: Theorem 4. If!Г(l), -r, v! is non~trivial, r is ап integer and р > 3 is а prime, then v (; Ь~) = vp (;р ~) апd vp (; Ь~) v (;Р ~). Proof: (2.05) implies w (; Ь~) - р. w (;Р ~) = О (mod 12) and as in this case Л 12 = ezяir - 1, the first equation is proved. Т aking the p~th powers Of both sides We find VP (; Ь~) = 2 vp (;р ;). From this the second equation follows because р 2 = 1 (mod 24) and all multipliers are 12~th roots of unity. If 3/r and л = ± е 6, these results are also true for р = 3, which can Ье proved using (2.06). Corollary: if f(т) е!г(l), -r, v!, r an integer, then ( f(т) е!г (12), -r. Ч а ь) (а ь) (Ь -а) Proof: If с d е Г(12) we have µ,.v с d :._ v d -с. Now (d, 6) = 1 so we can apply theorem 1.

12 13 We find, using Л 12 = µ, 4 = 1: ~) =! vva-d(~ -а:) = µ,d = µ, ( Ь ad) = µ, -Зd = µ, -1 -с From this follows v(; ~) = 1 if (; ~) е Г{12). (d >О) (d <О). Theorem 5~ lf, in jг(l), -r, v!, 2r = l is ап odd integer, and р > 3 is а prime, then v (; Ь~) = (:) vp (;р ~) and v (:р ~) = = ( : ) ор (; Ь~) Лi(/-1) Лi :Щ у _1_2_ Proof: Jn this case л = '>б. е е = ~12 е where ~6 = у12 = '>12= 1. The first equation now follows from (2.05) and the second сап Ье proved iп the same way as in theorem 4, if we remark that now all multipliers are 24~th roots of uпity. As in theorem 4, the result is also true for р = 3 if 3/l and л = ± е 6. Applying theorem 4 or theorem 5 twice, we find: Theorem 6. lf, in jг(l), -r, v!, 2r is ап. (а Ьр 2 ) (а а przme, then о с d = v ср 2 ~). integer, and р > 3 is Ье Again, this is also true for р = 3 if 3/2r and л = +е 6.. (ар2 Ь). (а Ь ) From these equations, others, e.g. v. с d = v с dp 2 сап proved easily {2r an integer). With these theorems the multi~ pliers can easily Ье determined iп those cases where (с, 6) = 1 or (d, 6) = 1. We shall поw prove analogous results for the groups Г0 (2) and Г1J.. First we remark that, Ьу (1.06), if f(т) е jг 0 {2), -r, о!, then: r(т~l) е {г{}. -r,vl(ь ;)}. (2.07)

13 The case Г = Г0 (2). Г 0 (2) is generated Ьу U and Т* = = (~ ~) for which the relation (U- 1 Т*) 2 = -1 holds. Writing v(u) = ;..* and v(t*) = µ,*, we have: -:тtir * + * -2- µ, =_Ле. (2.08) Using (2.08) we find that for every М е Г 0 (2) there are numbers w 1 (М) and w 2 (М). independent of r, so that: ("){\О\ \... V./ f Considering the multiplier systems of 11 (т) and 11 (2т), we now find two relations : е ~![ WJ (: Ь~) - 2w2 (~ Ъ~) ] - р [ wl ер ~) - 2w2 ер ~) ]! = (~ ). (2.1 О) е ~![ 2wl (: Ь~) - w2 (: Ь~) ] - Р [ 2wl ер ~) - "'2 (:Р ~) ]! = ( : ) (2.11) Applying these formulae in the same way as was done for we find: Г(l), Theorem 7. If we consider only multiplier systems in which ;.,* is а 24-th root of unity ( and therefoтe all multipliers are 24-th roots of unity), the theorems 4, 5 and 6 hold if Г ( 1) is replaced Ьу Г0 (2). 2.3 The case г = г ff г ff is generated Ьу U 2 and Т. W е write v(u2) = л* and v(t) = µ, = + е~. The calculations in this case are rather tedious. We use the functions 11 - W е shall only state the result: ( т + 1) - and &(т). 2 Theorem 8. If we consider only multiplier systems in which J.. * is а 24-th root of unity, the theorems 4, 5 and 6 hold if Г(l) is replaced Ьу Г {}

14 15 3. Some special operators T~(Q). In this paragraph we shall consider operators T~(Q) with Q = (6 ~). (~ ~) or (6 2 ~) If possiьle, we shall write these operators in the normal form (1.15). In the following we have К= jг 1, -r, v!, Л = jг 2, -r, v*! l = 2r. Furthermore р will always represent а prime number larger than Г 1 = Г2 = Г(l), r ап integer, Q = (6 ~) Ву (2.03) the multiplier systems are determined Ьу 2ni~ nir- 6 = v*(u). If we have л-= е е 6, we choose k 2 = k2 nir r(p- 1) 2ni6 (mod 6). Then л * = е е 6 satisfies л * = ;..,Р, and л* = pk л = v (И) and so Ьу (2.03) and theorem 4 we have : v* (;Р ~) = vp (;Р ~) = v (; Ь~) Therefore v* = v 1 Q on Г0 (р) = Г2 r\ Г1 1 Q. Both conditions in theorem 1 are satisfied. Hence we can de~ fine Т~ ( Q}. (Remark: (~ ( Q) can also Ье defined for these classes). In vlew of Iater applications we choose N = 24 to write Т~ (Q) in the normal form (see ( 1.15) ). The conditions of theorem 3 are satisfied. C'R_ consists of the р + 1 matrices We find: ( Р О) d (1 24Ь) О 1 an О р, (Ь = О,... р - 1). (3.01) j. 1 л (R) = r{p- 1) (- 1) 2 for R = (~ ~) for R = ( 0 1 2Р4Ь). (3.02) We shal1 write Т(р) for the operator we have defined. We found: When f(т) е! Г(I), -r, vj. r an integer, then f(т) 1 Т(р) е j Г(l), -r, v*j,

15 where 16 r(p-1) р-1 f(r)it(p) = p'f(pr)+(-1) 2 2: f(r+ 24 Ъ). (3.03) Ь=О Р If 3/r and л = ± е 6 we can define Т(З): f(r) \' Т(З) = 3' f(зr) + (-1)' ~ t(r + 3 ВЬ) (3.04) Ь=О If 1 = 2r is an odd integer, we cannot define an operator Т(р) as was done above for Г1 = Г2 = Г(l). Proof: А necessary condition is v * = v 1 Q, therefore л * = ЛР. Thenv*=vPforaH аь) ( с d е Г(l). - Wethenhavefor (а ер d Ъ) Е Г 0 {р): v* (:р ~) = v (: Ъ~) = ( ~) vp (;р ~), and this is equal to vp (;р ~) only if (:) = 1, therefore not for allшatrices in Г0 {р). We see that operators Т(р) can only Ье defined for these classes in the exceptional case mentioned in theorem 2 {see 5). 3.2 Г1 = Г2 = Г{l), r ал integer, Q = (~ 2 ~). W е now take v* = v. Then Ьу theorem 6: v* = v 1 Q on Г 0 (р 2 ) = Г2 Г) Г1 1 Q. The conditions of theorem 1 are satisfied. То write T~(Q), which operator we shall write as Т(р 2 ), in the normal form, we take : We find: 2 СУ) = {(р 0 о ) (Ро 21Рь) 1 -{L with (Ь = 1,... р-1); ( 1-24Ь). _ 2 } О Р 2 w1th (Ь - О,. р - 1). л(r) = r(p-1) (-1)_2_ for ( р 2 О) and (1 24Ь) р f or ( Р 24Ь) 0 р. (3.05) (3.06)

16 17 So we have: if f(т) Е!Г(l), -r, v!, r an integer, then f(т) 1 Т(р 2 ) Е!Г(l), -r, v! where f(r) 1 Т(р 2 ) = r(v- 1 J р-1 ( + 24Ь) Р 2-1 ( + 24Ь) =р 2 'f(р 2 т)+(-1)- 2 -Р' L f рт + L f ~- Ь= 1 Р Ь=О Р (3.07) lf we denote f(т) 1 T(p)I Т(р) Ьу f(т) 1 Т(р) 2 the following relation holds: r (р-1) f(т) 1 Т(р) 2 =f('г)1 Т(р 2 ) + (-1)_ 2 _ р'(р + 1) f(т). (3.08) ~ir If 3/r and Л = ± е 6 we can define Т(9) Ьу replacing 24 Ьу 8 in (3.07). Then (3.08) holds for р = 3. For Г1 = Г2 = Г(l), r an integer, Q = (~ ~) where(n,6)= 1, we can define an operator Т(п) in the same way Т(р) was de~ fined in 3.1. In the case v* = v = 1, these operators сап Ье compared with Hecke's operators Тп (Hecke [1]). Wohlfart [1] found the relation Тп = п- 1 L: h' т(~2). h>o,h2/n For the operators Т,(п) we can find а multiplication theorem. Wohifahrt [1] proved that:. Т (т) Т(п) = Т (тп) if (m, п) = 1. (3.09) In the same way as for Т(р) and Т(р 2 ) we find the normal form: k-1 = pkr { (рkт) -+. L v=l Р" - 1 r (р" - l) ( k-v + 24Ь L (-1)--2-.p(k-v)r.{ р Т )+ Ь=1 Р (Ь,р)= 1 pk-1 r(pk-1) + L (-1)-2- t(t + 24Ь) (3.10) Ь=О This gives us for k > 2 : r(p-1) T(pk) = Т(р) T(pk-1) _ (-1)_2_. р'+1 T(pk-2). (3.11) pk

17 18 Combining (3.08), (3.09) and (3.11) we find: r {р-1) ~ T(n} = П ( 1 - (-1)-2-pr-2s ) (3.12} п r(p-1) (п,6)=1 р>з l -T(p).p- +(-l)-2-pr+i-2s Here the series and the product are to Ье considered f ormally only. 3.3 Г 1 = Г2 = Г(i}, l = 2r ап odd integer, Q = (6 2 ~) As in 3.2 we take v* = v and Ьу the same reasoning as is. used there, we find that тji (Q) can Ье defined. We shall also call this operator Т(р 2 ). л(r)= 1 for R = (~ 2 _. ( 1> 1 е 2 - = ~ (24Ь) Р For CR, we choose (3.05) and find: ~) and R = (Ь 2 ~). ( 3 Ь) when р = 1 (mod 4) j (3.13) р i 1 ( 3 Ь) when р = 3 (mod 4) \ Р. fo, R = (~ 2 :). So we have: If f(т) е jг(l), -r, vj. l = 2r an odd integer, then f(т)it(p 2 ) е jг(l), -r, vj. where: f(т) 1 Т(р 2 ) = = р2'fр2т)+'урр' PL: 1 (3b)t(pт+24b)+P 2 L: 1 t(т+24ь) b=i Р Р Ь=О Р (3.14) with '/'р = 1 when р = 1 (mod 4) and '/'р = i-z when р = 3 (mod 4). For 3/r, л = + е 6, we can define Т(9): f(т) 1 Т(9) = = 32r f(32т) + i-z 3, ± (~) f(зт + 8Ь) + ~ f(т + ВЬ). Ь=О 3 3 Ь=О 9 (3.15)

18 ~~~~~ We can define T(n 2 ) for all п with (n, 6) = 1 in the same way. We find: f(т) J T(p2k) = p2kr f(p2kт) + 2k-1 pv-1 -nir(pv-1) (24b)v (р2k-ит + 24Ь) + 2: 2: е 2 - p(2k-v)г f, v. + Ь=О р Р v=l This gives us : and for k > 1: P 2 k- l (т + 24Ь) + 2: f 2k (3.16) Ь=О Т(р2) Т(р2) = Т(р4) + p2r+i(p + 1), (3.17) T(p2k) Т(р2) = Т(р2н2) + Р2,+2 T(p2k-2). (3.18) Fi:om (3.16), (3.17) and T(n 2 } T(m 2 ) = T(n 2 m 2 } if (n, m} = 1 we find: T(n2) _ ( 1 _ p2r ) (n, lr = 1 -;;;- р ~ Т(р2) p-s + p2r+2-2s (Series and product are to Ье Р considered formally). (3.19) 3.4 Г1 = Г2 = Г 0 (2) or Г{), 2r = 1 ап odd-integer, Q = (6 2 ~). The reasoning is the same as in 3.3. We now use theorem 7 and theorem 8. In the case Г 0 (2), we take (3.05) for Сfг, finding Л(R) = 1 for R = ( and 0 1 R = р 0 2, wh1le fqr R = = (~ 2 ~) we find: Р2 о) (1 24Ь). '(R) --! (~) when р = 1 (mod 4) " if л* 12 = -1, i 1 ( 3 :) when р = 3 (mod 4) j (6 :) when р = 1 (mod 4) л(r) = 6Ь i 1 (-р) when р = 3 (mod 4) if л* 12 = 1. (3.20) ln the case Г {} we take Cf( as in (3.05) but replace 24 Ьу 48.

19 We find: л(r) = 1 for R = (~ 2 ~) 4 = (~ ~) we have ~ (~) л (R) = ( ii (6:) 26 and R = ( 1 48Ь) 0 when р = 1 (mod 4), when р = 3 (mod 4). р 2, while for R :_ For the Т~ ( Q) we find formulae analogous to (3.14). (3.21) 3.5 Г1 = Г(l), Г2 = Г0 (р), 2r = l ап odd integer, Q = (1 0 Ро)., In 3.1 we proved that if l is an odd integer, we cannot de~ fine Т(р) as in (3.03). The ope~ator T~(Q) we now define, re~ sem.ыes Т(р) very much and is therefore of interest. We shall call this operator Т0 (р). For (; ~) е Го (р), we define v* с~ ~) = (:) vp с~ ~) = = v (; 7) Ву (1.06) this is а multiplier system for Г0 (р). ln fact we have: if f (т) е j Г (1), -r, v!, then f(рт) е j Г 0 (р), -r, v*!. Furthermore we have v* = vl Q on Г~ (р) = Г2 п Г 1 1 Q. There~ fore- T~(Q) = Т0 (р) can Ье defined. In this case we cannot find а normal form. For jг 0 (р): Г~ (р)! we take (~ 2 ;ь). (Ь =О, ".. р-1). We then have: If f(т) е j Г (1), -r, v!, 2r = 1 is an odd integer and v* is the multiplier system of f(рт), theп f(т) 1 Т 0 (р) е j Г 0 (р), -r, v*j. where f (т) 1 То (р) _: ~ 1 f (" + 24 Ь), Ь=О р and f(т)i Т 0 (3) == ь~о f (3.22) 2 ("+8Ь). ~ir if 3/r and Л = ± е. (3.23) 3

20 21 Remark : As was said in ( 1.13), we can also define Т~( Q *) in the case considered here. After some calculations we find : g(т)it~(q*) = r2 _ 1 ( 0) r - 1 (о _ ) = ь~о g(т)[ 2~Ь 1 + µ.-i ь~о g(т)[ 1 24Ь~ (3.24) An interesting property is : If f(т)e jг(l),-c,v!,!~ (\\ v then f(т)[ ~~ ~)ITЛ(Q*) = f(т)jt(p 2 ). (3.25) 3.6 Г1 = Г(l), Г 2 = r 0 (p), 2с = l ап odd integec, Q = (~ ~) This case is suggested Ьу the previous one. It is а Ьit more difficult in the calculations. That Т~ ( Q) can Ье defined is proved in exactly the same way as was done in 3.5. We sha11 call the operator Т 0 (р). For jr ( 1 0 (p): гg(р)! we choose 24 Ь о ), (Ь =О,... р-1). 1 This gives us : f(т)it~(q) = р'f(рт) + р' :~ 1 1 t( 24 :Ь~"+ 1 ).(24Ьт + 1)-r. Let Ь', d' Ье integers, so that 24 2 ЬЬ' = 1 (mod р) and pd' ЬЬ' = 1. Then we have : p'.f(24b~t+ 1 ). (24Ьт + 1)-r = ~ Р (" + 24Ь') - 24Ь'] = р'. { ( р+ 2 -!Ь') (24Ьт + 1)-r = 24Ь т + d' р _ (- 24Ь') -n~(p-i) (т + 24Ь') -.. е.f. р р

21 22 So we have: If f(т) е! Г (1), -r, v!, 2r = l is an odd integer and v* is the multiplier system of r(;). then f(т) 1 Т 0 (р) е! Го (р), -r, v*!, where f(т) 1 Т 0 (р) = -24) -.п;,(р-l) p- l ( Ъ) (т + 24Ь) ( = p f(p т) + -- е 2 L - f. Р Ь= 1 Р Р (3.26) The connection between {3.22) and (3.26) is given Ьу the fo1- lowing relation : f(т) 1Т 0 (р)1 Т = µ, f (т) 1 Т 0 (р). (3.27) The operators we have defined are of interest because of their applications which will Ье discussed in Applications; We shall now discuss some applications of the operators defined in 3, especially to powers of )j (т). Many of the resu] ts we find, have already been found Ьу М. Newman (cf. Newman [1], [3], [4]) and Н. Rademacher (cf. Rademacher [1]). The in" teresting fact is here, that а11 these resuits are consequences of the existence of Hecke"operators. If f(т) is an integral modular form е!г (1 ), -r, v!, where 2r is an integer, we see from (1.03) and (2.02) that the Fourier ех" pansion of f( т) has the form : 2ni-rn 00 L а(п) е 24 (4.01) п=о 00 2:тti'rn We can a1so write f(т) as 2: а'(п) е N, with N/24 and а'(п) =О, п=о if (п, N) > 1. The Dirich1et series connected with this Fourier series is: q> (s) = со 2: а (n) п-s. (4.02) n=l If, in {4.01) we have (п, 24) = k for all п with а(п) :;t: О, we shall write : - со. q>(s) = 2: a(nk) п-s. (4.03) n=i

22 23 For the first applications we consider: 4.1 Even powers of 11 (т). If l= 2r is even, then 11l(т) е!г(l), -r, vj. where v is determined Ьу л = е 6, r an integer. First we remark that these functions could also Ье discussed with Hecke's theory. applied to!г(12), -r, lj (cf. theorem 4, corollary). The results would not follow as easily as they do now. In this case we find for Hecke's operator ТР, the relation: r(p-1) Tr = (-1)_2_ Р-1 Т(р). (4.04) Now consider f(т) 1 Т(р) е!г(l). -r, v*j, where v* is determined лirp Ьу л* = еб. Ву (3.03) we have: f(т) 1 Т(р) = 00 2ni~np r(p-1) р л;(<+24Ь) п = р' L а(п) е------и' + (-1)_ 2 _ L L а(п) е 24 Р n=o Ь=Оп=О = n~o{pr.a(;) + (-l)<r;l) р.а(пр)} /~Zn. (4.05) (As usual, we define а(х) = О if х is not an integer.) If 3/1, formula (4.05) is a1so true for р = 3. We are interested in knowing if the Dirichlet series rp (s), con~ nected with 11l (т), are Euler products in regard to certain primes. ln the next paragraph we shall prove: If f( т) is eigenfunction for Т(р) with eigenvalue с, then: r(p-1) t:p(s)=( L а(п)п-s) (1-(-1)-2-c.p-s-I+ п =!=О (mod р) We now prove : + ( 1 r(p-1),-2- r-1-2s'-1 - } р ) (4.06) Theorem 9. lf О< 1~24, l is even and 1 (р - 1) =О (mod 24), then 11l(т) is eigenfunction о/ Т(р). Reщark: This is theorero 1 ~nd the9reщ 1' 9f Newman [3].

23 24 Proof: If l(p-1)=0 (mod24), we have ЬуЗ.1: л*=j.and 11i (т) 1 Т(р).. therefore g(т) = '11l(т) е jг(l), О, Ч Usшg the propert1es of '11 (т) stated in 1, we see that g(т) has no poles in the funda~ mental region of Г {l) except possiьly in т = i = Ii;i the Fourier expansion of 11l(т) 1 Т(р) we have for the first coefficient which is not zero: п = l (mod 24) and therefore п?: l. Hence g(т) has no pole in т = i со and so g ( т) has no poles at а11. W е now use the fact that an automorphic function f е jг, О, 1! which has no poles in the fundamental region of Г, is а constant (cf. L. R. Ford, Automorphic Functions, page 94). We have proved that g(т) is а constant. It is easily seen that this constant is р. a(lp) if we write: 11l(т) = ~ а(п) е 24. (We shall use this О < п l (mod 24) notation for the rest of this paragraph.) We have proved: 11i (т) 1 Т(р) = = р. а (lp). '11l (т) if [ is even, [ (р - 1) = О (mod 24). (4.07) Theorem 10. Define [ 1 = pl - 24 [f ~]. If l is even, О< l ~ 24 and [ 1 > [, then '11l (т) is eigenfunction о{ Т(р). In this case v* ~ v, and therefore the eigenvalue сап only Ье О. Remark: theorem 2 of Newman [3] is contained in this theorem. Proof: In this case g(т) = '11l(т~l/(~(p) Е jг{l),o,v*.v-1i, (л*. л- 1 ) 6 = e"'ir(p-l) = 1. Hence g 6 (т) е jг(l), О, 1! g 6 (т) has no poles in the fundamental region of Г (1) except possiьly in т = i =. In the Fourier expansion of 11l (т) 1 Т(р), 3""&i?:l1 the first coefficient which is not zero is the coefficient of е 24, and as [ 1 > l we find that g 6 (т) has а zero in т = i со. Ве~ cause g 6 (т) has no poles in the fundamental region of Г (1), g 6 (т) -must Ье а constant. Therefore we have g (т) = О. We have found : If l is even, О < l ~ 24 and -pl - 24 [~~] >!, then 11~(т)/Т(р) =О. (4.08)

24 25 Theorem 11.!{ О < l::::::; 24, l even, then 11i (т) is eigenfunction о{ Т (р 2 ) for all р > 3, and {or р = 3 if 3/ l. Proof: This theorem is proved in the same way as theorem 9 was proved. From theorems 9 and 1 О we find that 11 1 (т) is eigenfunction of Т(р) for all р > 3 if l = 2, 4, 6, 8, 12 or 24, i.e. if l is an even divisor of 24. Denoting the Dirichlet series connected with (т) Ьу lfz (s) we have ф2 \/ ")\ - тr { 1 - 'р\ - -s 1 ( - 1 \ - - 2s \ -1 1 ~ - ll \ 1 -., ] 2 \ } f! -г- 1 --; f! } р>з ' р q;4 (s) = П (1 _ 'Тб (р). p-s + p1-2s)-1 р>з ~б (s) = fi (i - T4(p).p-s+ (-l\p2-2s1-j р>2 р! J о/8 (s) = П (1 - т 3 (р). p-s + p3-2s)-1 р>з ~12 (s) = П (1 - т 2 (р). p-s + p5-2s)-1 р>2 ~24 (s} = П (1 - т (р). p-s + pll -2s)-1 р (4.09) (4.10) (4.11) (4.12) (4.13) (4.14) For the coefficier1ts тп (р), the following expressions are known, (cf. Schoeneberg [2], [3]). Let 7r 1 Ье а Gaussian prime, 7r1 its conjugate.,3, if and 1 (mod 1/-3) i з 71"1 + 7r1 'Т3 ( р ) - р = 7r17r1 7r1 - о if р =!= 1 (mod 3) if, р = 7r17r1 and 7r1-1 (mod 1/-4) if р =!= 1 (mod 4) 'Т5 (р) = 71"1 + 71" 1 1 if р = 7r 1 7r' 1 and X(7r 1 ) = 1, where Xis the character mod 2 1/ - 3 of order 6, for which Х (/:;) = е -;"'. О if р =/= 1 (mod 6)

25 26 These expressions and an analogous expression for т12 (р) are given in Schoeneberg [3]. Ву (1.08) we have т12 (р) р = 1 (mod 12), р =/= 1 (mod 12). Expressions for т2 {р) and т (р) are given in Schoeneberg [2]. (Remark: for some of the functions we have discussed, operators Т(2) and Т(3) can also Ье defined as we see from the products (4.09) to (4.14).) We now discuss the even powers of '1 (т) which are not eigen~ function for all Т(р), (1~24). From Hecke's theory of the Тр (Hecke [1]) and the formula (4.04) we see that 11 1 (т) where the modular forms f; (т) = L: f;(т}, i are eigenfunctions of all Т (р). We can take the f; (т) liвearly independent. Applying Т (р) we find ii 1 (т)1 Т(р) = L: с;(р)f;(т), where the с;(р) are the eigenvalues. i Applying Т (р) once more we find '1 1 (т)1т(р)1 Т (р) = L: с; 2 (р) f; (т). Ву (3.08) and theorem 11 : 11 1 (т)1 Т (р) 1 Т (р) = r(p-1) = c(p)ii 1 (т) + (-1)_ 2 _pr(p+ l)ii 1 (т) = c'(p}ii 1 (т). Because the f; (т) i are linearly independent, we have: с;2 (р) = с' (р) for all i. Taking the functions {; (т) with с; (р) = V с' (р) and those with С; (р) = -V с' (р) together, we find: '1 1 (т) is the sum of 2 functions which are eigenfunction for all Т(р). We refer to this as (4.15) W е now consider ( т). This is an eigenfunction of Т (р) for р =/= 5 (mod 12) Ьу theorems 9 and 10. We write ii 10(т)1 Т (р) = 2niтn = L: Ь (п) е For р = 5 (mod 12) we find Ь (О) = п =О = Ь (1) = О, Ь (2) = р. а (2 р).

26 27 From this we see that the function g (т) = = J1 10 (:~i(~(p) е fг(l),-4,1! has по poles in the fundaшental region of Г (1 ). Hence g (т) must Ье а multiple of G 4 (т). Comparing coefficients we find: J1 10 (т)it(p) = р.а(2р)g4 (т)j1 2 (т) for р = 5 (mod 12). (4.16) The two functions {; (т) шentioned in (4.15) are linear com~ Ьinations of J1 10 (т) and G 4 (т) J1 2 (т). If we write 00 2пiтп G 4 (т) J1 2 (т) = 2: с(п) е 24, we have for р = 5 (mod 12), p-f:-5: n=o G 4 (т) 11 2 (т) 1 Т (р) = р с (1 Ор) J1 10 (т). If J1 10 (т) + х G 4 (т) 11 2 (т) is eigenfunction for all р then р. а (2р) = а 2 р. с (lop). For the decomposition of J1 10 (т) into eigenfunctions of all Т(р) we find: J1 10 (т) =tj11 10 (т) + i8 G4 (т)11 2 (т)! +tj11 10 (т) G 4 (т)11 2 (т)!. (4.17) The eigenvalues are: р. а (lop) for р = 1 (mod 12), 48 р. а (2р) and - 48 р. а (2р) resp. for р = 5 (mod 12), О for р = 7 (mod 12) and р. 11 (mod 12). The Oirichlet series connected with the two functions, have Euler products. ComЬining these we find with (4.17): 1 rp1a(s)=- П (l-p4-2s)-i. П (1-a(l0p).p-s+P4-2s) р-7,11 р=1 (mod 12) (mod 12)! П (1-48 а (2р). р- + p4-2s)-i p-s (mod 12) - fl (1 + 48а(2р).р- + p4-2s)-i! р_5.. (4.18) (mod 12) Another proьlem we are interested in, is finding the zeros of Pz(n) where П (1 - хпу = 2: Pz(n)xn. n=l п=о

27 28 Writing '11i = (т) = :2: а{п) е 24 we have Pz(n) = а(24п + l). п=о ТаЫеs of the pz (п) were given Ьу Newman (Newman [1]). If 1/24, the zeros of Р1 (п) are found easily from the well known Euler products (4.9) to (4.14). For l = 10 we have Ьу (4.18): р 10 (п) is zero if at least one of the primes = 7 or 11 (mod 12) divides 24п + 10 in an odd power, and also if all primes = 5 (mod 12) divide 24п + 10 in even powers. This last condition does not give us any zeros not covered Ьу the first condition. W е shall now consider '11 14 (т). Ву theorems 9 and 1 О this is an eigenfunction of Т(р) for р = 1 (mod 12) and р - 5 {mod 12). '11 14 (т) is also eigenfunction of Т (Р) for р = 11 (mod 12). '11 14 (т)1тfр) Proof: g (т) = (т)' is а modular form е 1 Г (1), -2, 1 j 1110 without poles in the fundamental region of Г (1). The only function satisfying these conditions is g (т) = О. We remark that this same proof can Ье applied in the case of '11 25 (т) and р = 11 (mod 12). Ву the same same method as was used for (т) we find: '11 14 (т) / Т(р) = -р.а (2р) G 6 (т)'11 2 (т) for р = 7 (mod 12). (4.19) '111;(т) = t j'1114(т) + 3б;V3 G5(т)'112(т)! + i + t j '1114 (т) - 360V3 Gб (т)'112(т) j (4.20) = fi (т) + { 2 (т) with {; (т) eigenfunction for all Т (р), (р > 3). From this we find ~14 (s) as sum of two Euler products: i4(s) = 72oivз _!I (1 +(-1(;1рб-2s)-1 Р=5. 11 (mod 12) П p:=l (mod 12) { pg.? ( iV3.a(2p)p- ) - p 5 - (mod 12) (l -a(l4p)p- +p 6-2s)-1. 2 )-l p:g 7 (1-360i vз. а (2p)p-s - р 6-2 )- 1 ~ (mod 12) (4.21)

28 29 So р14 (п) = О if one of the primes = 5 or 11 (mod 12) divides 24п + 14 in an odd power; or if all primes = 7 (mod 12) divide 24п + 14 in even powers. Again this second condition does not give us any zeros not covered Ьу the first condition. То discuss (т) completely we must introduce an operator Т(2). То do this we choose Л = j Г (1). -8, v*!, where v* is the multiplier system of 11 8 (т). Ву using one of the kпоwп ех~ plicit expressions for the multiplier system of 11(т), it is easily sееп that we have v* = v 1 Q if Q = ( ~ ~). Therefore Т(2) сап Ье defined. In the same way as (4.16) апd (4.19) we now fiпd: (т)i Т(2) = 2 G4 (т)11 8 (т), and for all primes р = 2 (mod 3) the analogous resu]t: (т) 1 Т(р) = р. а (8р) G 4 (т) 11 8 (т). (4.22) (4.23) Ву theorem 9, (т) is eigenfunctioп of Т(р) for р = 1 (mod 3). The decomposition into eigenfuпctions of all Т(р) is: (т) 1 (f";5(s)= П (l-a(lбp)p-s+p7-2s)-l. 12VIO р-1 (mod3) п (1-6 v10. a(8p)p-s + p7-2s)-1 (4.24) { p_2(mod3) - п (1 +6VIO.a(8p)p-s+p s)- 1 }. (4.25) р=2 (mod3) From (4.25) we fiпd: р16 (п) = О if all primes = 2 (mod 3) divide 3п + 2 iп even powers, Ьut this is not possiьle. In this way we do not find zeros of р 16 (п). lt is поt known if р16 (п) has zeros. Ву the taьles in Newman [1], р16 (п) ~О for п::::;:; 400. The fuпctions (т), (т) and (т) сап also Ье treated as was done for the other even powers of 11 (т). For (т) the for~ mulae Ьесоmе much more complicated. We shall give the results for (т) and (т) (т)IТ(р)=р.а(l8р)11 18 (т) for v= 1 (mod 4), ( ) (т) 1 Т(р) = -р. а(бр) G6 (т)11 6 (т) for р = 3 (mod 4).

29 30 i ;8 (s) = п (1 - а {18р) p-s + p8-2s)-1. 48V 35 р = 1 (mod 4) { п (1 + 24iV35 а{6р) p-s - ps-2s)-1 p=3(mod4) П {1-24iV 35 а(6р). p-s - p 8-2s)- 1 }. '(4.27) р =3(mod4) As was the case for '1 16 (т), this formula gives us no zeros of Р1в (п). '120(т)1 Т(р) = р. а (20р) '1 20 (т) for р = 1 {mod 6), '1 20 (т)iт(р)=р.а(4р)g8 (т)'1 4 (т) for р=5 {mod 6) ff 20 (s) = П (1-a(20p)p-s+P9-2s)-I. 576V70 р = 1 (mod б) { П (1-288V70 а (4р). p-s + p9-2s)-1 Р=5 (modб) (4.28) - П ( V70 а (4р).p-s + p 9-2s)- 1 }. (4.29) p=5(modб) We find no zeros of р 20 (п). 4.2 Odd, positive powers of '1 (т). Consider '1i ( т) е jг (1 ), - r, v! where 2r = l is odd, v deter~ nir mined Ьу Л = еб. As we have seen, operators Т(р) generally cannot Ье defined for these functions. W е write 00 2ni_?:n '1~(т) = f(т) = L а (п) е ---U-. n=l We find for р > 3 from (3.13) for every f(т) е jг(i), -r, v!: 00 2nirnp 2 f(т)it(p 2 )=p 2 ' L a(n)e-----u- + n=l 00 (р-1 ( 3Ь) 2nibn) 2ni~n + rp P' L а(п) L - е-р- е2г + n= l b=i Р оо (р2-1 2nibn) 2"'i~n + L а(п) L е р2, е24р2 = Ь=О n=l = ~ {p2r а (--;-) + (- 1 )z;t. (Зп )рг+t а(п) + р2 a(np2)} е zп;_;п. n=l Р Р Р (4.30)

30 If 3/l we have f(т) 1 Т(3 2 ) = 3i n~l {з 2 1 'а:( ~) + (- 1 {-; (n~з) 3r+t a(n) + 9a(9n)}e 2 ~:n. (4.31) Theorem 12. If О < l < 24, l odd, then 11 1 (т) is eigenfunction о{,т(р 2 ) for all р > 3, and for р = 3 if 3/ (т) 1 Т(р 2 ) Proof: Consider g (т) = (т). Ву the same reasoning 111 i:1 -" -w--:;" u ""'u.j 1'v "--- ~& "(,.,.\ ~,~ ~~~ +J..~+,-, (,.,-\ с Sr (1 \ () 11 v -...,...,.._ J.: t::vt::ll _l.juwt::l" U.L 'i\ jt VV~.;J"'-~ LJ..J.U\. 'tf\"j""' (.._ \..1.Jt'-'t А~ and that g(т) has no poles in the fundamental region of Г(l). Therefore g(т) is а constant. Ву computing g(т) in т = i = we find ( ) / \-2 (3 l ) 1 2 f ~) g т = \ р}. р р'+ + р а,lp'- for р > 3 and 1-1 (l/3) g(т) = (- 1) '+.! + 9а(91) for р = 3 if 3/1. With this theorem we can deduce relations for the numbers pz(n). We have ( п) (-1) 1 ; 1 (3n), pzr а pi + Р. р. р'+. a(n) + p 2.a(np 2 ) -.. = j( / )';'(~ 1 ).,,,.Н + р'. a(lp~} а (п). We now write п = 24т + 1, др= and we use ( п - l) a(n)=p 24 Then ( 4.32) becomes: р2". pz ( т -lд) р2 Р + р (4.32) { (- l)l-l( 3 ) r+_!_[(24m + l) ( l )] } + р 2 р.р z р. - р. - p2.pz(lдp) pz(m)+ + p 2.p 1(mp 2 + [др)= О for р>3. (4.33)

31 ' 32 For instance: l = 5, р = 5, т = 5п gives us ( 1) 5 п - 3 р бр 5 (5п) + р5 (125п + 5) =О. In the same way: l = 5, р = 7, т =}п + 3 gives us ( 1) 7 п - 3 р бр 5 (7п + 3) + р5 (343п + 157) = О. These, and many other relations are mentioned in Newman [1]. As we have seen they are all special cases of the general for~ mula (4.33). For 3/1, р = 3, formula (4.33) is true if we replace (! ) Ьу р ( 1 ~ 3 ). The author did not succeed in finding zeros of pz(n) Ьу using these formulae as was done for even l. Of course if а zero is known, others сап Ье found' from (4.32). We have Ьу induc~ tion: If а (п) = О and. р 2 / п (р > 3) thert а (np 2 k) = О. If 3/1, а (п) = О and 27 / п then а (3 2 k n) = О. lt has been shown that р15 (53) =О (cf. Newman [1]). So we find: р15 ( 53п 2 k + 15 п2k_ 1) 24 = О for all odd п. (4.34) Whether there are other zeros than these and whether the fact that р15 (53) = О has а deeper significance or not remains an unsolved proьlem. The formulae we can find Ьу applying the operators Т0 (р) and Т 0 (р) are generally too complicated to Ье of any use in finding relations for the coefficients pz (п). То illustrate this, we shall give one example (here the result is still relatively simple) (т) 1 Т0 (7) Cons1der g(т) = 115 ( т). Ву (3.22) we have: niт:n 24 7 L: а(7п) е g(т) = :0-0 2~.:гс;~,п-.7 е jго(р), О, 11. L а(п)е 24 n=o

32 33 Applying the operator Т and using (3.27) we find : g(т) = 2л:i п -2:п;i 49п _7з ~ (.!!_) ~ (п) е ~ a(n) е---;;;;- 24 п=о 7 п=о = -2ni n L а(п) е~ 24 п=о W е see that the function g (т) has а pole of order 1 in e 2 7 "'i" in т = i = and а zero of order 1 in е " in т = О and по other poles or zeros in the fundamental region of Г 0 (р). The 11 (т) }4 function { 11 ( т) has the same properties and therefore we have 7 g (т) = с. { (~;)} 4 Comparing coeffici~nts -2ni we find с = 7 а (77) = 70, an d so: 11 5 (т)i Т0 (7) = (т)11(7т) (7т). (4.35) Even this simple result does not give us linear relations for the р5 (п). 4.3 Applications to 11-1 (т). The operators we defined сап also Ье used for negative values of r. An interesting case is the function 11-1 (т). -:тti-c 11-I(т) = e---i-2 L р(п} e2"'inт' ()О п=о where р(о) = 1 and р(п) is the number of partitions of п. Ву proving formulae as 11-1 (т) 1 Т0 (5) = (5т)11-6 (т) and 11-1 (т) 1 Т 0 (7) = (7т)11-4 (т) (7т)11-8 (т), we can give proofs of the Ramanujan congruences р(5п + 4) =О (mod 5) and р(7п + 5) =О (mod 7) and others. These proofs were given Ьу Rademacher (Rademacher [1]). The theorems he used were special cases or applications of the the~ orems we have proved. We could also apply the operators Т(р 2 ) to 11-1 (т). As 11-1 (т) is not eigenfunction of these operators, the results do not give us linear relations for р(п).

33 34 We give one example: (. = 11-1(т) 1 Т(р'1) {r(1) _р2-1 1} g 'Т) J1-Р 2 (т) е ' 2 ' and has no poles in the fundamentaj region of Г(l). Therefore g(т) can Ье written as п~о xnдn(т)g12 {k-n)(r), (k=p 2 2 ~ 1 ) (т)jт(25) For шstance 1 11 _25 (т) д(r) G12 (т). 4.i Applications: to.э- ( т) We can aiso use the methods of this:paragraph to prove some results of Hurwitz [l] and Sandham [1] on the representation of а square as а sum of squares. We apply Т(р 2 ) to.э-z(т), (l'> О,. { 1 1 l odd). We find.э-z(т~j Т(р 2 ) е Гff. -z Vfff where Vff is.the multiplier system of.э-(т)..э-z(т) 1 Т(р 2 ) has no poles in the fundamental region of Гff. b.ecause.э-(т) has this property. Therefore the behaviour in т = -1 is described Ьу : (т + 1)1;21.э-z (т) 1 Т(р2)! = e2;i С:1) z ~- cv'. e2"'i С:1)". 'P=-k l wh ere k ~ 8 " (4.36).,э.z (т) 1 Т(р 2 ) Let l Ье 3, 5 or 7. Cons1der g(r) =,Э.l(т) We have: g(т)'е jгff, О, Ч As if(т) has no zeros in the fundamental region of Гff except in the limit points т = ± 1, we see that g ( т) has no poles in the: fiшdamental regioп of Г {} except possiьly in т = ±J. From (1.10) and (4:36) we see that g(т) is regular in т = ± 1. Therefore g (т) has no poles in the fundamental region of Г {} Hence g (т) is а constant. Le1 r1(n) Ье the number of representations of п as а sum of l squares. Then ifl(т) = 00 2:< rz(n) е"iтп and n=o

34 Ву 35 comparing the constant terms in these series we find g(т) = р 1 + р 2 This gives us the re1ation: ( п) (-l)z-1 z+l(n) irz р 2 + р 2.р 2 р rz(n)+p 2.rz(np 2 )= Taking п = 1.we find: rз(р2) = 6 { р { р, 1 )} r5(p2) = 10 / рз - р + 1! r7(p2) = 14 { р ( р 1 ) р2} 5. Linear relations and Euler products. = (р 1 + р 2 } rz (п). (4.37) 1 \ {4.38) ln this paragraph we shall study modular forms for which the associated Dirichlet series have an Euler product. Hecke [1, 3] and Petersson [2] studied this proьlem for modular forms of in~ tegral dimension. For the many results concerning these forms, we refer to their papers. W е are interested in the proьlem for nonintegral dimension. Some special cases were discussed Ьу Hecke [2]. First we shall prove some generalizations of theorems in Hecke [1], 9. Theorem 13. Let f(т) Ье а modular form е jг(q), -r. v!. and suppose that for some primitive transformation Sn о{ order п, with (п, Q) = 1, we have f(т) 1 Sn е \Г(Q), -r, v*!.then f(т) must Ье identically zero. Proof: The proof is nearly the same as in Hecke [1]. With~ out loss of generality we may take Sn to Ье (~ ~). Now we have and therefore f! Sп UQ = v* (UQ) fl Sп. fl Sп U 0 Sп- 1 = fl Sп U 0 1 Sп - 1 = v* (U 0 ) fl Sп 1 Sп - 1 = v*(u 0 )f, and so:

35 36 1 thoose А= (; ~) Е Г{Q), with а 1 = с1 = 1 (mod п). 1 Then V = А(~ ~) = (~ g) (mod п), and fl V = = f 1 А 1 (~ ~) = v(a)v*(uq)n' {. All powers of V are primitive matrices. Choose l so that ni = 1 (mod Q), and find matrices В, с Е г ( 1) so that vi = BSn 21 с. Then V 1 =BC=l(modQ).Forg(т)=f(т)IB е jг(q),-r,vlщ '. - J we now have: gls~ 1 = flbl~ 1 =flbs~ 1. (J(V 1,C- 1 )flvi1c- 1 = = (J(V 1, С- 1 ) v 1 (A)v* 1 (UQ)n' 1 fl с- 1 and g = f! в = () (ВС, с- 1 ) fl вс 1 с- 1 = () (ВС, с- 1 ) v (ВС) fl с- 1 Hence gl~ 1 = rп' 1 g, where r = (J(VZ, с- 1 ) (J- 1 (ВС, с- 1 ) v- 1 (ВС) v 1 (A)v* 1 (UQ). As g ( т) only changes Ьу а factor 'У n' 1 when transformed Ьу S?.1. g (т) must Ье ide~tically zero. Hence f (т) is identically zero. 00 2niт:n Corollary: If f(т) = L: а(п) е N е jг (Q), -r, v/, and р is а п=о. prime for which (р, N) = 1, and а(п) =О if р.1 п, then f(т) is identicaily zero. Proof: We have f(т) = f(т + N), and the trans~. р formation matrix (Ь N) is primitive. р. 00 2ni-cn Theorem И. If f(т) = L: а(п)е N, and а relation п=о f(p т) + а : ~~ f (" +р Nb) = (З f (т) holds, we have formally: 00 L а(п)п-s = n=l = ( L а (п)п-s) (1 _ (З. а-1 p-1-s + ог1р-1-2s)-1. (5.01) (п,р)= 1 Proof: From the linear relation we find а(;) + ар а(пр) = = (З а(п}, and therefore we have for (п,р) = 1 :

36 37 а(прk) = a(n)c(pk) where c(l) = 1, с(р) = {З. (ар)- 1, and apc(pk+i) + c(pk-1) = {З. c(pk). From this (5.01) follows Ьу dividing Ьу pks and summing over k. Modular forшs е (Г(l), -r, v( with integral т:, which are eigen~ functions of the operators Т(р), satisfy the conditions of theorem 14. For nonintegral r we generally do not have an operator of the type of Т(р). We сап define these operators for functions. f(т) for which the exception mentioned in theorem 2 holds. We shall try to find modular forms f(т) for which the transformed functions f(pт),f (т ~ Ь), (Ь =О,... р-1), are linearly dependent. W е can consider the linear relation as an operator of order р, mapping f(т) onto О. We shall study this proьlem for the groups Г(l), Г 0 (2) and Гn.. Let f(т) Ье an integral modular form е!г (1), -r, v!, and sup~ ~ 1 pose (т+ь) а linear relation аf(рт) + L аь f -- = О (5.02) Ь=О Р holds. We may suppose а ;;z:= О, for application of а unimodular transformation changes (5.02) into an other relation, in whicb the coefficients аь have been permuted and multiplied Ьу cer~ tain multipliers v. (If all the coefficients are zero, the relation is trivial). We may take а = 1. We now apply the transformation UP to (5.02). This gives us: лр 2 f(рт)+лрllаьf(т+ъ) =О (where Л = v(u)). Ь=О Р Combining this with (5.02) we find (ЛР ) f(рт) = О. If f(рт) is not identically zero we have ЛР 2-1 = 1, that is: т:(р2-1) 12 is an integer. We have found: Theorem 15. If f(т) е!г(l), -r, v!, and а linear relation (5.02) holds fот: almost all primes, then f(т) is identically zero or 2r is ап integer. As we are interested in functions е jг(l), -r, v! for which the linear relations hold for all р > 3, we only have to consider the

37 38 case where 2r is an integer. First we sha11 simplify the form of the linear relation. We have seen in (4.01) that we can write 00 Zni-rn f(т) as L а(п) е N, where N/24, and а (n) =О if (n, N) > 1. п=о W е write the linear relation as : We write this as : {(рт) + р L 1 аь f (т + Nb) = О Ь=О Р (5.03) 00 2пiтпр 00 ( р - 1 2ninb) ' 2пiтп п~о а(п) е N + п~о а(п) Ь~О аье Р е Np =О. From this we find, with rx = 2: аь, p-i the linear re1ation: Ь=О p-i (т+ Nb) f (рт) + а L f = О. Ь=О Р (5.04) If а linear relation (5.02) holds, then the relation (5.04) holds. We shall now prove that if а relation (5.04) holds, no essentially different transformation of order р can Ье defined. Suppose that (5.04) holds and also {(рт) + р 2: 1 аь f (т + Nb) = g (т), with Ь=О Р g(т) е jг(l), -r, v*! То this we apply the transformation им, and then sum over k =О,... р - 1. We find: f(рт) + {ЗРL 1 f (т + Nb) = g(т), where {З = _!_ р 2: 1 аь. This Ь=О Р _р Ь=О comьined with (5.04) gives us: (а -- {З) f(рт) = аg(т) е jг{l), -r, v*! If f(т) is not identically zero we find, applying theorem 13, а= {З and g(т) =О. We have proved: Theorem 16. If f(т)е jг(l),-r,v! and а relation(5.04)holds, p-l (" + Nb) then а linear comblnation f(рт) + 2: аь f сап only Ь=О Р Ье а modular form е j Г{l), -r, v*! if it is identically zeto.

38 39 Corollary: If f(т) е! Г(l), -r:, о!, r: an integer, and а relation (5.04) holds, then we have f(т) 1 Т(р) = О. ln 4 we have seen examples in which {(т) 1 Т(р) = О for some primes, but not for all р > 3. Hecke [2] proved that for the _functions 11 (т) and 11 3 (т) relations (5.04) exist for all р > 3 (for 11 3 (т) also for р = 3). W е shall prove that there are no other integral modular forms е jг(l), -r, v!, for which the relation (5.04) holds for all р with (р, N) = 1. For this proof we shall need а theorem proved Ьу Hecke [4]. Definition : Let Л and и Ье positive constants, and ;v = ± 1. We say f(т) is an automorphic form with signature ) Л, и,?' \ if: 1. f(т) is regular for lm т >О, and f(т) is represented Ьу а 00 2л:i'rn series 2: а (п) е л, which implies f(т + л) = f(т) if lm т >О. п=о 2. f ( т 1 ) = у(-iт)" f(т), (Im т >О). 3. а(п) = О(пК) for some К> О. Definition : А complex function rp (s) is called а Dirichlet series with signature! л, "?'! if: 1. In some right half~plane rp (s) is represented Ьу а Dirichlet = series 2: а(п)п-s. n=l 2. (s - х,) rp(s) is an integral function of s of finite genus. 27') -s ( 3. The function R(s) = Л Г(s) rp(s) satisfies the functional equation: R (s) =?' R (х, - s). 00 2ni'rn Theorem 17. lf f(т) = 2: а (п) е л is ап automor:phic form п=о with signatur:e! = л, " у!, then the Dirichlet ser:ies 2: а ( п) п -s n=l conver:ges somewher:e and is а Dir:ichlet series wiф signatur:e ) }., "?'!. f Qr а proof we i;efei: to Hecke' s paper.