Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

manuscripta mathematica manuscript No. (will be inserted by the editor) Robert S. Maier Nonlinear differ ential equations satis ﬁed b y certain classical modular f orms Abstract. A un iﬁed treatment is gi ven o f lo w-weight modu lar forms on Γ 0 ( N ) , N = 2 , 3 , 4 , that hav e Eisenstein series representations. For each N , certain weight-1 forms are sho wn to satisfy a coup led system of nonlinear dif ferential equations, which yields a single non - linear third-order equa tion, called a generalized Chazy equ ation. As byproducts, a table of di visor function and theta identities is gen erated by means of q -exp ansions, and a transfor- mation la w under Γ 0 ( 4 ) for the secon d complete elliptic integral is deri ved. M ore generally , it is sho wn how Picard–Fuchs equ ations of t riangle subg roups of PSL ( 2 , R ) , which are hyper geometric eq uations, yield systems of nonlinear equation s for weight-1 forms, and generalized Chazy equations. Each triangle group commensu rable wi th Γ ( 1 ) is treated. 1. General introduction In this article we systematically deri ve ordinary dif ferential equations (ODEs) that are satisﬁed by certain elliptic modular forms and t heir roots. The latter are re- spectiv ely single-valued h olomor phic functions, and potentially multi valued ones, on the u pper h alf plane H = { ℑ τ > 0 } . Amo ng the classical mo dular g roup s that will ap pear are th e full modular group Γ ( 1 ) = PSL ( 2 , Z ) , the He cke cong ruence subgrou ps Γ 0 ( N ) , N = 2 , 3 , 4 , an d their Fricke e xtensions Γ + 0 ( N ) < PSL ( 2 , R ) . There are two distinct sorts o f ODE satisﬁed b y forms on classical mo dular group s, distinguished by their independ ent v ariables. 1. If the independent v ariable is the period ratio τ ∈ H , the ODE will typically be nonlinear . Classical examples inclu de Ramanu jan’ s cou pled ODEs fo r Eisen- stein series on Γ ( 1 ) , Rank in’ s fourth -orde r O DE for the modular discrimi- nant ∆ (a weight- 12 fo rm on Γ ( 1 ) ), a nd Jacobi’ s th ird-ord er ODE f or the theta-null function s ϑ 2 , ϑ 3 , ϑ 4 (which are weight- 1 2 forms on Γ 0 ( 4 ) ). 2. If the independent v ariable is a P 1 ( C ) -valued Hauptmodu l ( function ﬁeld gen- erator) fo r a modu lar grou p Γ , th e ODE will be linear [42]. Classical exam- ples include Jaco bi’ s hy pergeometric ODE f or the ﬁrst com plete e lliptic inte- gral K (a weight-1 f orm on Γ 0 ( 4 ) ), vie wed as a function of the modulus k or its square k 2 (a Hauptmo dul for Γ 0 ( 4 ) ); and Picard–Fuc hs equation s satisﬁed b y periods of elliptic families rationally parametrized by other Hauptmoduls. Depts. of Mathematics and Physics, Uni versity of Arizona, T ucson AZ 85721, US A e-mail: rsm@math.arizona.edu 2 Robert S. Maier The stress in th is article is o n deriving ODEs of the ﬁrst (non linear) class, which are less clo sely tied than the secon d to the classical theory of special functio ns. The computations wil l make hea vy use of q -expansions and the theory of modular forms. Howev er , special f unction metho ds such as hypergeometr ic transform ations will prove useful in dealing with forms on the Fricke extensions Γ + 0 ( N ) . The deriv a tion of ODEs for classical modular forms has recently been consid- ered by Ohy ama (e.g., in [3 0]) and Zud ilin [47]. Ou r ap proach differs from th eirs by being extensiv ely ‘mod ular , ’ in that it exploits dimension fo rmulas ( coming ultimately fro m the Rieman n–Roch theor em), explicit q -expansion s and nu mber- theoretic interpr etations of their coefﬁcients, etc. Also, Ohyama fo cuses on de riv- ing systems of nonlinear ODEs satisﬁed by weight-2 quasi-modu lar forms, analo- gous to the Eisenstein series E 2 on Γ ( 1 ) . His cou pled ODEs are of an in teresting quadra tic ty pe, called Darboux –Halphen systems [1, 22]. In the the present ar ti- cle the fund amental depen dent variables are weight-1 modular fo rms, sometimes multiv alued, which are analogo us to E 1 / 4 4 , E 1 / 6 6 , and ∆ 1 / 12 ; and we regard th e resulting differential systems as mor e fundam ental than Da rboux –Halphen on es, though quasi-mod ular forms play a role. Zudilin focuses on deri ving systems, and also linear ODE s o f the seco nd ty pe disting uished above, which are satisﬁed by forms on Γ 0 ( N ) , Γ + 0 ( N ) , or on subg roups of Γ + 0 ( N ) . W e are able to c larify the modular underpin nings of his ODEs, and deriv e se veral more such equations. The weigh t-1 forms stu died below in clude trip les of forms on Γ 0 ( N ) , N = 2 , 3 , 4 , which we den ote A r , B r , C r , r = 4 , 3 , 2. They were intr oduced as fu nctions on H by the Borweins [7] in their study of alternati ve A GM (arith metic-geom etric mean) iterations. Their work was inspired by Ramanujan’ s theory of elliptic func- tions to alternati ve bases, the base being speciﬁed by the ‘signature’ r . (Cf. Berndt et al. [3].) Our appr oach places these fu nctions ﬁr mly in a modu lar setting (see also [27]). As a byp roduc t o f the an alysis o f these mo dular for ms a nd th eir po wers, we derive many di v isor function and theta id entities. A mino r example is Jacobi’ s Six Squares Theo rem; o ur proof of it may be the most explicitly m odular one to date. (See Th m. 3.8.) W e also give a modular interpretation of the second com - plete ellip tic integral E , prob ably for th e ﬁrst time, by iden tifying it a s a weigh t-1 form on Γ 0 ( 4 ) with an explicit, quasi-modular transformation law . (See Prop. 5.1 .) The fundamental goal of this article, however , is the dev elopmen t of a mod ular theory of ‘nonlinear’ special functions, b y determining which in tegrable nonlinear ODEs ( gener alized Cha zy equations , in our terminology ) can arise in certain well- speciﬁed modular contexts. (See Thms. 2.3 and 7.1; and the discussion in § 7.4.) 2. Motivation and the ﬁrst theorem As initial motiv ation, consider forms on the full modular group Γ ( 1 ) = P SL ( 2 , Z ) and their differential relations. V an der Pol [32, § 13] and Rank in [37] p roved that the modu lar discrimin ant f unction ∆ on H = { ℑ τ > 0 } , which when viewed as a function of q : = exp ( 2 π i τ ) , | q | < 1 , is deﬁned by ∆ ( q ) = q ∞ ∏ n = 1 ( 1 − q n ) 24 , (2.1) Nonlinear dif ferential equations for classical modular forms 3 satisﬁes the nonlinea r , fourth-or der h omog eneous dif feren tial equation 2 ∆ 3 ∆ ′′′′ − 10 ∆ 2 ∆ ′ ∆ ′′′ − 3 ∆ 2 ∆ ′′ 2 + 24 ∆ ∆ ′ 2 ∆ ′′ − 13 ∆ ′ 4 = 0 , (2.2) where ′ signiﬁes the derivation q d / d q = ( 2 π i ) − 1 d / d τ . (The deriv ation d / d τ will be indica ted by a d ot; the prime s in ( 2.2) can be optionally r eplaced by do ts.) This ODE is fairly well known, as is Jaco bi’ s thir d-ord er one for h is theta-nu ll function s ϑ i , i = 2 , 3 , 4 [24]. (F or remarks on the latter , see [15 ].) In the following, the parallels between them will be brough t out. One approach to understanding the rather complicated Eq. (2.2) i s to treat i t as a corollar y of a much nicer nonlinear third-ord er ODE [39, 32], nam ely ... u − 12 u ¨ u + 18 ˙ u 2 = 0 , (2.3a) i.e., 2 E ′′′ 2 − 2 E 2 E ′′ 2 + 3 E ′ 2 2 = 0 . (2.3b) Here, u = ( 2 π i / 12 ) E 2 = π i E 2 / 6 , an d E 2 is the secon d (n ormalized) Eisenstein series on the full modular group. The Eisenstein series E k = E 1 , 1 k on Γ ( 1 ) ar e E k ( q ) = 1 + a k ∞ ∑ n = 1 σ k − 1 ( n ) q n = 1 + a k ∞ ∑ n = 1 n k − 1 q n 1 − q n , σ k ( n ) = ∑ d | n d k , a k = 2 ζ ( 1 − k ) = 2 L ( 1 − k , 1 ) = − 2 k B k , where B k is the k th Bern oulli num ber; so a 2 , a 4 , a 6 , . . . are − 24 , 240 , − 5 04 , . . . . Th e nonlinear ODE ( 2.3a) is a so- called Chazy e quation, with the interesting analytic proper ty of h aving solutions with a natural boun dary (e. g., ℑ τ = 0 or | q | = 1), beyond which they cannot be continued ; much as is the case with a lacunary series. (See [2, pp. 342–3] and [11].) Substituting E 2 = ∆ ′ / ∆ into (2.3b) yields (2.2). Equation (2.3b), in tu rn, f ollows f rom a result of Ram anujan. He introd uced function s P , Q , R on the disk | q | < 1 , deﬁned by co n vergent q -series, which ar e identical to E 2 , E 4 , E 6 . That is, the y are respectively a q uasi-mod ular for m of weight 2 and depth 6 1 , and modular forms of weights 4 and 6. He d etermined the differ - ential structure on the ring C [ E 2 , E 4 , E 6 ] b y showing that ( E 4 3 ) ′ = E 2 · E 4 3 − E 4 2 E 6 , (2.4a) ( E 6 2 ) ′ = E 2 · E 6 2 − E 4 2 E 6 , (2.4b) ∆ ′ = E 2 · ∆ , (2.4c) 12 E 2 ′ = E 2 · E 2 − E 4 , (2.4d) where Eqs. (2. 4abc) are linea rly d epende nt, since E 4 3 = E 6 2 + 12 3 ∆ , which is an equality between weight-12 modular forms. B y rewriting the system (2.4abd) into 4 Robert S. Maier a single third -order equ ation fo r E 2 , one obtains Eq . (2 .3b). It is worth n oting for later use that the system (2.4abcd) can be rewritten as ( A 12 ) ′ = E · A 12 − A 8 B 6 , (2.5a) ( B 12 ) ′ = E · B 12 − A 8 B 6 , ( 2.5b) ( C 12 ) ′ = E · C 12 , (2.5c) 12 E ′ = E · E − A 4 , (2. 5d) where A , B , C ; E are respectively E 1 / 4 4 , E 1 / 6 6 , ( 12 3 ∆ ) 1 / 12 ; E 2 . Of these, A , B , C are fo rmally weight-1 form s for Γ ( 1 ) ; but the ﬁrst tw o are m ultiv alued on H . (T heir q -expansion s, the integer coefﬁcients of which lack an arith metical interpretation , do not conver ge on all of | q | < 1 .) Some o f Ramanujan’ s results alo ng this lin e were sub sequently extend ed by Ramamani ([35]; see also [ 36]). Sh e intro duced three q -series somewhat similar to his P , Q , R , and derived a coupled system of ﬁrst-orde r ODEs tha t they satisfy . Recently , Ablowitz, Chakrav arty and Hahn ([1]; see also [21]) sh owed that h er q -series deﬁne m odular fo rms on the Hecke su bgrou p Γ 0 ( 2 ) < Γ ( 1 ) , inc luding a we ight-2 qu asi-modu lar form analog ous to E 2 , an d de riv ed a sing le no nlinear third-or der ODE that it satisﬁes. This turn s o ut to be a Chazy-like equ ation, of a general type ﬁrst studied by Bureau [9]. One may wonder whether these results can be generalized , by extending them to other mo dular subgrou ps. The question is answered in the afﬁrmati ve by Theo- rem 2.3 below , which provides a un iﬁed treatmen t of certain Eisenstein series o n the subg roup s Γ 0 ( 2 ) , Γ 0 ( 3 ) , Γ 0 ( 4 ) . For the latter two as well as for Γ 0 ( 2 ) , a n onlinear third-or der ODE is satisﬁed by a quasi-mod ular form of weight 2. A uniﬁed treat- ment is facilitated by the fact th at up to isomorp hism, these ar e th e only genus-ze ro proper sub group s of Γ ( 1 ) that have e xactly thr ee inequiv alent ﬁxed points on H ∗ ; with the exception o f the prin cipal mod ular subg roup Γ ( 2 ) , which is co njugated to Γ 0 ( 4 ) by the 2-isogeny τ 7→ 2 τ in PSL ( 2 , R ) ; and also the i ndex-2 subgroup Γ 2 , which is a bit anomalo us. The statement of the theorem requires Deﬁnition 2.1. If u is a holomorphic f unction on H , deﬁne functions u 4 , u 6 , u 8 , . . . by u 4 : = ˙ u − u 2 and u k + 2 : = ˙ u k − kuu k . Thus, u 4 = ˙ u − u 2 , u 6 = ¨ u − 6 u ˙ u + 4 u 3 , u 8 = ... u − 12 u ¨ u − 6 ˙ u 2 + 48 u 2 ˙ u − 24 u 4 . A gener alized Chazy equation C p for u is a differential equatio n of the form p = 0 , where p ∈ C [ u 4 , u 6 , u 8 ] is a non zero polynomial, homogen eous in that t he weights of its monom ials are equal. Here, the weight of u a 4 u b 6 u c 8 is 4 a + 6 b + 8 c . Remark. The classical Chazy e quation, Eq. (2.3a), has p = u 8 + 24 u 2 4 . Th e so- called Chaz y–XII class [11] in cludes equation s C p with p = u 8 + con st · u 2 4 . This further genera lization is preﬁgured by the t reatmen t of Clarkson and Olv er [12]. Nonlinear dif ferential equations for classical modular forms 5 Deﬁnition 2.2. For any χ : Z / N Z → C , deﬁne the χ -weighted di visor and conju- gate divisor functions σ k ( n ; χ ) = ∑ d | n χ ( d mod N ) d k , σ c k ( n ; χ ) = ∑ d | n χ (( n / d ) mo d N ) d k . Such weighted di v isor functions, with χ not necessarily a Dirichlet character , have been considered by Glaisher [19], Fine [17, §§ 32 an d 3 3], and others. The argu- ment χ will usually be written out in full, as χ ( 0 ) , . . . , χ ( N − 1 ) . Results attached to Γ 0 ( 2 ) , Γ 0 ( 3 ) , Γ 0 ( 4 ) will be referred to as belong ing to Ra- manujan’ s theories o f sign ature 4 , 3 , 2 , respectively . The ﬁxed points o n H ∗ of each gr oup includ e (the eq uiv alence classes of) two cusps, nam ely the inﬁnite cusp τ = i ∞ (i.e., q = 0) and the cusp τ = 0; and a lso a third ﬁxed poin t, which for Γ 0 ( 2 ) is the quad ratic ellip tic p oint τ = i , fo r Γ 0 ( 3 ) is the cubic ellip tic point τ = ζ 3 : = exp ( 2 π i / 3 ) , and for Γ 0 ( 4 ) is an additional cusp, namely τ = 1 / 2. (F or a revie w of these facts, and fo r trian gular fun damental d omains th e v ertices of which are these ﬁxed points, see, e.g., [41].) Theorem 2.3. On each modu lar subgr o up Γ 0 ( N ) , N = 2 , 3 , 4 , i.e., for each o f the corr esponding signatures r = 4 , 3 , 2 , the following are true. 1. Ther e is a quasi-modular form E r of weight 2 and depth 6 1 , eq ualing unity a t the in ﬁnite cusp, such that u = ( 2 π i / r ) E r satisﬁes a generalized Cha zy equa- tion C p r , fo r some polynomial p r . Namely , E 4 ( q ) = 1 3  4 E 2 ( q 2 ) − E 2 ( q )  = 1 + 8 ∞ ∑ n = 1 σ 1 ( n ; − 1 , 1 ) q n = 1 + 8 ∞ ∑ n = 1 σ c 1 ( n ; − 3 , 1 ) q n , E 3 ( q ) = 1 8  9 E 2 ( q 3 ) − E 2 ( q )  = 1 + 3 ∞ ∑ n = 1 σ 1 ( n ; − 2 , 1 , 1 ) q n = 1 + 3 ∞ ∑ n = 1 σ c 1 ( n ; − 8 , 1 , 1 ) q n , E 2 ( q ) = 1 3  4 E 2 ( q 4 ) − E 2 ( q 2 )  = 1 + 4 ∞ ∑ n = 1 σ 1 ( n ; − 1 , 0 , 1 , 0 ) q n = 1 + 8 ∞ ∑ n = 1 σ c 1 ( n ; − 3 , 0 , 1 , 0 ) q n , so that E 2 ( q ) = E 4 ( q 2 ) . The polynomia ls p r ∈ C [ u 4 , u 6 , u 8 ] a r e p 4 = u 4 u 8 − u 2 6 + 8 u 3 4 , (2.6) p 3 = u 4 u 2 8 − u 2 6 u 8 + 24 u 3 4 u 8 − 15 u 2 4 u 2 6 + 144 u 5 4 , (2.7) p 2 = u 4 u 8 − u 2 6 + 8 u 3 4 , (2.8) so that p 2 = p 4 . 2. Ther e is a triple o f weig ht- 1 mod ular forms A r , B r , C r (allowed to have non - trivial [i.e ., non-Dirichlet] multiplier s ystems, and also allowed t o be multival- ued in the above sense of being r oo ts of con vention al [ single-va lued] modular forms), such that 6 Robert S. Maier (a) A r r = B r r + C r r , ea ch ter m being a single-valued weight-r form. (b) A r , B r , C r vanish respectively at (the e quivalenc e c lasses of) the ab ove- mentioned thir d ﬁxed po int, the cusp τ = 0 , an d the cusp τ = i ∞ ; and they vanish no wher e else. In each ca se, the or der of va nishing (comp uted with r espect to a local parameter for Γ 0 ( N ) ) is 1 / r . (c) A r r , B r r , C r r , together with E r , satisfy the co upled system o f non linear ﬁrst- or der equations ( A r r ) ′ = E r · A r r − A r 2 B r r , ( B r r ) ′ = E r · B r r − A r 2 B r r , ( C r r ) ′ = E r · C r r , r E ′ r = E r · E r − A r 4 − r B r r , fr om which the generalized Chazy equ ation C p r for E r can be de rived by elimination. (The thir d equation says that u = ˙ C r / C r .) Remark. The resu lts of van der Pol– Rankin and Raman ujan, attached to Γ ( 1 ) , cannot be subsumed into Thm. 2.3; but s ee the mor e general Theorem 7.1 below . Remark. For the subg roup Γ 0 ( 2 ) , i.e., wh en r = 4 , the co upled ODEs of The- orem 2.3(2) are equ iv ale nt to those of Rama mani [35], Ablowitz et al. [1], and Hahn [21]. (Their P , e [or f P ] , Q are the E 4 , A 4 2 , B 4 4 of the th eorem.) The non- linear third-order ODE of Jaco bi [24], which is satisﬁed by h is theta-null f unction s ϑ 2 , ϑ 3 , ϑ 4 on H , turns out to be a co rollary of the r = 2 case of th e theore m, since A 2 , B 2 , C 2 can be chosen to equal ϑ 3 2 , ϑ 4 2 , ϑ 2 2 . The body of th is article is laid out a s follo ws. I n § 3, the modular fo rms A r , B r , C r are deﬁned as eta p roduc ts an d q -series. (These functio ns on | q | < 1 were in- troduced by the Borweins [7] as the theta f unction s of ce rtain quadr atic for ms; see the Ap pendix . They play a ro le in Ramanujan’ s alternative theories of elliptic function s [3]. I n [2 7], we interpreted them as forms on Γ 0 ( 2 ) , Γ 0 ( 3 ) , Γ 0 ( 4 ) .) In pass- ing, we g enerate a tab le of q - expansions and d i visor-function identities (T able 1), of indep endent in terest, and give a modu lar p roof of Jacob i’ s Six Squares Th eo- rem. In § 4, we prove T heorem 2 .3(2) by exploiting the d imensionality of spaces of modular forms, i.e., by applying linear algebra to the graded ring C [ A r , B r , C r ] . Section 5 is a digression . From the r = 2 system, we der iv e a n elliptic in- tegral transfo rmation law , and differential relations for the ta-nulls th at imply Ja- cobi’ s no nlinear third- order ODE . Deriving intere sting identities is facilitated by the quasi-modu lar form E 2 ( q ) equaling (up to a transcendental constant factor) the ev en function K ( q ) E ( q ) , i.e., the pr oduct of the classical ﬁrst and second com plete elliptic integrals, viewed as fu nctions of the no me q . No satisfactorily ‘ modular ’ transform ation la w for E = E ( q ) has p reviously been deri ved. In § 6, we g iv e a direct pro of of the ge neralized Chazy eq uations o f Theo- rem 2.3(1). Th ey too can b e d erived by linear algebra. (Ind eed, fo r each r , the function s u 4 , u 6 , u 8 are modu lar form s of th e speciﬁed weight, with trivial m ulti- plier system s; cf. [45, Lemma 5].) W e give a second pr oof that is less explicitly modular, based on r esults of [ 27]. Each o f A r , B r , C r satisﬁes a ‘h ypergeome tric’ Nonlinear dif ferential equations for classical modular forms 7 Picard–Fuch s eq uation, which is a linear second-o rder ODE with three singula r points, th e in depend ent variable of which is a Hau ptmod ul for th e corre spond- ing grou p Γ 0 ( N ) . Moreover , τ is a ratio of solutions of this equa tion (cf . [18]). These facts m ake possible the second proof. Theorem 6.4 is an e xtension of Theo- rem 2.3(1), or equ iv a lently , a general result on solution s of Gauss hyp ergeometric equations. It r ev eals which generalized Chazy equations can arise fro m genus-zero subgrou ps of PSL ( 2 , R ) with thre e inequiv alent ﬁxed points. In § 7, a comp arable extension o f Th eorem 2.3(2) is o btained. Theo rem 7 .1, derived u sing ODE man ipulations like those of Oh yama [30], presents the sy s- tem of nonlinear ﬁrst-orde r ODEs, s atisﬁed by a trip le of weight-1 modular forms A , B , C , that arises from any speciﬁed triang le subgroup of PS L ( 2 , R ) ; i.e., fro m its Picard –Fuchs equation. As examples, we tr eat the nine tr iangle gro ups co m- mensurab le with Γ ( 1 ) . (Gen eralized Darbou x–Halph en system s on these g roups have been obtained by Harnad and McKay [ 22].) The systems we derive in §§ 7.2 and 7.3 include a ‘T ype I I’ one that subsume s Ram anujan’ s sy stem (2 .5abcd) on Γ ( 1 ) , and also applies to the Fricke extension s Γ + 0 ( N ) , N = 2 , 3 . A ‘ T ype III ’ system, associated to index-2 subgroup s of these three groups, is deriv ed as well. 3. Modular forms and divisor function identities The modular forms A r , B r , C r , r = 4 , 3 , 2 , of wh ich o nly A 4 is multiv alued on H , will b e deﬁn ed here in term s of the Dedekind eta functio n, r ather than un iv a riate or multiv ariate theta functions. In the Appendix, s everal of the original deﬁnitions of the Borweins [7] are reprod uced, as are A GM identities these forms satisfy . Being a (single-valued) form h as its usual meaning . On H ∗ = H ∪ P 1 ( Q ) , i.e., H ∪ Q ∪ { i ∞ } , a holom orphic functio n f is mo dular o f integral weigh t k on some Γ < Γ ( 1 ) if f ( a τ + b c τ + d ) = ˆ χ ( a , b , c , d )( c τ + d ) k f ( τ ) fo r all ±  a b c d  ∈ Γ . Here, ˆ χ is a C × -valued m ultiplier system, with ˆ χ ( − a , − b , − c , − d ) equaling ( − 1 ) k χ ( a , b , c , d ) . The simplest case, occurring if Γ < Γ 0 ( N ) for some N , is when ˆ χ ( a , b , c , d ) equals χ ( d ) , the extension to Z of som e Dirichlet ch aracter χ : ( Z / N Z ) × → C × , satis- fying χ ( − 1 ) = ( − 1 ) k . By d eﬁnition, χ ( d ) = 0 if ( d , N ) > 1 , whe re ( · , · ) is the g.c.d. The n otation 1 N for th e p rincipal ch aracter mod N , satisfy ing 1 N ( d ) = 1 if ( d , N ) = 1 , will be used. The trivial character of period 1 will be denoted 1 . In terms o f q , the Dedek ind eta functio n equals q 1 / 24 ∏ ∞ n = 1 ( 1 − q ) n . On Γ ( 1 ) , it transform s as [34 ] η ( a τ + b c τ + d ) = (  d c  ζ 3 ( 1 − c )+ bd ( 1 − c 2 )+ c ( a + d ) 24 [ − i ( c τ + d )] 1 / 2 η ( τ ) , c odd ,  c d  ζ 3 d + ac ( 1 − d 2 )+ d ( b − c ) 24 [ − i ( c τ + d )] 1 / 2 η ( τ ) , d odd , (3.1) if c > 0 , wh ere ζ 24 : = exp ( 2 π i / 24 ) , an d the Jaco bi symb ol is taken to satisfy  c − d  =  c d  . Fine’ s notation [ δ ] for the f unction τ 7→ η ( δ τ ) o n H ∗ will be used, so that, e.g ., ∆ = [ 1 ] 24 . At any cu sp s = a d ∈ Q ∪ { i ∞ } ( in lowest term s, w ith 1 0 signifying i ∞ ), the ord er of vanishing of η ( δ τ ) , deno ted o rd s ([ δ ]) , is given by a well-known formula stated in Ref. [28], ord s ([ δ ]) = 1 24 ( δ , d ) 2 / δ . (3.2) 8 Robert S. Maier Here, ord s ( · ) is com puted with respe ct to a local par ameter on the quo tient c urve X ( 1 ) = Γ ( 1 ) \ H ∗ , such as the Klein –W eber j -inv ariant (which equals E 4 3 / ∆ = 12 3 E 4 3 / ( E 4 3 − E 6 2 ) and is a Hauptm odul for Γ ( 1 ) ). As usual, or d i ∞ ( f ) is the lowest po wer of q in the Fourier e xpan sion of f . If f is a mo dular form on Γ , its order of vanishing at a cusp s ∈ H ∗ , compu ted with respect to a local parameter for Γ (i.e., on the quotient curve X = Γ \ H ∗ ) is Ord s , Γ ( f ) : = h Γ ( s ) · o rd s ( f ) , (3.3) Here, h Γ ( s ) is the multip licity with which the image of s in X is map ped to X ( 1 ) , i.e., th e wid th of the cusp s . If s ∈ H ∗ is n ot a cusp but rathe r a quad ratic or c ubic elliptic ﬁxed point of Γ (implying that s ∈ H ), then by deﬁnitio n s will be mapped doubly , resp. triply to X . In this case, ord s ( f ) = ( 2 , resp. 3 ) · Ord s , Γ ( f ) , (3.4) where ord s ( f ) is th e order of vanishing of f at the point s ∈ H in the con ventional sense of analytic fun ctions. If f has n o po les and is single-valued on H , i.e., h as no branch points, then this order must be a non-negative integer . In the case Γ = Γ 0 ( N ) , the inequiv alent cusps τ = a d on H ∗ may be taken to be the fr actions a d ∈ Q with d | N , 1 6 a 6 N , and with a reduce d m odulo ( d , N / d ) while remainin g copr ime to d . (E .g., the cusps of Γ 0 ( N ) would b e 1 1 , 1 2 if N = 2; 1 1 , 1 3 if N = 3; a nd 1 1 , 1 2 , 1 4 if N = 4. Note that 1 1 ∼ 0 and 1 N ∼ i ∞ und er Γ 0 ( N ) .) If this co n vention is ad hered to, then each inequivalent cusp a d will have width h Γ 0 ( N ) ( a d ) = e d , N : = N / d ( d , N / d ) . Deﬁnition 3.1. A r , B r , C r , r = 4 , 3 , 2 , ar e certain functions on H ∗ , d eﬁned to ha ve the eta-prod uct representations A 4 = ( 2 6 · [ 2 ] 24 + [ 1 ] 24 ) 1 / 4 / [ 1 ] 2 [ 2 ] 2 , B 4 = [ 1 ] 4 / [ 2 ] 2 , C 4 = 2 3 / 2 · [ 2 ] 4 / [ 1 ] 2 ; A 3 = ( 3 3 · [ 3 ] 12 + [ 1 ] 12 ) 1 / 3 / [ 1 ][ 3 ] , B 3 = [ 1 ] 3 / [ 3 ] , C 3 = 3 · [ 3 ] 3 / [ 1 ] ; A 2 = ( 2 4 · [ 4 ] 8 + [ 1 ] 8 ) 1 / 2 / [ 2 ] 2 , B 2 = [ 1 ] 4 / [ 2 ] 2 , C 2 = 2 2 · [ 4 ] 4 / [ 2 ] 2 , so that by d eﬁnition, A r r = B r r + C r r . A t the in ﬁnite cusp ( i.e., at q = 0 ), each A r and B r equals un ity , and each C r vanishes. The A r , deﬁned as r oots of single- valued modu lar forms, ar e p otentially mu lti valued, but it will be sho wn that A 2 , A 3 are single-valued. One notes t hat B 4 = B 2 and C 4 ( q ) = 2 − 1 / 2 · C 2 ( q 1 / 2 ) . Remark. Connections to th eta fun ctions, such as Jacobi’ s theta-nu lls ϑ 2 , ϑ 3 , ϑ 4 , will be discussed i n § 5. (Also, see the App endix.) For the mo ment, observe th at by theta identities ﬁrst proved by Euler, or altern ativ ely by the Jacob i triple p roduct formu la, A 2 , B 2 , C 2 equal ϑ 3 2 , ϑ 4 2 , ϑ 2 2 . Sim ilarly , A 4 2 = ϑ 2 4 + ϑ 3 4 , B 4 = ϑ 4 2 , and C 4 = √ 2 A 2 C 2 = 2 1 / 2 · ϑ 2 ϑ 3 . Nonlinear dif ferential equations for classical modular forms 9 Proposition 3. 2. A r , B r , C r , r = 4 , 3 , 2 a r e weight - 1 modu lar forms o n the sub - gr o ups Γ 0 ( N ) , N = 2 , 3 , 4 , r espectively , w ith each bein g single-valued save fo r A 4 , the squa r e o f which is sing le-valued . Each has exactly o ne equivale nce cla ss of zer oes on H ∗ , at which its order of vanishing is 1 / r (compu ted with respect to a local parameter for Γ 0 ( N ) ), lo cated as stated in Theo r em 2 .3 . Under the F ricke in volution W N : τ 7→ − 1 / N τ for Γ 0 ( N ) , B r and C r ar e inter changed in the sense that B r 2 | W N = − C r 2 , and A r 2 is ne ga ted. Th er e is an alternative, e xplicitly s ingle- valued r epr esentation for A 2 , n amely A 2 = [ 2 ] 10 / [ 1 ] 4 [ 4 ] 4 . Pr oo f. It follows fr om ( 3.2 ) that fo r each r , ord i ∞ ( B r ) = 0 , ord i ∞ ( C r ) = 1 / r , and ord 0 ( C r ) = 0; and for r = 4 , 3 , 2 , that ord 0 ( B r ) = 1 / 9 , 1 / 8 , 1 / 9. Also, ord 1 / 2 ( B 2 ) = ord 1 / 2 ( C 2 ) = 0. The cu sps τ = 0 , i ∞ of Γ 0 ( 2 ) , Γ 0 ( 3 ) h av e width s 2 , 1 and 3 , 1 , and th e cusps τ = 0 , 1 2 , i ∞ of Γ 0 ( 4 ) have wid ths 4 , 1 , 1. It follows from ( 3.3) that the ord er of B r , C r at each cusp is zero, e xcept at τ = 0 , i ∞ re spectiv ely , where the big-O order in each case equals 1 / r , as claimed. T o prove the claim abou t the zeroes of A r , note tha t t 2 = 2 12 · [ 2 ] 24 / [ 1 ] 24 , t 3 = 3 6 · [ 3 ] 12 / [ 1 ] 12 , t 4 = 2 8 · [ 4 ] 8 / [ 1 ] 8 are Ha uptmod uls for Γ 0 ( 2 ) , Γ 0 ( 3 ) , Γ 0 ( 4 ) , i.e., ra tional p arameters fo r the associated quotient cu rves X 0 ( N ) . Each vanishes at the cusp τ = i ∞ and has a pole at the cusp τ = 0. (See [27]; the normal- ization factors are unimpo rtant here.) By construction , A 4 / B 4 = ( 1 + t 2 / 2 6 ) 1 / 4 , A 3 / B 3 = ( 1 + t 3 / 3 3 ) 1 / 3 , and A 2 / B 2 = ( 1 + t 4 / 2 4 ) 1 / 2 . Hence, Ord 0 , Γ 0 ( N ) ( A r ) = Ord 0 , Γ 0 ( N ) ( B r ) + Ord 0 , Γ 0 ( N ) ( A r / B r ) = 1 / r − 1 / r = 0 , (3.5) i.e., each A r must be r egular an d n onzero at th e cusp τ = 0. Also, each of these quotients A r / B r is zero at th e third ﬁxed poin t; see [27, T able 2]. It fo llows that A r must ha ve big-O order at the third ﬁx ed p oint eq ual to 1 / r . The thir d ﬁxed p oint is quadratic, r esp. cubic, for r = 4 , resp. r = 3; hence by (3.4), the small-o order of vanishing there will be 2 · ( 1 / 4 ) = 1 / 2 , resp. 3 · ( 1 / 3 ) = 1. One concludes that A 4 has quadratic branch points on H , but its square and A 3 are single-valued. The statements abou t th e Fricke inv olution f ollow readily from the tran sforma- tion law η ( − 1 / τ ) = ( − i τ ) 1 / 2 η ( τ ) and the deﬁnitions of A r , B r , C r . T o prove th at A 2 = [ 2 ] 10 / [ 1 ] 4 [ 4 ] 4 , observe that A 2 / { [ 2 ] 10 / [ 1 ] 4 [ 4 ] 4 } has zero order of vanishing at each of the three inequiv alent cusps of Γ 0 ( 4 ) .  Proposition 3.3. 1. On Γ 0 ( 2 ) , A 4 2 and A 4 4 , B 4 4 , C 4 4 have trivial character 1 2 ( d ) , which ta kes d ≡ 1 ( mod 2 ) to 1 . 2. On Γ 0 ( 3 ) , A 3 and A 3 3 , B 3 3 , C 3 3 have quad ratic character χ − 3 ( d ) : =  − 3 d  =  d 3  , which takes d ≡ 1 , 2 ( mod 3 ) to 1 , − 1 , and A 3 2 has trivial character 1 3 ( d ) , which takes d ≡ 1 , 2 ( mod 3 ) to 1 . 3. On Γ 0 ( 4 ) , A 2 has quad ratic character χ − 4 ( d ) : =  − 4 d  , which takes d ≡ 1 , 3 ( mod 4 ) to 1 , − 1 , and A 2 2 , B 2 2 , C 2 2 have trivial c haracter 1 4 ( d ) , which takes d ≡ 1 , 3 ( mod 4 ) to 1 . 10 Robert S. Maier Pr oo f. T o prove ea ch statement, v erify it on a generating set for the speciﬁed sub- group , using the transfor mation law (3. 1). For example, Γ 0 ( 3 ) has (m inimal) gen- erating set ±  1 1 0 1  , ±  1 1 − 3 − 2  , an d for each of the associated maps τ 7→ a τ + b c τ + d , the power of ζ 24 appearin g in the transformation law for B 3 3 , d educed from (3.1 ), is consistent with the Dirichlet character χ − 3 . Th e sam e is tru e for C 3 3 ; he nce for A 3 3 as well, since A 3 3 = B 3 3 + C 3 3 . Hence, the claim in volving A 3 3 , B 3 3 , C 3 3 is proved. Further details are left to the reader .  The fo rmulas fo r dim M k ( Γ 0 ( N )) an d dim S k ( Γ 0 ( N )) , the dim ensions of the vector spa ces of a ll modular fo rms an d o f cu sp form s on Γ 0 ( N ) of weigh t k , with trivial character, are well kno wn [13, 14]. For Γ 0 ( 2 ) , Γ 0 ( 3 ) , Γ 0 ( 4 ) , the s paces M 2 , M 4 have d imensions 1 , 2; 1 , 2 ; 2 , 3 respectiv ely; and there are no cusp forms of weight 2 or 4 . Also, dim M 6 ( Γ 0 ( 2 )) = 2 , and th ere are no cusp for ms of weig ht 6 on Γ 0 ( 2 ) . Similarly , M 1 ( Γ 0 ( 3 ) , χ − 3 ) , M 3 ( Γ 0 ( 3 ) , χ − 3 ) , M 5 ( Γ 0 ( 3 ) , χ − 3 ) have dimen sions 1 , 2 , 2 , and M 1 ( Γ 0 ( 4 ) , χ − 4 ) , M 3 ( Γ 0 ( 4 ) , χ − 4 ) have dimensions 1 , 2 , cusp forms being ab- sent in all cases. In the absence of cusp f orms, all mod ular forms in the prec eding spaces are combin ations of Eisenstein s eries. Proposition 3.4. The following spanning r elation s hold. 1. M 2 ( Γ 0 ( 2 )) =  A 4 2  , M 4 ( Γ 0 ( 2 )) =  A 4 4 , B 4 4  , M 6 ( Γ 0 ( 2 )) =  A 4 6 , A 4 2 B 4 4  . 2. M 1 ( Γ 0 ( 3 ) , χ − 3 ) =  A 3  , M 2 ( Γ 0 ( 3 )) =  A 3 2  , M 3 ( Γ 0 ( 3 ) , χ − 3 ) =  A 3 3 , B 3 3  , M 4 ( Γ 0 ( 3 )) =  A 3 4 , A 3 B 3 3  , M 5 ( Γ 0 ( 3 ) , χ − 3 ) =  A 3 5 , A 3 2 B 3 3  . 3. M 1 ( Γ 0 ( 4 ) , χ − 4 ) =  A 2  , M 2 ( Γ 0 ( 4 )) =  A 2 2 , B 2 2  , M 3 ( Γ 0 ( 4 ) , χ − 4 ) =  A 2 3 , A 2 B 2 2  , M 4 ( Γ 0 ( 4 )) =  A 2 4 , B 2 4 , A 2 2 B 2 2  . Pr oo f. Immediate , by Proposition 3.3 and dimension consideration s.  Let M e ven ( Γ ) denote the graded ring of even-weight modu lar form s on Γ . By exploiting the v alence formula one can prove the follo wing generalization. Proposition 3.5. 1. M e ven ( Γ 0 ( 2 )) = C [ A 4 2 , B 4 4 ] = C [ A 4 2 , B 4 4 − C 4 4 ] . 2. M e ven ( Γ 0 ( 3 )) = C [ A 3 2 , A 3 B 3 3 ] = C [ A 3 2 , A 3 ( B 3 3 − C 3 3 )] . 3. M e ven ( Γ 0 ( 4 )) = C [ A 2 2 , B 2 2 ] = C [ A 2 2 , B 2 2 − C 2 2 ] . In th e sequ el, som e stand ard Eisenstein machinery will be used. (Cf. [14, Thms. 4 .5.2, 4 .6.2, 4 .8.1]. ) Let a sub group Γ 0 ( N ) , N > 2 , and an in teger weigh t k > 1 be spe ciﬁed. L et a Dirichlet c haracter χ : ( Z / N Z ) × → C × , extended to Z , satisfying χ ( − 1 ) = ( − 1 ) k , also be given. The c onduc tor ( primitive p eriod) o f χ will divide N . For each p air ψ , φ of Dirichlet charac ters, the con ductors u , v of which satisfy uv | N and for which ψ φ = χ (the e quality being o ne of ch aracters mod N ) , there is an Eisenstein series E ψ , φ k ∈ M k ( Γ 0 ( N ) , χ ) , namely E ψ , φ k ( q ) : = ( 1 + 2 L ( 1 − k , φ ) · ˆ E k ( 1 , φ ) , if ψ = 1 , 2 · ˆ E k ( ψ , φ ) , if ψ 6 = 1 , (3.6) Nonlinear dif ferential equations for classical modular forms 11 where ˆ E k ( ψ , φ ) : = ∞ ∑ n = 1 " ∑ 0 < d | n ψ ( n / d ) φ ( d ) d k − 1 # q n = ∞ ∑ e = 1 ∞ ∑ d = 1 ψ ( e ) φ ( d ) d k − 1 q ed , (3.7) and the L -series value L ( 1 − k , φ ) lies in the extension o f Q by the values of φ . In the case when χ = 1 N , the principa l char acter m od N , these Eisen stein series are o f the f orm E ψ , ψ − 1 k , wher e ψ ranges over the ch aracters mod N with condu c- tor u , subject to u 2 | N . The subcase ψ = 1 is special: E 1 , 1 k reduces to E k , the k th Eisenstein series on Γ ( 1 ) . If k > 3 , the co llection of all E ψ , φ k ( q ℓ ) , where E ψ , φ k is of the ab ove fo rm and 0 < ℓ | N / ( uv ) , is a ba sis for M k ( Γ 0 ( N ) , χ ) / S k ( Γ 0 ( N ) , χ ) . For instance, if χ = 1 N then these series are of the form E ψ , ψ − 1 k ( q ℓ ) with 0 < ℓ | N / u 2 , and are eq uinum er- ous with the cusps of Γ 0 ( N ) , of which there are ∑ 0 < d | N ϕ (( d , N / d )) in all. ( Here, ( · , · ) an d ϕ ( · ) are th e g .c.d. and totient fu nctions.) But if k 6 2 , the precedin g statements mu st be mod iﬁed. When k 6 2 , the Eisenstein series E ψ , φ k ( q ℓ ) are q uasi- modular but in g eneral are n ot mo dular, so the q uotient M k ( Γ 0 ( N ) , χ ) / S k ( Γ 0 ( N ) , χ ) is a prope r subspace of their span. Proposition 3.6. One h as the Eisenstein series an d divisor-function r epr esenta- tions sho wn in T able 1 , for mon omials in A r , B r , r = 4 , 3 , 2 , with multiplier sys- tems of qua dratic Dirichlet-char acter type. (Th e o nes involving C r ar e included for completeness; they follow fr om A r r = B r r + C r r .) Pr oo f. For each of the monomials in Proposition 3.4, by working out the ﬁrst fe w coefﬁcients in its q -expan sion one determines th e Eisenstein representa tion given in the rightmo st column, and hence the full q -expansion . This is a matter of linear algebra, since M k ( Γ 0 ( 2 )) ⊆ h E k ( q 2 ) , E k ( q ) i , k = 2 , 4 , 6 , M k ( Γ 0 ( 3 )) ⊆ h E k ( q 3 ) , E k ( q ) i , k = 2 , 4 , M k ( Γ 0 ( 4 )) ⊆ h E k ( q 4 ) , E k ( q 2 ) , E k ( q ) i , k = 2 , 4 , M k ( Γ 0 ( 3 ) , χ − 3 ) ⊆ h E 1 , χ − 3 k , E χ − 3 , 1 k i , k = 1 , 3 , 5 , M k ( Γ 0 ( 4 ) , χ − 4 ) ⊆ h E 1 , χ − 4 k , E χ − 4 , 1 k i , k = 1 , 3 , where ⊆ signiﬁes ⊂ if k 6 2 and = if k > 2. T he L -series values L ( 1 − k , χ − 3 ) and L ( 1 − k , χ − 4 ) ar e compu ted f rom L ( 1 − k , φ ) = − B k , φ / k an d the gen eralized Bernoulli formula for any Dirichlet character φ to the modu lus N [26], ∞ ∑ k = 0 B k , φ x k k ! = x e N x − 1 N − 1 ∑ a = 0 φ ( a ) e ax .  12 Robert S. Maier T able 1. M 2 ( Γ 0 ( 2 )) : A 4 2 = 1 + 24 ∑ σ 1 ( n ; 0 , 1 ) q n = 2 E 2 ( q 2 ) − E 2 ( q ) M 4 ( Γ 0 ( 2 )) : A 4 4 = 1 + 24 ∑ σ 3 ( n ; 3 , 2 ) q n = 1 5  4 E 4 ( q 2 ) + E 4 ( q )  B 4 4 = 1 − 16 ∑ σ 3 ( n ; − 1 , 1 ) q n = 1 15  16 E 4 ( q 2 ) − E 4 ( q )  C 4 4 = 8 ∑ σ 3 ( n ; 7 , 8 ) q n = − 4 15  E 4 ( q 2 ) − E 4 ( q )  M 6 ( Γ 0 ( 2 )) : A 4 6 = 1 + 18 ∑ σ 5 ( n ; 3 , 4 ) q n = 1 7  8 E 6 ( q 2 ) − E 6 ( q )  A 4 2 B 4 4 = 1 + 8 ∑ σ 5 ( n ; − 1 , 1 ) q n = 1 63  64 E 6 ( q 2 ) − E 6 ( q )  A 4 2 C 4 4 = 2 ∑ σ 5 ( n ; 3 1 , 32 ) q n = 8 63  E 6 ( q 2 ) − E 6 ( q )  M 1 ( Γ 0 ( 3 ) , χ − 3 ) : A 3 = 1 + 6 ∑ σ 0 ( n ; 0 , 1 , − 1 ) q n = E 1 , χ − 3 1 ( q ) M 2 ( Γ 0 ( 3 )) : A 3 2 = 1 + 12 ∑ σ 1 ( n ; 0 , 1 , 1 ) q n = 1 2  3 E 2 ( q 3 ) − E 2 ( q )  M 3 ( Γ 0 ( 3 ) , χ − 3 ) : A 3 3 = B 3 3 + C 3 3 (see belo w) = E 1 , χ − 3 3 ( q ) + 27 2 E χ − 3 , 1 3 ( q ) B 3 3 = 1 − 9 ∑ σ 2 ( n ; 0 , 1 , − 1 ) q n = E 1 , χ − 3 3 ( q ) C 3 3 = 27 ∑ σ c 2 ( n ; 0 , 1 , − 1 ) q n = 27 2 E χ − 3 , 1 3 ( q ) M 4 ( Γ 0 ( 3 )) : A 3 4 = 1 + 8 ∑ σ 3 ( n ; 4 , 3 , 3 ) q n = 1 10  9 E 4 ( q 3 ) + E 4 ( q )  A 3 B 3 3 = 1 − 3 ∑ σ 3 ( n ; − 2 , 1 , 1 ) q n = 1 80  81 E 4 ( q 3 ) − E 4 ( q )  A 3 C 3 3 = ∑ σ 3 ( n ; 2 6 , 27 , 27 ) q n = − 9 80  E 4 ( q 3 ) − E 4 ( q )  M 5 ( Γ 0 ( 3 ) , χ − 3 ) : A 3 5 = A 3 2 B 3 3 + A 3 2 C 3 3 (see belo w) = E 1 , χ − 3 5 ( q ) + 27 2 E χ − 3 , 1 5 ( q ) A 3 2 B 3 3 = 1 + 3 ∑ σ 4 ( n ; 0 , 1 , − 1 ) q n = E 1 , χ − 3 5 ( q ) A 3 2 C 3 3 = 27 ∑ σ c 4 ( n ; 0 , 1 , − 1 ) q n = 27 2 E χ − 3 , 1 5 ( q ) M 1 ( Γ 0 ( 4 ) , χ − 4 ) : A 2 = 1 + 4 ∑ σ 0 ( n ; 0 , 1 , 0 , − 1 ) q n = E 1 , χ − 4 1 ( q ) M 2 ( Γ 0 ( 4 )) : A 2 2 = 1 + 8 ∑ σ 1 ( n ; 0 , 1 , 1 , 1 ) q n = 1 3  4 E 2 ( q 4 ) − E 2 ( q )  B 2 2 = 1 − 8 ∑ σ 1 ( n ; 0 , 1 , − 2 , 1 ) q n = 1 3  8 E 2 ( q 4 ) − 6 E 2 ( q 2 ) + E 2 ( q )  C 2 2 = 8 ∑ σ 1 ( n ; 0 , 2 , − 1 , 2 ) q n = − 2 3  2 E 2 ( q 4 ) − 3 E 2 ( q 2 ) + E 2 ( q )  M 3 ( Γ 0 ( 4 ) , χ − 4 ) : A 2 3 = A 2 B 2 2 + A 2 C 2 2 (see belo w) = E 1 , χ − 4 3 ( q ) + 8 E χ − 4 , 1 3 ( q ) A 2 B 2 2 = 1 − 4 ∑ σ 2 ( n ; 0 , 1 , 0 , − 1 ) q n = E 1 , χ − 4 3 ( q ) A 2 C 2 2 = 16 ∑ σ c 2 ( n ; 0 , 1 , 0 , − 1 ) q n = 8 E χ − 4 , 1 3 ( q ) M 4 ( Γ 0 ( 4 )) : A 2 4 = 1 + 4 ∑ σ 3 ( n ; 4 , 4 , 3 , 4 ) q n = 1 15  16 E 4 ( q 4 ) − 2 E 4 ( q 2 ) + E 4 ( q )  B 2 4 = 1 − 16 ∑ σ 3 ( n ; − 1 , 1 ) q n = 1 15  16 E 4 ( q 2 ) − E 4 ( q )  C 2 4 = 4 ∑ σ 3 ( n ; 7 , 0 , 8 , 0 ) q n = − 16 15  E 4 ( q 4 ) − E 4 ( q 2 )  A 2 2 B 2 2 = 1 − 2 ∑ σ 3 ( n ; − 1 , 0 , 1 , 0 ) q n = 1 15  16 E 4 ( q 4 ) − E 4 ( q 2 )  A 2 2 C 2 2 = 2 ∑ σ 3 ( n ; 7 , 8 ) q n = − 1 15  E 4 ( q 2 ) − E 4 ( q )  B 2 2 C 2 2 = 2 ∑ σ 3 ( n ; − 7 , 8 , − 9 , 8 ) q n = 1 15  16 E 4 ( q 4 ) − 17 E 4 ( q 2 ) + E 4 ( q )  Nonlinear dif ferential equations for classical modular forms 13 Remark. Each q -expansion in T ab le 1 of a modula r form of even weight k can alternatively be written in terms of a σ c k − 1 conjuga te d ivisor f unction, rather than a σ k − 1 divisor function. F or instance, A 4 2 = ϑ 2 4 + ϑ 3 4 = 1 + 24 ∞ ∑ n = 1 σ 1 ( n ; 0 , 1 ) q n = 1 + 24 ∞ ∑ n = 1 σ c 1 ( n ; − 1 , 1 ) q n , (3.8a) C 4 4 = 4 ϑ 2 4 ϑ 3 4 = 8 ∞ ∑ n = 1 σ 3 ( n ; 7 , 8 ) q n = 64 ∞ ∑ n = 1 σ c 3 ( n ; 0 , 1 ) q n . (3.8b) Using (3.8ab), one can check that A 4 2 , C 4 4 / 64 are identical to the forms C , D used by Kaneko and K oike [25] as generators of M e ven ( Γ 0 ( 2 )) . Remark. The modular f orm E 1 , χ − 3 1 ( q ) = 1 + 6 ∑ ∞ n = 1 σ 0 ( n ; 0 , 1 , − 1 ) q n ﬁgured in W iles’ proof of the Mod ularity Theorem; for a sketch, see [14, Ex. 9.6.4 ]. T able 1 reveals that this m odular form is identical to A 3 , the Borwein s’ cubic theta fu nction in the spirit of Ramanujan. This observation may be new . Remark. E ach representation i n T able 1 can be re w ritten as a Lambert series iden- tity . Of the resulting identities, se veral were recorded by Ramanujan and ha ve been giv en non-mo dular proofs by Berndt and others [4, 5]. Remark. Th e d ifﬁculty in extending T ab le 1 to hig her-degree monom ials in th e triples A r , B r , C r , i.e. , in deriving simple expressions for their Fourier coef ﬁcients in terms of di visor functions, is of course that one begins to encounter cusp forms. T o some e xtent one ca n work aro und this. F or instance, V an d er P ol [33] expr essed the coefﬁcients of A 2 12 , B 2 12 , C 2 12 , i.e., ϑ 3 24 , ϑ 4 24 , ϑ 2 24 , with the aid of Ramanu- jan’ s tau function . Recently Hahn [21, Thm. 2.1], for each e ven k > 4 , worked out the comb ination of the b asis mon omials { A 4 2 a B 4 4 b , 2 a + 4 b = k } of M k ( Γ 0 ( 2 )) , i.e., the theta polynom ials { ( ϑ 2 4 + ϑ 3 4 ) a ϑ 4 8 b , 2 a + 4 b = k } , which equals E i ∞ k ( q ) : = 1 2 k − 1  2 k E k ( q 2 ) − E k ( q )  = 1 + 2 k ( 2 k − 1 ) B k ∞ ∑ n = 1 σ k − 1 ( n ; − 1 , 1 ) q n . This is a weight- k Eisenstein for m on Γ 0 ( 2 ) which vanishes at the cu sp τ = 0 and is nonzero at τ = i ∞ , like B 4 k . In ef f ect, her combination of monomials ( unlike the single m onomial B 4 k for even k > 6) h as no cusp-for m component, and th erefore has a q -expansion with coefﬁcients e xpressible in terms of divisor functions. Proposition 3 .7. On e ha s th e supplemen tary q -expansions shown in T able 2 , for certain powers of B r , C r , r = 4 , 3 , 2 , the mu ltiplier systems of which a r e no t o f Dirichlet-character typ e. (In eac h, k deno tes the weight.) Pr oo f. Each q -expansion comes from an Eisenstein r epresentation com puted b y linear algebra, like those of T able 1. The starting points are B 3 ( q ) , C 3 ( q 3 ) ∈ M 1 ( Γ 0 ( 9 ) , χ − 3 ) , (3.9) B 2 ( q ) , C 2 ( q 2 ) ∈ M 1 ( Γ 0 ( 8 ) , χ − 4 ) , (3.10) which follow from the deﬁnitions of B 3 , C 3 and B 2 , C 2 , like Proposition 3.3. (The statements abo ut C 3 ( q 3 ) , C 2 ( q 2 ) here are equiv alent to C 3 ∈ M 1 ( Γ ( 3 ) , χ − 3 ) and 14 Robert S. Maier T able 2. Γ 0 ( 2 ) , k = 1 : B 4 = 1 − 4 ∑ σ 0 ( n ; 0 , 1 , − 2 , − 1 , 0 , 1 , 2 , − 1 ) q n C 4 = 2 3 / 2 ∑ σ 0 ( n ; 0 , 1 , − 1 , − 1 , 0 , 1 , 1 , − 1 ) q n / 4 k = 2 : B 4 2 = 1 − 8 ∑ σ 1 ( n ; 0 , 1 , − 2 , 1 ) q n C 4 2 = 4 ∑ σ 1 ( n ; 0 , 2 , − 1 , 2 ) q n / 2 Γ 0 ( 3 ) , k = 1 : B 3 = 1 − 3 ∑ σ 0 ( n ; 0 , 1 , − 1 , − 3 , 1 , − 1 , 3 , 1 , − 1 ) q n C 3 = 3 ∑ σ 0 ( n ; 0 , 1 , − 1 , − 1 , 1 , − 1 , 1 , 1 , − 1 ) q n / 3 Γ 0 ( 4 ) , k = 1 : B 2 = 1 − 4 ∑ σ 0 ( n ; 0 , 1 , − 2 , − 1 , 0 , 1 , 2 , − 1 ) q n C 2 = 4 ∑ σ 0 ( n ; 0 , 1 , − 1 , − 1 , 0 , 1 , 1 , − 1 ) q n / 2 k = 3 : B 2 3 = 1 + 2 ∑ σ 2 ( n ; 0 , 2 , − 1 , − 2 , 0 , 2 , 1 , − 2 ) q n − 16 ∑ σ c 2 ( n ; 0 , 1 , − 8 , − 1 , 0 , 1 , 8 , − 1 ) q n C 2 3 = ∑ σ 2 ( n ; 0 , − 4 , 1 , 4 , 0 , − 4 , − 1 , 4 ) q n / 2 + 4 ∑ σ c 2 ( n ; 0 , 1 , − 4 , − 1 , 0 , 1 , 4 , − 1 ) q n / 2 C 2 ∈ M 1 ( Γ 0 ( 4 ) ∩ Γ ( 2 ) , χ − 4 ) .) T o derive the given expan sions of B 4 , C 4 and B 4 2 , C 4 2 , one simply uses the facts that B 4 = B 2 and C 4 ( q ) = 2 − 1 / 2 · C 2 ( q 1 / 2 ) . (The latter fact incidentally implies that C 4 ∈ M 1 ( Γ ( 4 ) , χ − 4 ) .) The k = 1 exp ansions in T able 2 h av e previously been been derived by non- modular method s, in [17, §§ 32 and 33] and [7]. The ﬁnal two expansions, of weight-3 fo rms on Γ 0 ( 4 ) , may possibly be classical (since B 2 = ϑ 4 2 and C 2 = ϑ 2 2 ), but are more likely to be ne w . They come from B 2 3 ( q ) = − E 1 , χ − 4 3 ( q ) + 2 E 1 , χ − 4 3 ( q 2 ) − 8 E χ − 4 , 1 3 ( q ) + 64 E χ − 4 , 1 3 ( q 2 ) , (3.11) C 2 3 ( q 2 ) = E 1 , χ − 4 3 ( q ) − E 1 , χ − 4 3 ( q 2 ) + 2 E χ − 4 , 1 3 ( q ) − 8 E χ − 4 , 1 3 ( q 2 ) , (3.12) in which the four E 3 ’ s span M 3 ( Γ 0 ( 8 ) , χ − 4 ) .  The (form al!) weight-1 mod ular form A 4 = p ϑ 2 4 + ϑ 3 4 on Γ 0 ( 2 ) does not ﬁt into th e precedin g Eisenstein fra mew ork, since it is mu ltiv alued. This is why A 4 and its odd powers are not e xpand ed in T able 1 or 2. A bit of compu tation yields A 4 = 1 + 12  q − 5 q 2 + 64 q 3 − 917 q 4 + 1485 0 q 5 + · · ·  , (3.13) but there is no obvious arithmetical interpretation of the (inte gral, see [23]) coefﬁ- cients of this q -expansion , a ny more than there is for the q -expansions E 4 1 / 4 = 1 + 60  q − 81 q 2 + 1100 8 q 3 − 1751 057 q 4 + · · ·  , (3.14a ) E 6 1 / 6 = 1 − 84  q + 243 q 2 + 7878 4 q 3 + 2982 6307 q 4 + · · ·  (3.14b ) of the multivalued weight-1 form s E 4 1 / 4 , E 6 1 / 6 on Γ ( 1 ) , introd uced in § 2. It shou ld be noted that the form A 4 2 ∈ M 2 ( Γ 0 ( 2 )) is the theta fun ction of the D 4 lattice. Nonlinear dif ferential equations for classical modular forms 15 The divisor -functio n representations of T able s 1 and 2 can be viewed as theta identities; inclu ding even the r = 3 o nes, since A 3 , B 3 , C 3 too can be expressed in terms of ϑ 2 , ϑ 3 , ϑ 4 . (See [7] and § 5, below .) They imply , inter alia , Theorem 3.8. Let r 2 s ( n ) , n > 1 , r esp. t 2 s ( n ) , n > 0 , deno te the number of ways of r epr esenting a n in te ger n a s th e sum of 2 s sq uar es, resp. triangles. (These terms signify m 2 , r esp. m ( m + 1 ) / 2 , with m ranging over Z .) Then in te rms of divisor and conjuga te divisor functions, r 2 ( n ) = 4 σ 0 ( n ; 0 , 1 , 0 , − 1 ) , r 4 ( n ) = 8 σ 1 ( n ; 0 , 1 , 1 , 1 ) = 8 σ c 1 ( n ; − 3 , 1 , 1 , 1 ) , r 6 ( n ) = 16 σ c 2 ( n ; 0 , 1 , 0 , − 1 ) − 4 σ 2 ( n ; 0 , 1 , 0 , − 1 ) , r 8 ( n ) = 4 σ 3 ( n ; 4 , 4 , 3 , 4 ) = 16 σ c 3 ( n ; 15 , 1 , − 1 , 1 ) ; t 2 ( n ) = 4 σ 0 ( 4 n + 1; 0 , 1 , − 1 , − 1 , 0 , 1 , 1 , − 1 ) = 4 σ 0 ( 8 n + 2; 0 , 1 , 0 , − 1 ) , t 4 ( n ) = 8 σ 1 ( 2 n + 1; 0 , 2 , − 1 , 2 ) = 1 6 σ c 1 ( 2 n + 1; 0 , 1 , − 2 , 1 ) = 16 σ 1 ( 2 n + 1; 1 ) = 16 σ c 1 ( 2 n + 1; 1 ) , t 6 ( n ) = σ 2 ( 4 n + 3; 0 , − 4 , 1 , 4 , 0 , − 4 , − 1 , 4 ) + 4 σ c 2 ( 4 n + 3; 0 , 1 , − 4 , − 1 , 0 , 1 , 4 , − 1 ) = 8 σ 2 ( 4 n + 3; 0 , − 1 , 0 , 1 ) , t 8 ( n ) = 4 σ 3 ( 2 n + 2; 7 , 0 , 8 , 0 ) = 25 6 σ c 3 ( 2 n + 2; 0 , 0 , 1 , 0 ) = 32 σ 3 ( n + 1; 7 , 8 ) = 256 σ c 3 ( n + 1; 0 , 1 ) . Pr oo f. A 2 = ϑ 3 2 and ϑ 3 ( q ) = ∑ m ∈ Z q m 2 ; h ence r 2 s ( n ) is the coefﬁcient of q n in the q - expansion of A 2 s . Sim ilarly , C 2 = ϑ 2 2 and ϑ 2 ( q ) = q 1 / 4 ∑ m ∈ Z q m ( m + 1 ) ; h ence t 2 s ( n ) is the coe fﬁcient of q 2 n in the q -expansion of  q − 1 / 2 C 2  s . Each r 2 s ( n ) formula is tak en directly from T a ble 1 or 2, and if possible, re w rit- ten in an a lternative form based on a conju gate divisor func tion. The same is true of the ﬁrst line of each o f the t 2 s ( n ) f ormulas. Th e second, simpler lin es o f the latter follow by elementary arithmetic arguments.  Theorem 3.8 is a restatement o f Jacobi’ s T wo, Four, Six, and Eight Square s Theorem s, and th e known for mulas fo r t 2 , t 4 , t 6 , t 8 [31]. But the p resent mod ular proof of the f ormulas fo r r 6 ( n ) , t 6 ( n ) , in particu lar , is illumin ating. (For the his- tory of these difﬁcult formulas, see [29 , p. 80].) The p resent proof, un like pre vious arithmetical o r elliptic one s, makes it clear for the ﬁrst time how the two terms in the rather awkward formula for r 6 ( n ) co me from E χ − 4 , 1 3 , E 1 , χ − 4 3 ∈ M 3 ( Γ 0 ( 4 ) , χ − 4 ) . In contrast, a modular deriv a tion of the seemingly simple formula for t 6 ( n ) has al- ready been given b y Ono et al. [31]; but the present derivation, based on Eq. (3.12), reveals its complicated underpin nings. Difﬁculties arise in extend ing any Eisenstein ap proach to s > 4 , of course. A s Rankin [38] showed, the po wer ϑ 3 2 s (i.e., A 2 s ) for each s > 4 h as a nonzero cusp- form compon ent. 16 Robert S. Maier 4. Proof of Theor em 2.3(2) Using the results obtained in the last section, one can d erive the d ifferential sys- tems of Theorem 2.3(2) as an exercise i n linear algebr a, as follows. The d eﬁnition of quasi-mo dular form used here is standar d. On H ∗ , a ho lo- morph ic f unction f is q uasi-mod ular o f weigh t 2 and depth 6 1 on a subgro up Γ < Γ ( 1 ) , with trivial multiplier system, if f  a τ + b c τ + d  = ( c τ + d ) 2 f ( τ ) + ( s / 2 π i ) c ( c τ + d ) , (4.1) for all ±  a b c d  ∈ Γ and some s ∈ C . One writes f ∈ M 6 1 2 ( Γ ) . The constant s is called the coefﬁcient of af ﬁnity of f . Lemma 4.1 . If F ∈ M k ( Γ , ˆ χ F ) and G ∈ M ℓ ( Γ , ˆ χ G ) , i.e., F , G are m odular forms on Γ with mu ltiplier systems no t r equired to b e of Dirichlet-character type, and F vanishes only at cusps, then 1. E : = F ′ / F ∈ M 6 1 2 ( Γ ) , an d E has coefﬁcient of af ﬁnity k. 2. k E ′ − E · E ∈ M 4 ( Γ ) . 3. k G ′ − ℓ E · G ∈ M ℓ + 2 ( Γ , ˆ χ G ) . Pr oo f. By differentiation of the transformation laws for F , G and E .  Pr oo f of Theor em 2.3(2). Gi ven A r , B r , C r , r = 4 , 3 , 2 , as in Deﬁnition 3.1, deﬁn e the function E r of Theorem 2.3 as ( C r r ) ′ / C r r . By part ( 1) of th e lemma, it is quasi- modular of weight 2 and depth 6 1 on Γ 0 ( 2 ) , Γ 0 ( 3 ) , Γ 0 ( 4 ) , respectively , wit h tri vial multiplier system. The space of such quasi-modular forms is spanned respectiv ely by E 2 ( q 2 ) , E 2 ( q ) ; by E 2 ( q 3 ) , E 2 ( q ) ; and by E 2 ( q 4 ) , E 2 ( q 2 ) , E 2 ( q ) . By working out the ﬁrst few Fourier coefﬁcients of C r and ( C r r ) ′ / C r r , and co mparin g them with those of these basis functions, one deri ves the Eisenstein and di v isor-function rep- resentations of E r stated in the theorem. By par t (2 ) of the lem ma, r E ′ r − E r · E r must lie in M 4 ( Γ 0 ( 2 )) , M 4 ( Γ 0 ( 3 )) , M 4 ( Γ 0 ( 4 )) for r = 4 , 3 , 2 . By work ing out the ﬁrst two Fourier coefﬁcients of r E ′ r − E r · E r , an d co mparing th em with the q -expansions of the basis monom i- als of these vector spaces, given in T able 1, one pr oves that in each case this for m equals − A r 4 − r B r r , as claimed. A single example (the r = 3 case) will sufﬁce. By direct computation, 3 E ′ 3 − E 3 · E 3 = − 1 + 3 q + . . . , (4.2) and accord ing to the table, M 4 ( Γ 0 ( 3 )) is span ned by A 3 4 = 1 + 24 q + . . . , (4.3a) A 3 B 3 3 = 1 − 3 q + . . . . (4.3b) The id entiﬁcation of 3 E ′ 3 − E 3 · E 3 with − A 3 B 3 3 is justiﬁed by the agr eement to ﬁrst order in q . Nonlinear dif ferential equations for classical modular forms 17 By part ( 3) of the lemma, ( A r r ) ′ − E r · A r r must lie in the spaces M 6 ( Γ 0 ( 2 )) , M 5 ( Γ 0 ( 3 ) , χ − 3 ) , M 4 ( Γ 0 ( 4 )) , fo r r = 4 , 3 , 2. By e xpand ing in q again, and comparing coefﬁcients with the q -expansions of the spanning monomials listed in T able 1, one proves that this form equals − A r 2 B r r , as claimed. The details are elementary .  One can derive the g eneralized Chazy e quations o f Theor em 2 .3(1) by elimi- nating A r , B r , C r from the differential systems satisﬁed by A r , B r , C r ; E r . But the computatio ns are undesirably length y , especially for r = 3. Altern ativ e, more struc- tured proofs of Theorem 2.3(1) will be giv en in § 6. 5. Elliptic integral and differential theta identities This section is a dig ression, in wh ich the s ystems of Theorem 2.3(2) are employed to derive an elliptic integral tran sformation fo rmula and differential identities in- volving Jacobi’ s theta-nulls. The latter are deﬁned on H ∋ τ , i.e. , on | q | < 1 , by ϑ 2 ( q ) = ∑ m ∈ Z q ( m + 1 2 ) 2 = 2 · [ 4 ] 2 / [ 2 ] , ϑ 3 ( q ) = ∑ m ∈ Z q m 2 = [ 2 ] 5 / [ 1 ] 2 [ 4 ] 2 , ϑ 4 ( q ) = ∑ m ∈ Z ( − 1 ) m q m 2 = [ 1 ] 2 / [ 2 ] , the g i ven eta representatio ns following fr om classical q -series identities. Each ϑ i is a weig ht- 1 2 modular fo rm o n Γ 0 ( 4 ) with a no n-Dirichlet multip lier system [ 34, § 8 1]. They satisfy ϑ 3 4 = ϑ 2 4 + ϑ 4 4 . As noted in § 3 , A 2 , B 2 , C 2 equal ϑ 3 2 , ϑ 4 2 , ϑ 2 2 , and moreover [7], e.g., A 3 ( q ) = ϑ 3 ( q ) ϑ 3 ( q 3 ) + ϑ 2 ( q ) ϑ 2 ( q 3 ) . The theta-nulls ϑ 2 , ϑ 3 , ϑ 4 vanish respecti vely at q = 0 , − 1 , 1 , i.e., at the points τ = i ∞ , 1 / 2 , 0 , which are the in equiv alent cusp s o f Γ 0 ( 4 ) . Infor mally , each ϑ i has a simp le zero at the resp ectiv e cusp, and is n onzero a nd r egular elsewhere. T his does no t mean that in the conv entiona l an alytic sense, ϑ i is bou nded as either of the other two cusps is app roached . For instanc e, ϑ 3 ( q ) → ∞ log arithmically as q → 1 − , i.e., as τ → 0 alon g the po siti ve imaginar y axis. Having zero or der of vanishing at a ﬁnite cusp does not preclude a logarithmic di vergence. The reader is cautioned that i n the classical literature, and in the applied math- ematics literature to this day , the argument o f eac h ϑ i is taken to be q 2 : = √ q = exp ( π i τ ) rather than q = exp ( 2 π i τ ) . Using q 2 rather tha n q is equ iv a lent to view- ing the theta-nulls as modular forms on Γ ( 2 ) r ather than Γ 0 ( 4 ) , since the two sub- group s of Γ ( 1 ) are conjug ates under th e 2-isogeny τ 7→ 2 τ in PSL ( 2 , R ) . In this article the Γ 0 ( 4 ) co n vention is adhered to. The following is a brief revie w of how theta-nulls arise from elliptic integrals. Consider the param etric family E of elliptic plane curves E α / C deﬁne d by the equation y 2 = ( 1 − x 2 )( 1 − α x 2 ) , where α ∈ P 1 ( C ) \ { 0 , 1 , ∞ } . The ( ﬁrst) c omplete elliptic integral K = K ( α ) is deﬁn ed by K ( α ) = 1 2 Z 1 0 x − 1 / 2 ( 1 − x ) − 1 / 2 ( 1 − α x ) − 1 / 2 d x , (5.1) which makes sense if 0 6 α < 1 , and can be continu ed to a holomo rphic fu nc- tion o n P 1 ( C ) α , slit betwe en α = 1 and α = ∞ to ensure single-valuedness. The 18 Robert S. Maier fundam ental perio ds of th e cur ve E α are pr oportio nal to K ( α ) , i K ( 1 − α ) , so its period ratio τ = τ 1 / τ 2 ∈ H is i K ( 1 − α ) / K ( α ) . Since K ( 0 ) = π / 2 , it is con venient to normalize by deﬁning ˆ K = K / ( π / 2 ) . One can show (e.g., b y comp aring q -series) that if K is regard ed as a fu nction of the no me q = exp ( 2 π i τ ) , i.e. , ˆ K = ˆ K ( q ) , then ˆ K ( q ) equals ϑ 3 2 ( q ) , which is holomo rphic and single-valued on H ∗ . The re ason for this equality is that in mo d- ern lang uage, E is the elliptic family attached to Γ 0 ( 4 ) . Th e parame ter α can a lso be viewed as a function o f q , i.e., as a Γ 0 ( 4 ) -stable holom orphic fun ction on H , with a zero at τ = i ∞ and a pole at τ = 0: it is a Hauptmodu l fo r Γ 0 ( 4 ) . So, ˆ K = ϑ 3 2 = A 2 . By T ab le 1, ˆ K ∈ M 1 ( Γ 0 ( 4 ) , χ − 4 ) , and ˆ K ha s the Eisen stein series representatio n ˆ K ( q ) = 1 + 4 ∞ ∑ n = 1 σ 0 ( n ; 0 , 1 , 0 , − 1 ) q n = E 1 , χ − 4 1 ( q ) . (5.2) This expansion is well known, as is the presence of the character χ − 4 in the trans- formation law of ˆ K un der τ 7→ a τ + b c τ + d with ±  a b c d  ∈ Γ 0 ( 4 ) . But, analo gous expan- sions and tr ansforma tion prop erties for the second com plete elliptic in tegral are not. This function E = E ( α ) is deﬁned locally (on α 6 0 < 1 ) by E ( α ) = 1 2 Z 1 0 x − 1 / 2 ( 1 − x ) − 1 / 2 ( 1 − α x ) 1 / 2 d x . (5.3) Since E ( 0 ) = π / 2 also, one no rmalizes by letting ˆ E = E / ( π / 2 ) . As with ˆ K , ˆ E can be vie wed as ˆ E ( q ) , a h olomor phic and single-valued f unction on H . It is a class ical result (see [16, p. 218] and [20, § 31 ]) that ˆ K ( q ) ˆ E ( q ) = 1 + 8 ∞ ∑ n = 1 q 2 n ( 1 + q 2 n ) 2 = 1 + 4 ∞ ∑ n = 1 σ 1 ( n ; − 1 , 0 , 1 , 0 ) q n = 1 3  4 E 2 ( q 4 ) − E 2 ( q 2 )  . (5.4) Hence ˆ E = ( ˆ K ˆ E ) / ˆ K , i.e., ˆ E ( q ) = 1 + 4 ∑ ∞ n = 1 σ 1 ( n ; − 1 , 0 , 1 , 0 ) q n 1 + 4 ∑ ∞ n = 1 σ 0 ( n ; 0 , 1 , 0 , − 1 ) q n = 4 E 2 ( q 4 ) − E 2 ( q 2 ) 3 E 1 , χ − 4 1 ( q ) (5.5) (cf. [ 20, § 38]). Remarkab ly , the d i visor-function representatio n of (5. 4) is id entical to that of the quasi-mo dular form E 2 ∈ M 6 1 2 ( Γ 0 ( 4 )) , given in Theorem 2.3. So, E 2 = ( C 2 2 ) ′ / C 2 2 = ( ϑ 2 4 ) ′ / ϑ 2 4 = ˆ K ˆ E (5.6) and ˆ K ˆ E ∈ M 6 1 2 ( Γ 0 ( 4 )) . Also, one can write ˆ E = E 2 / A 2 . Proposition 5.1. The forms ˆ K , ˆ E , ˆ K ˆ E have the transformation laws ˆ K ( q 1 ) = χ − 4 ( d )( c τ + d ) ˆ K ( q ) , ˆ E ( q 1 ) = χ − 4 ( d )  ( c τ + d ) ˆ E ( q ) + ( π i ) − 1 c ˆ K ( q ) − 1  , ˆ K ˆ E ( q 1 ) = ( c τ + d ) 2 ˆ K ˆ E ( q ) + ( π i ) − 1 c ( c τ + d ) , for all ±  a b c d  ∈ Γ 0 ( 4 ) . Her e, q = e xp ( 2 π i τ ) , q 1 = exp ( 2 π i τ 1 ) with τ 1 = a τ + b c τ + d . Nonlinear dif ferential equations for classical modular forms 19 Pr oo f. ˆ K = A 2 ∈ M 1 ( Γ 0 ( 4 ) , χ − 4 ) , he nce its law is kno wn. The quasi-modu lar la w for ˆ K ˆ E = E 2 = ( C 2 2 ) ′ / ( C 2 2 ) , of the type (4.1 ), follows from Lemma 4.1(1). T aking the quotient yields the law for ˆ E .  Remark. This tra nsformatio n law unde r Γ 0 ( 4 ) for the (norm alized) secon d com- plete elliptic integral ˆ E ( q ) is arguably th e most informative obtained to date. Tri- comi [4 6, Chap . IV , § 2 ] has some related results, but it is difﬁcult to compa re them, since h e (i) u sed homogene ous modu lar forms, i.e. , functions of τ 1 , τ 2 rather than τ , (ii) worked in terms of K ( α ) , E ( α ) , and espec ially , (iii) treated only  a b c d  =  1 1 0 1  ,  0 − 1 1 0  , wh ich are generator s of Γ ( 1 ) rather of than Γ 0 ( 4 ) ( or Γ ( 2 ) ). Remark. One can similarly deﬁne quasi-mod ular forms ˆ K ˆ G , ˆ K ˆ I ∈ M 6 1 2 ( Γ 0 ( 4 )) by ( A 2 2 ) ′ / A 2 2 = ( ϑ 3 4 ) ′ / ϑ 3 4 = : ˆ K ˆ G , (5.7a) ( B 2 2 ) ′ / B 2 2 = ( ϑ 4 4 ) ′ / ϑ 4 4 = : ˆ K ˆ I (5.7b) (cf. Glaisher [20]) , and work out the transform ation laws of ˆ G , ˆ I . Each of ˆ G , ˆ I has a represen tation as a complete elliptic integral, analogous t o (5.3) for ˆ E . Proposition 5.2. The theta -nulls ϑ 2 , ϑ 3 , ϑ 4 , together with ˆ K ˆ E , satisfy a d iffer entia l system on the half-plan e H , nam ely 4 ϑ ′ 2 / ϑ 2 = ˆ K ˆ E , 2 ( ˆ K ˆ E ) ′ = ( ˆ K ˆ E ) 2 − ϑ 3 4 ϑ 4 4 , 4 ϑ ′ 3 / ϑ 3 = ˆ K ˆ E − ϑ 4 4 , 4 ϑ ′ 4 / ϑ 4 = ˆ K ˆ E − ϑ 3 4 , wher e ′ signiﬁes q d / d q = ( 2 π i ) − 1 d / d τ . Pr oo f. Substitute ϑ 3 2 , ϑ 4 2 , ϑ 2 2 , ˆ K ˆ E for A 2 , B 2 , C 2 , E 2 in the r = 2 system of T he- orem 2.3(2).  Remark. Th is system of couple d ODEs may be new , thou gh it can be d educed from identities of Glaisher and o f the Borwe ins [6, § 2.3]. For i = 2 , 3 , 4 , one can derive from it a nonlinear th ird-ord er ODE satis ﬁed by ϑ i , by eliminating th e o ther three depend ent v ariables. F or each ϑ i this turns out to be ( ϑ 2 ϑ ′′′ − 15 ϑ ϑ ′ ϑ ′′ + 30 ϑ ′ 3 ) 2 + 32 ( ϑ ϑ ′′ − 3 ϑ ′ 2 ) 3 = ϑ 10 ( ϑ ϑ ′′ − 3 ϑ ′ 2 ) 2 . (5.8) This is the 1847 equation of Jacobi [24], which was m entioned in § 2. His deriv a- tion used differentiation with respect to Hauptmodu ls for Γ 0 ( 4 ) ( his k 2 and k ′ 2 ). For an easy pro of that each of ϑ 2 , ϑ 3 , ϑ 4 must satisfy the same third -order ODE, reason as follo ws. First, work out the dif ferential systems for ϑ 2 , ϑ 3 , ϑ 4 ; ˆ K ˆ G and ϑ 2 , ϑ 3 , ϑ 4 ; ˆ K ˆ I that are analo gues of th e system for ϑ 2 , ϑ 3 , ϑ 4 ; ˆ K ˆ E in Proposi- tion 5.2. Then no tice that up to cyclic permutatio ns of the ordered pairs ( ϑ 2 , ˆ K ˆ E ) , ( ϑ 3 , ˆ K ˆ G ) , ( ϑ 4 , ˆ K ˆ I ) , th e three systems are the same. Hen ce, eliminating all dep en- dent variables except a single ϑ i must yield the same equation , irrespectiv e of i . Brezhnev [8, § 7] h as recently derived a different but related differential sys- tem, symmetr ic and elegant, in which the depend ent variables are ϑ 2 , ϑ 3 , ϑ 4 , and (in the notation used here) the element ( ˆ K ˆ E + ˆ K ˆ G + ˆ K ˆ I )( q ) of M 6 1 2 ( Γ 0 ( 4 )) , which by e xamina tion is proportion al to E 2 ( q 2 ) . His system can b e o btained by averaging together the three precedin g ones; and this a veraging ensures symmetry . 20 Robert S. Maier 6. Proofs of Theorem 2.3(1); Hypergeometric id entities Direct deri vations of the generalized Chazy equations of Theorem 2.3(1) will no w be gi ven. They will not employ , e xcept superﬁcially , the dif ferential systems s atis- ﬁed by the weight-1 modular forms A r , B r , C r . T wo proof s of Theor em 2.3(1) are supp lied. The ﬁrst is an explicitly mod- ular , lin ear-algebraic one. It is mod eled after Resnikoff ’ s proo f [40] of E q. ( 5.8), Jacobi’ s nonlinear third-o rder ODE (for ϑ = ϑ 3 ). Equation (2.3a), the C hazy equa- tion satisﬁed by u = ( 2 π i / 12 ) E 2 , can b e proved similarly . The secon d proof emp loys analytic man ipulations of Picar d–Fuchs equa tions, and relies o n results of [27]. It is based on a sort o f nonlinear hy pergeometric iden- tity , stated as Proposition 6.3, which ho lds for certain v ery special parameter v alues that appear in Picard–Fuchs equations attached to Γ 0 ( N ) , N = 2 , 3 , 4 . Remarkably , this identity has an extension to all parameter v alues, namely Theorem 6.4. Rankin [3 9] gives an altogether different sort of pro of of the Chazy equa- tion ( 2.3b), based on eleme ntary arithmetic metho ds. On e may speculate that the generalized Chazy equations can also be derived b y such methods. 6.1. A modu lar pr oof of Theor em 2.3(1) Deﬁne A r , B r , C r as in § 3, and let E r = ( C r r ) ′ / C r r , as in the pro of of Theo- rem 2 .3(2). For r = 4 , 3 , 2 , E r is quasi-mo dular of weigh t 2 a nd depth 6 1 on Γ 0 ( N ) , N = 2 , 3 , 4 , respectively . F or k = 4 , 6 , 8 , . . . , de ﬁne u ( r ) k by u ( r ) 4 = r E ′ r − E r · E r , u ( r ) k + 2 = u ( r ) k ′ − ( k / r ) E r · u ( r ) k . By Lemma 4.1, u ( r ) k ∈ M k ( Γ 0 ( N )) . If u : = ( 2 π i / r ) E r and u k is deﬁned in ter ms of u as in § 2 , then one has that u k = ( 2 π i ) k / 2 u k / r 2 for all k . By th e last d ifferential equatio n in Th eorem 2. 3(2c), u ( r ) 4 equals − A r 4 − r B r r . By Theor em 2.3(2b), Ord 0 ( A r ) = 0 and Ord 0 , Γ 0 ( N ) ( B r ) = 1 / r ; hence one has that Ord 0 , Γ 0 ( N ) ( u ( r ) 4 ) = 1. It is e vident that Ord 0 , Γ 0 ( N ) ( u ( r ) k ) > 1 for k > 4. According to the valence f ormula [41, Chap. V], the total nu mber of z eroes of a no nzero element f ∈ M k ( Γ 0 ( N )) , coun ted with r espect to local paramete rs, is equ al to ( k / 1 2 )[ Γ ( 1 ) : Γ 0 ( N )] . It follows that if at any s ∈ H ∗ , it is the case that Ord s , Γ 0 ( N ) ( f ) > ( k / 12 )[ Γ ( 1 ) : Γ 0 ( N )] , then f = 0. Her e, the sub grou p ind ex [ Γ ( 1 ) : Γ 0 ( N )] equals 3 , 4 , 6 when N = 2 , 3 , 4. In the following analyses, the superscript ( r ) will be omitted for readability . • r = 4 , Γ 0 ( N ) = Γ 0 ( 2 ) . One sets k = 1 2 , i.e., uses linear algebr a on M 12 ( Γ 0 ( 2 )) . For e ach g ∈ V = { u 4 u 8 , u 6 2 , u 4 3 } , it is the case th at g ∈ M 12 ( Γ 0 ( 2 )) a nd Ord 0 , Γ 0 ( 2 ) ( g ) > 2 . There i s a linear combination f of the thr ee monomials in V for which Ord 0 , Γ 0 ( 2 ) ( f ) > 4. But if f ∈ M 12 ( Γ 0 ( 2 )) vanishes with order greater than ( k / 12 )[ Γ ( 1 ) : Γ 0 ( 2 )] = 3 , then f = 0. Nonlinear dif ferential equations for classical modular forms 21 This comb ination can be foun d by dir ect co mputation , using q -series (even though q -series are expansions at the inﬁnite cusp, not at τ = 0). T o O ( q 1 ) , u 4 u 8 = 3 2 − 8 q + . . . , u 6 2 = 1 + 16 q + . . . , u 4 3 = − 1 + 48 q + . . . . There is a unique combination (up to scalar multiples) that is zero to this or der, and must therefor e v anish identically; nam ely , 2 u 4 u 8 − 2 u 6 2 + u 4 3 . Its vanish- ing is equiv alent to u 4 u 8 − u 2 6 + 8 u 3 4 = 0. • r = 3 , Γ 0 ( N ) = Γ 0 ( 3 ) . One sets k = 2 0 , i.e., uses linear algebr a on M 20 ( Γ 0 ( 3 )) . For each g ∈ V = { u 4 u 8 2 , u 6 2 u 8 , u 4 3 u 8 , u 4 2 u 6 2 , u 4 5 } , it is th e c ase that g ∈ M 20 ( Γ 0 ( 3 )) an d Ord 0 , Γ 0 ( 3 ) ( g ) > 3 . There is a linear combina tion f o f the ﬁve monom ials in V fo r which Ord 0 , Γ 0 ( 3 ) ( g ) > 7. But if f ∈ M 20 ( Γ 0 ( 3 )) vanishes with order greater than ( k / 12 )[ Γ ( 1 ) : Γ 0 ( 3 )] = 2 0 / 3 , then f = 0. As in th e r = 4 case, this combina tion can be fou nd by a d irect comp utation (a tedious one). T o O ( q 3 ) , u 4 u 8 2 = − 64 9 − 112 3 q + 23 q 2 − 7123 3 q 3 + . . . , u 6 2 u 8 = − 128 27 − 368 9 q − 944 3 q 2 + 7381 9 q 3 + . . . , u 4 3 u 8 = 8 3 − 13 q − 2 01 q 2 + 2075 q 3 + . . . , u 4 2 u 6 2 = 16 9 − 8 3 q − 71 q 2 − 2654 3 q 3 + . . . , u 4 5 = − 1 + 15 q + 45 q 2 − 2145 q 3 + . . . . There is a unique combination (up to scalar multiples) that is zero to this or der, and therefore must vanish identically; namely , 9 u 4 u 2 8 − 9 u 2 6 u 8 + 24 u 3 4 u 8 − 15 u 2 4 u 2 6 + 16 u 5 4 . Its v anishing is equiv alent to u 4 u 2 8 − u 2 6 u 8 + 24 u 3 4 u 8 − 15 u 2 4 u 2 6 + 144 u 5 4 = 0. • r = 2 , Γ 0 ( N ) = Γ 0 ( 4 ) . No linear algeb ra is needed , since as no ted in the state- ment of Theorem 2.3(1), E 2 ( q ) = E 4 ( q 2 ) . By comp aring u ( 4 ) 4 = 4 E ′ 4 − E 4 · E 4 u ( 4 ) k + 2 = u ( 4 ) k ′ − ( k / 4 ) E 4 · u ( 4 ) k , u ( 2 ) 4 = 2 E ′ 2 − E 2 · E 2 u ( 2 ) k + 2 = u ( 2 ) k ′ − ( k / 2 ) E r · u ( 2 ) k , one deduces that u ( 2 ) k ( q ) = 2 ( k − 4 ) / 2 u ( 4 ) k ( q 2 ) . But in the r = 4 case, 2 u 4 u 8 − 2 u 6 2 + u 4 3 = 0 (see the treatment above). Hence, for r = 2 , u 4 u 8 − u 6 2 + 2 u 4 3 = 0 . This is equiv alent to u 4 u 8 − u 2 6 + 8 u 3 4 = 0. 22 Robert S. Maier 6.2. A hypergeometric pr oof of Theo r em 2.3(1) This pro of is in the spirit of Jacob i, since it employs Hauptmo duls an d deriva- ti ves with respe ct to them. I t u ses the results of [27], which were inspired by the following stan dard theorem on s ubgr oup actions of PSL ( 2 , R ) [18, § 44, Thm. 15]. Theorem 6 .1. Let Γ < PSL ( 2 , R ) be a Fuchsian gr ou p o f M ¨ obius tr ansformations of H ( o f the ﬁrst kind ) that has a Hauptmodu l t = t ( τ ) , i.e., a non-con stant si mple automorp hic fun ction with a sing le simple zer o on a fun damental r e gion of Γ . Then τ can be expr essed as a ( multivalu ed ) function of t as f 1 / f 2 , a ratio of indepen dent solutions f 1 , f 2 of some second- or der differ ential equation L ( Γ ) f : =  D t 2 + P ( t ) · D t + Q ( t )  f = 0 (6.1) on P 1 ( C ) t , in which P , Q ∈ C ( t ) . Equation (6. 1) is called a Picar d–Fuch s equatio n (th e term b eing histor ically most accur ate when Γ < Γ ( 1 ) ). It is an ODE on the ge nus-zero curve P 1 ( C ) t , which is e ssentially the fun damental region of Γ with bound ary identiﬁcatio ns, i.e., the (co mpactiﬁed) q uotient of H by Γ . It fo llows fro m a second theo rem o n automor phic func tions [18, § 110, Thm. 6] that Eq. (6. 1) mu st be a ‘Fuchsian ’ ODE, i.e. , all its singular p oints on P 1 ( C ) t must be regular . These points are bi- jectiv e with the v ertices of the fundamental re gion of Γ . The difference of t he two characteristic expon ents of the op erator L ( Γ ) will b e 0 at a cusp, and 1 / n at an order- n elliptic ﬁxed p oint. That is, it will be the recipro cal of the or der o f the associated stabilizing subgro up. The Picard–Fu chs equ ation h as solution space C f 1 ⊕ C f 2 , i.e., ( C τ ⊕ C ) f 2 . It will shortly be of in terest to determ ine wh ether the lo garithmic derivati ve u : = ˙ f 2 / f 2 also satisﬁes an ODE, in this case with respect to τ ∈ H . (As al ways, the dot signiﬁes differentiation with respect to τ .) For this, the following will be useful. Let u k , k = 4 , 6 , . . . , be deﬁned as in Theo rem 2.3, i. e., u 4 = ˙ u − u 2 and u k + 2 = ˙ u k − kuu k , an d let dif feren tiation wit h respect to t be denoted by a subscripted t . Lemma 6. 2. One can write u k = ˆ u k ˙ t k / 2 , where the sequ ence ˆ u 4 , ˆ u 6 , . . . follows fr om ˆ u 4 = − Q and ˆ u k + 2 = ( ˆ u k ) t + ( k / 2 ) P ˆ u k . Thus u 4 = − Q ˙ t 2 , u 6 = − ( Q t + 2 PQ ) ˙ t 3 , u 8 = − ( Q tt + 5 PQ t + 2 QP t + 6 P 2 Q ) ˙ t 4 . Pr oo f. ˙ t = ( d τ / d t ) − 1 = 1 / ( f 1 / f 2 ) t = f 2 2 / w , wh ere w = w ( f 1 , f 2 ) is th e Wronskian. Similarly , ¨ t = P ˙ t 2 + 2 u ˙ t comes fro m (6.1) b y differential calculus. The r ecurrenc e ˆ u k + 2 = ( ˆ u k ) t + ( k / 2 ) P ˆ u k comes fr om u k + 2 = ˙ u k − kuu k by substituting d / d τ = ˙ t D t , and exploiting these f acts.  Picard–Fuch s dif ferential operator s L ( Γ ) that illustrate Theorem 6.1 were ob- tained in [2 7] f or the grou ps Γ = Γ 0 ( N ) , N = 2 , 3 , 4 , among other s. Th e corre- sponding Hauptmod uls t = t N = t N ( τ ) were chosen to be t 2 ( τ ) = 2 12 · [ 2 ] 24 / [ 1 ] 24 , t 3 ( τ ) = 3 6 · [ 3 ] 12 / [ 1 ] 12 , t 4 ( τ ) = 2 8 · [ 4 ] 8 / [ 1 ] 8 , Nonlinear dif ferential equations for classical modular forms 23 where th e prefactors are of ar ithmetical signiﬁcance but are n ot impo rtant here (they could equally well be set equal to unity). In each of these three cases, t N = 0 correspo nds to the inﬁnite cusp, and t N = ∞ to the cusp τ = 0. For N = 2 , 3 , 4 , th e operator s L N = D t N 2 + P ( t N ) · D t N + Q ( t N ) were com puted to be L 2 = D t 2 2 +  1 t 2 + 1 2 ( t 2 + 64 )  D t 2 + 1 16 t 2 ( t 2 + 64 ) , (6.2a) L 3 = D t 3 2 +  1 t 3 + 2 3 ( t 3 + 27 )  D t 3 + 1 9 t 3 ( t 3 + 27 ) , (6.2b) L 4 = D t 4 2 +  1 t 4 + 1 t 4 + 16  D t 4 + 1 4 t 4 ( t 4 + 16 ) . (6.2c) Each is a Gauss hy pergeometr ic operator, up to a scaling of the indep endent vari- able. That is, each has three (regular) sin gular points, located at t = t ∗ N , ∞ , 0 , where t ∗ N (respectively − 6 4 , − 27 , − 16) is the third ﬁxed po int of Γ 0 ( N ) on the quotien t curve X 0 ( N ) = Γ 0 ( N ) \ H ∗ ∼ = P 1 ( C ) t N . It is respec ti vely a qua dratic elliptic point, a cubic one, and a third cusp (the image of τ = 1 / 2), as mentioned in § 2. For each N , th ere is a solution f = h N ( t N ) of L N f = 0 that is h olomor phic and equal to unity at t N = 0. It was sho wn in [27] that if h N ( τ ) : = ( h N ◦ t N )( τ ) , then h 2 ( τ ) = [ 1 ] 4 / [ 2 ] 2 , h 3 ( τ ) = [ 1 ] 3 / [ 3 ] , h 4 ( τ ) = [ 1 ] 4 / [ 2 ] 2 . That is, the h olomorp hic lo cal solution of (6.1) at the inﬁnite cusp, in each c ase, can be continued to a weight-1 modular form on H ∗ . In fact, h 2 , h 3 , h 4 are respectively equal to B 4 , B 3 , B 2 in the notation of the present article (see Deﬁnition 3.1). For N = 2 , 3 , 4 , a weight-1 modular form ¯ h N ( τ ) = ( ¯ h N ◦ t N )( τ ) that v anishes at the inﬁnite cusp, and has zero ord er of vanishing at the cusp τ = 0 , is obtained by multiplying h N ( τ ) by an appropriate power of the Hauptmodul t N ( τ ) . Let ¯ h 2 ( t 2 ) = 2 − 3 / 2 t 1 / 4 2 h 2 ( t 2 ) , ¯ h 3 ( t 3 ) = 3 − 1 t 1 / 3 3 h 3 ( t 3 ) , ¯ h 4 ( t 4 ) = 2 − 2 t 1 / 2 4 h 4 ( t 4 ) . Then by Deﬁnition 3.1, ¯ h 2 ( τ ) , ¯ h 3 ( τ ) , ¯ h 4 ( τ ) are identical to C 4 , C 3 , C 2 . It follows by changing (dependent) v ariables in the equations L N h N = 0 that ¯ h N satisﬁes the slightly modiﬁed Picard–Fuch s equation ¯ L N ¯ h N = 0 , wher e ¯ L 2 = D t 2 2 +  1 2 t 2 + 1 2 ( t 2 + 64 )  D t 2 + 4 t 2 2 ( t 2 + 64 ) , (6.3a) ¯ L 3 = D t 3 2 +  1 3 t 3 + 2 3 ( t 3 + 27 )  D t 3 + 3 t 2 3 ( t 3 + 27 ) , (6.3b) ¯ L 4 = D t 4 2 +  0 t 4 + 1 t 4 + 16  D t 4 + 4 t 2 4 ( t 4 + 16 ) . (6.3c) The ﬁxed points of ea ch Γ 0 ( N ) on th e corr espondin g q uotient X 0 ( N ) ∼ = P 1 ( C ) t N are visible in (6.3), just as in (6.2). Each modiﬁed equation ¯ L N f = 0 is of the for m L α , β , γ f = 0 , wh ere L α , β , γ : = D t 2 +  α + β t + 1 − α t − t ∗  D t + [ γ 2 − ( 1 − α − β ) 2 ] t ∗ 4 t 2 ( t − t ∗ ) . (6.4) 24 Robert S. Maier The operato r L α , β , γ is the gener al second-order Fuchsian operator on P 1 ( C ) t that has singular points at t = t ∗ , ∞ , 0 with respective exp onent dif ference s α , β , γ , an d with on e expon ent at each of t = t ∗ , ∞ constrain ed to b e zero . It is o f hy pergeo- metric but not Gauss-hypergeometric typ e. The solutions of L α , β , γ f = 0 inclu de t ( 1 − α − β − γ ) / 2 2 F 1  1 − α − β − γ 2 , 1 − α + β − γ 2 ; 1 − γ ; t / t ∗  , (6.5) which is the local solutio n at t = 0 associated to the exp onent ( 1 − α − β − γ ) / 2. (Here, 2 F 1 ( λ , µ ; ν ; x ) is the Gau ss hy pergeometric f unction, d eﬁned and sing le- valued on the disk | x | < 1 .) This is the r epresentation of th e form ¯ h N = C r as a (multiv alued) fun ction of th e Hau ptmodu l t = t N . For N = 2 , 3 , 4 , the param eters ( α , β , γ ) , which are the reciprocals of the orders of elements of Γ 0 ( N ) that stabilize the correspon ding ﬁxed points, are respectiv ely ( 1 2 , 0 , 0 ) , ( 1 3 , 0 , 0 ) , ( 0 , 0 , 0 ) . By The orem 2.3(2), u = ( 2 π i / r ) E r equals ˙ C r / C r . Henc e, if one takes the o p- erator L ( Γ ) = L ( Γ 0 ( N )) to eq ual ¯ L N rather th an L N , the fun ction u of Th eo- rem 2.3(1 ) will ag ree with the fu nction u = ˙ f 2 / f 2 , in the no tation of Th eorem 6.1 and Lemma 6.2. Therefor e Theorem 2.3(1) will follow immediately from Proposition 6.3. Let τ = f 1 / f 2 , a ratio of indep endent s olution s of the hyper geo - metric ODE L α , β , γ f = 0 o n P 1 ( C ) t . Let u = ˙ f 2 / f 2 , a nd let u k , k = 4 , 6 , . . . be deﬁned by u 4 = ˙ u − u 2 and u k + 2 = ˙ u k − kuu k . Then u , r e garded a s a function of τ , will satisfy a nonlin ear thir d -or der ODE: the generalized Chazy equation u 4 u 8 − u 2 6 + 8 u 3 4 = 0 , if ( α , β , γ ) = ( 1 2 , 0 , 0 ) , u 4 u 2 8 − u 2 6 u 8 + 24 u 3 4 u 8 − 15 u 2 4 u 2 6 + 144 u 5 4 = 0 , if ( α , β , γ ) = ( 1 3 , 0 , 0 ) , u 4 u 8 − u 2 6 + 8 u 3 4 = 0 , if ( α , β , γ ) = ( 0 , 0 , 0 ) . Pr oo f. By dire ct c omputatio n, using th e expressions o f Lem ma 6 .2 for u 4 , u 6 , u 8 in term s o f the coefﬁcient function s P , Q ∈ C ( t ) , which can be read off fr om th e formu la (6.4 ) for L α , β , γ .  So, in each o f the cases N = 2 , 3 , 4 , i.e ., r = 4 , 3 , 2 , the qu asi-modu lar form u = ˙ C r / C r satisﬁes a generalized Chazy equation. This proof of Theorem 2.3(1) is more analytic than the proof giv en in § 6.1, an d less explicitly modular . The reader m ay wond er wheth er this seco nd, altern ativ e p roof was nece ssary . It requ ired extra mach inery , su ch as the Picard –Fuchs e quations L N h N = 0 and ¯ L N ¯ h N = 0 , and the Hauptmod uls t N , the q -expansions o f which are relativ ely com- plicated and are not discussed here. Also, Proposition 6.3 is restricted to very spe- cial triples of parameter values. In fact, Proposition 6.3 is th e tip of a n interesting iceb erg. The follo wing is its extension to arbitrary triples ( α , β , γ ) . Theorem 6.4 . Let τ = f 1 / f 2 , a ratio of indepen dent solution s o f the hyp er geo- metric ODE L α , β , γ f = 0 o n P 1 ( C ) t . Let u = ˙ f 2 / f 2 , a nd let u k , k = 4 , 6 , . . . be deﬁned by u 4 = ˙ u − u 2 and u k + 2 = ˙ u k − kuu k . Then u , r e garded a s a function of τ , will satisfy a nonlin ear thir d -or der ODE: a generalized Chazy equation C 88 u 2 4 u 2 8 + C 86 u 4 u 2 6 u 8 + C 84 u 4 4 u 8 + C 66 u 4 6 + C 64 u 3 4 u 2 6 + C 44 u 6 4 = 0 Nonlinear dif ferential equations for classical modular forms 25 with coefﬁcients symmetric under α ↔ β , n amely C 88 = ( 2 α − 1 )( 2 β − 1 )( α + β − γ − 1 ) 2 ( α + β + γ − 1 ) 2 , C 86 = −  ( 2 α − 1 )( 3 β − 1 ) + ( 3 α − 1 )( 2 β − 1 )  ( α + β − γ − 1 ) 2 ( α + β + γ − 1 ) 2 , C 84 = − 16 ( 2 α − 1 )( 2 β − 1 )( α + β − 1 )( α + β − γ − 1 )( α + β + γ − 1 ) , C 66 = ( 3 α − 1 )( 3 β − 1 )( α + β − γ − 1 ) 2 ( α + β + γ − 1 ) 2 , C 64 = 4  2 ( 2 α − 1 ) 2 ( 3 β − 1 ) + 2 ( 3 α − 1 )( 2 β − 1 ) 2 − 3 ( α − β ) 2  × ( α + β − γ − 1 )( α + β + γ − 1 ) , C 44 = 64 ( 2 α − 1 )( 2 β − 1 )( α + β − 1 ) 2 . Pr oo f. W ith the aid of a compu ter alge bra system, eliminate t f rom the expres- sions for u 8 / u 2 4 and u 2 6 / u 3 4 that follow from Lemm a 6.2. As in the pr oof of Pro po- sition 6.3, P , Q ∈ C ( t ) come from Eq. (6.4).  Theorem 6.4 is a nonlinear hyperg eometric identity , re lev ant e ven to hypergeo- metric equ ations withou t modu lar applications. It belong s to th e theory of special function s, but as one sees, in a loose sense it is a relation of linear d ependen ce among modu lar forms of weight 24 (since each monomial has that weight). Rational expo nent differences α , β , γ that are not member s of { 0 , 1 2 , 1 3 } occur in th e th eory of auto morph ic fun ctions on subgr oups of PSL ( 2 , R ) that a re n ot subgrou ps of Γ ( 1 ) = PSL ( 2 , Z ) . Th is will be illustrated in the next section. 7. Differential systems f or weight- 1 f orms The systems of Theorem 2.3(2), satisﬁed by triples A r , B r , C r , will no w be greatly generalized . As sociated to any ﬁr st-kind Fuchsian subgroup Γ < PSL ( 2 , R ) that is a triang le group , i.e. , that ha s a hy perbo lic triangular fu ndamen tal domain in H ∗ and a Hauptmodu l, there are weight-1 modular forms A , B , C (possibly multi val- ued) th at vanish respectively at the three vertices. The f orms will satisfy a system of coupled nonlin ear ﬁ rst-ord er ODEs with independen t variable τ . The key result o n this is Th eorem 7 .1, which is proved by hy pergeometr ic manipulatio ns related to those of Ohyama [3 0]. It deals with the ca se when Γ has at least one cusp, wh ich with out loss o f g enerality can be taken to be τ = i ∞ . In §§ 7.2 and 7.3, as illustrations, the triang le subgrou ps Γ < PSL ( 2 , R ) th at are commensu rable with Γ ( 1 ) = P SL ( 2 , Z ) a re examined. These in clude Γ ( 1 ) itself; Γ 0 ( N ) , N = 2 , 3; the Fricke extensions Γ + 0 ( N ) , N = 2 , 3; the ind ex-2 s ubgr oup Γ 2 of Γ ( 1 ) ; and two others [43]. For each such Γ , th e forms A , B , C are w orked out explicitly , as are the hypergeom etric representatio n of A , the d ifferential system the forms satisfy , and the generalized Chazy equation that the system implies. 7.1. Hyper geometric manipulations Let Γ < PSL ( 2 , R ) be a Fuchsian subg roup (of the ﬁrst kind), regarded as a group of M ¨ obius tran sformation s of H ∗ ∋ τ . If it h as a Hauptm odul t = t ( τ ) , i.e., is 26 Robert S. Maier of genu s zero, then Γ \ H ∗ ∼ = P 1 ( C ) t and τ can be expressed as a ratio τ 1 / τ 2 of two solution s f = τ 1 , τ 2 of a Picar d–Fuchs equ ation L ( Γ ) f = 0 on P 1 ( C ) t , as stated in Theorem 6.1. Its solution space will b e C τ 1 ⊕ C τ 2 = ( C τ ⊕ C ) τ 2 . The solution τ 2 can be viewed as a weight-1 form on Γ . Th is fo llows b y co nsidering the h omoge neous counterpart Γ < SL ( 2 , R ) to Γ , which acts on vectors  τ 1 τ 2  , and the associated homog eneous forms, which are functions of τ 1 , τ 2 . Suppose that Γ is a triangle gr oup, i.e., is of genus zero w ith a triangular fun da- mental d omain and hence with thr ee ineq uiv alent classes of ﬁxed po ints on H ∗ , say classes A,B,C. They correspond to three conjugacy classes of stabilizing elements of Γ , eith er elliptic or parabolic. T he group Γ is speciﬁed up to conjugacy by their orders, i. e., b y a signatu re ( n A , n B , n C ) such as the signa ture ( 3 , 2 , ∞ ) of Γ ( 1 ) . It will be assum ed that at le ast one of these ord ers is ∞ , i.e., at least o ne of th e three classes is parab olic, corr espondin g to a cusp. Without lo ss of generality one can take n C = ∞ , a nd the cu sp to b e τ = i ∞ . This inﬁn ite cusp will b e ﬁxed by some τ 7→ τ + υ , where by deﬁnition, υ ∈ R + is its width. By a M ¨ obius transf ormation the H auptmod ul t c an b e redeﬁned , if n ecessary , so that t = 0 a t the inﬁnite cu sp, and t = t ∗ , ∞ , for some t ∗ ∈ C \ { 0 } , on the ﬁxed points in th e classes A,B. The Picard –Fuchs equation, bein g hypergeom etric, will then ha ve t = t ∗ , ∞ , 0 as its (regular) singu lar points. Their respectiv e expo nent d if- ferences ( α , β , γ ) will equal ( 1 / n A , 1 / n B , 1 / n C ) . T hese are v ertex angles in terms of π radians, and necessarily α + β + γ < 1. (If the convention of the last paragraph is adop ted then n C = ∞ and γ = 1 / ∞ = 0 , but for reason s of symmetry n C and γ will be kept as independen t parameters.) Fuchs’ s relation on character istic expo nents implies that the six exponents o f any second-orde r ODE of hypergeometric typ e must sum to unity . This lea ves two degrees of freedom in the choice of L ( Γ ) , as in § 6.2. (Cf. L N vs. ¯ L N .) Let L ( Γ ) be chosen to have e xpone nts { 0 , α } at t = t ∗ , { 0 , β } at t = ∞ , { 1 − α − β − γ 2 , 1 − α − β + γ 2 } at t = 0 , (7.1) i.e., so that there is a zero exponent at each of t = t ∗ , ∞ . W ith this choice, L ( Γ ) = L α , β , γ : = D t 2 +  α + β t + 1 − α t − t ∗  D t + [ γ 2 − ( 1 − α − β ) 2 ] t ∗ 4 t 2 ( t − t ∗ ) , (7.2) deﬁned as in (6 .4). Let the local solu tion of L ( Γ ) f = L α , β , γ f = 0 at the singu lar point t = 0 (i. e., th e inﬁnite cusp) , associated to th e exponent ( 1 − α − β − γ ) / 2 , be deno ted C = C ( t ) . Then the lifte d function C ( τ ) : = C ( t ( τ )) will be a weight- 1 form on Γ , which vanishes at cusps in class C because α + β + γ < 1. Also, deﬁne (potentially multiv alued) functions A ( τ ) , B ( τ ) that vanish at the ﬁxed points in the classes A,B, at which t = t ∗ , resp. t = ∞ , by A = [( t − t ∗ ) / t ] 1 / ρ C , (7.3a) B = [ − t ∗ / t ] 1 / ρ C , (7.3b) where ρ : = 2 1 − α − β − γ = 2 1 − n − 1 A − n − 1 B − n − 1 C . (7.4) Nonlinear dif ferential equations for classical modular forms 27 W ith these deﬁnitions, A ρ = B ρ + C ρ . (7.5) The correspo nding quotients t / ( t − t ∗ ) = C ρ / A ρ , (7.6a) t / t ∗ = − C ρ / B ρ , (7.6b) are normalized Hauptmod uls, the respecti ve v alues of which on the classes A,B,C are ∞ , 1 , 0 and 1 , ∞ , 0. The Γ -speciﬁc qu antity ρ ∈ Q + of (7 .4) gen eralizes the ‘sig- nature’ r that par ametrizes Ramanuja n’ s alter native theo ries of ellip tic in tegrals. (Recall that r = 4 , 3 , 2 corr espond to Γ = Γ 0 ( 2 ) , Γ 0 ( 3 ) , Γ 0 ( 4 ) , i.e., to ( n A , n B , n C ) = ( 2 , ∞ , ∞ ) , ( 3 , ∞ , ∞ ) , ( ∞ , ∞ , ∞ ) .) The func tions A ( τ ) , B ( τ ) could also be deﬁned as A ( t ( τ )) and B ( t ( τ )) , wher e A ( t ) , B ( t ) are so lutions o f Picard– Fuchs equatio ns having ap propr iately m odiﬁed exponents, b ut the same expon ent dif fer ences as L ( Γ ) α , β , γ , i.e. , α , β , γ . (Cf. the rela- tion between L N , ¯ L N in § 6 .2.) Their respecti ve e xpon ents w ould be { 1 − α − β − γ 2 , 1 + α − β − γ 2 } at t = t ∗ , { 0 , β } at t = ∞ , { 0 , γ } at t = 0 , (7.7a) { 0 , α } at t = t ∗ , { 1 − α − β − γ 2 , 1 − α + β − γ 2 } at t = ∞ , { 0 , γ } at t = 0 . (7.7b) Gauss-hypergeo metric repr esentations of A , B , C in term s of 2 F 1 are A ( τ ) = A ( t ( τ )) = 2 F 1  1 − α − β − γ 2 , 1 + α − β − γ 2 ; 1 − γ ; t ( τ ) / [ t ( τ ) − t ∗ ]  , (7.8a) B ( τ ) = B ( t ( τ )) = 2 F 1  1 − α − β − γ 2 , 1 − α + β − γ 2 ; 1 − γ ; t ( τ ) / t ∗ ) , (7.8b) C ( τ ) = C ( t ( τ )) = ( − t / t ∗ ) 1 / ρ 2 F 1  1 − α − β − γ 2 , 1 − α + β − γ 2 ; 1 − γ ; t ( τ ) / t ∗ ) , (7.8 c) in which the normalizations of A , B , C , not pre viously speciﬁed , ha ve been set by requirin g that A , B equ al u nity at th e inﬁnite cusp , a t which t = 0 . The p arameters and arguments of the 2 F 1 ’ s are d etermined b y 2 F 1 ( λ , µ ; ν ; x ) having expo nents { 0 , 1 − ν } at x = 0 , { 0 , ν − λ − µ } at x = 1 , and { λ , µ } at x = ∞ . The representations A = 2 F 1  1 − α − β − γ 2 , 1 + α − β − γ 2 ; 1 − γ ; C ρ / A ρ  , (7.9a) B = 2 F 1  1 − α − β − γ 2 , 1 − α + β − γ 2 ; 1 − γ ; − C ρ / B ρ  (7.9b) follow fr om (7.8ab) with th e aid of ( 7.6ab). The id entities (7. 9ab) ar e equiv alent: they ar e related by Pfaff ’ s transform ation of 2 F 1 . T he functio n 2 F 1 ( λ , µ ; ν ; x ) is deﬁned on the d isk | x | < 1 , so (7. 9ab) hold in a neig hbor hood of the inﬁnite cusp, at which C = 0. In fact each will hold n ear any cusp in the class C, if an appro priate constant of propor tionality is included. Similarly , there follow C ∝ 2 F 1  1 − α − β − γ 2 , 1 − α − β + γ 2 ; 1 − α ; A ρ / C ρ  , (7.10a ) C ∝ 2 F 1  1 − α − β − γ 2 , 1 − α − β + γ 2 ; 1 − β ; − B ρ / C ρ  , (7.10b ) which ho ld near any ﬁxed poin t in the class A, r esp. B, with th e con stant of pro- portion ality dependent on the choice of ﬁxed point. 28 Robert S. Maier As d eﬁned, A , B , C ar e for mally we ight-1 mo dular fo rms on Γ , with some multiplier system s; and they vanish respectively on the cla sses A,B,C of ﬁxed points of Γ on H ∗ . Howe ver , if A, r esp. B com prises elliptic ﬁxed p oints, then A , resp. B may be a m ultiv alued function of τ . Th is is because of the f ractional powers in their deﬁnitions (7.3ab). The test for multi valuedness on H is as follows. By construc tion, each of A , B , C has order o f v anishing ( compu ted with respect to a local parame ter for Γ , e.g., t ) equal to 1 / ρ . Fixed points in c lasses A,B,C are m apped to Γ \ H ∗ ∼ = P 1 ( C ) t with multiplicities n A , n B , n C . If n A < ∞ , resp. n B < ∞ , signaling ellipticity , th en the order of vanishing of A , B at the associated elliptic po ints on H , in clas ses A,B, will b e n A / ρ , n B / ρ ∈ Q + . If this is n ot a n in teger then A , resp. B will be mu l- ti valued, i.e., the k ’th roo t of a true modu lar form (of weight k ), where k eq uals the numerato r of the fraction ρ / n A = ρ α , re sp. ρ / n B = ρ β , expr essed in lo we st terms. The generalizatio n of Theor em 2.3(2) to a rbitrary triangle grou ps can now b e stated and proved. As alw ays, ′ signiﬁes q d / d q = ( 2 π i ) − 1 d / d τ . Theorem 7.1 . Let Γ < PSL ( 2 , R ) be a triangle g r ou p with signatur e ( n A , n B , n C ) and exponent parameters ( α , β , γ ) = ( 1 / n A , 1 / n B , 1 / n C ) , with α + β + γ < 1 , and let ρ : = 2 1 − α − β − γ . Assume that γ = 0 , i.e., th at the thir d vertex is a cusp , and deﬁne the formal weight- 1 modular forms A , B , C as above, vanishing at the classes A,B,C of ﬁ xed po ints of Γ on H ∗ (the last con taining the inﬁ nite cusp, of width υ ), an d satisfying A ρ = B ρ + C ρ . Then A ρ , B ρ , C ρ , alon g with the weight- 2 , depth - 6 1 qua si-modular form E : = υ ( C ρ ) ′ / ( C ρ ) tha t is associated with class C, satisfy the coupled system of nonlin ear ﬁrs t-order equations υ ( A ρ ) ′ = E · A ρ − A ρ ( 1 − α ) B ρ ( 1 − β ) , (7.11a ) υ ( B ρ ) ′ = E · B ρ − A ρ ( 1 − α ) B ρ ( 1 − β ) , (7.11b) υ ( C ρ ) ′ = E · C ρ , (7.11c ) υ ρ E ′ = E · E − A ρ ( 1 − 2 α ) B ρ ( 1 − 2 β ) , (7. 11d) fr om which a generalized Chazy equ ation C p for u = ( 2 π i / υ ρ ) E , parametrized by α , β ; γ as in Theorem 6. 4 , ca n b e derived by eliminatio n. ( The third equation says that u = ˙ C / C .) Pr oo f. Equation (7. 11c) is true by deﬁnition , an d (7.11 a) is implied b y (7.11 b ), (7.11c) and A ρ = B ρ + C ρ . It r emains to prove (7.11b),(7.11 d ). They will come from a useful formula for ˙ t = d t / d τ , dedu ced as follo ws. The s olution space of the Picard–Fuchs equation L α , β , γ f = 0 is C f 1 ⊕ C f 2 : = C τ C ⊕ C C = ( C τ ⊕ C ) C , wher e τ is viewed as a (mu ltiv alued) f unction o n th e quotient curve P 1 ( C ) t , and C is deﬁned in terms o f 2 F 1 by (7.8c). From the exp res- sion for L α , β , γ giv en in (7.2), the Wron skian w = w ( f 1 , f 2 ) = w ( τ C , C ) must eq ual a multiple of 1 / t α + β ( t − t ∗ ) 1 − α . The constant of proportio nality can be calculated by taking the t → 0 limit, in which the inﬁnite cusp is approach ed. In this limit, τ ∼ ( υ / 2 π i ) log t , C ∼ ( − t / t ∗ ) 1 / ρ , (7.12) Nonlinear dif ferential equations for classical modular forms 29 the form er co ming f rom t ∼ const · q 1 / υ , which is tr ue since υ is the width of the inﬁnite cusp. One readily deduces that 1 / w ( t ) = h 2 π i ( − t ∗ ) − β / υ i · t α + β ( t − t ∗ ) 1 − α . (7.13) As in th e pro of o f Lemma 6. 2, ˙ t = f 2 2 / w , i.e., ˙ t = C 2 ( τ ) / w ( t ( τ )) on H . T aking account of (7.3ab), one can rewrite this in the useful form ˙ t = [ 2 π i ( − t ∗ ) / υ ] A ρ ( 1 − α ) B − ρ ( 1 + β ) C ρ . ( 7.14) Now consider the logarithmic deri vati ve of the equality B ρ = ( − t ∗ / t ) C ρ , i.e. , ( B ρ ) ′ / B ρ = ( C ρ ) ′ / C ρ − t ′ / t . ( 7.15) By emp loying (7.14) to expand t ′ = ˙ t / 2 π i , one ob tains Eq. ( 7.11b). Equ ation (7.11d) follows by similar manipulation s.  Let E A and E B denote the weight-2 , d epth- 6 1 quasi-mo dular forms associated with classes A an d B, i.e., υ ( A ρ ) ′ / A ρ and υ ( B ρ ) ′ / B ρ , just as the for m E C = E = υ ( C ρ ) ′ / C ρ is associated with class C. More over , let u A , u B , u C denote the normalized forms ( 2 π i / υ ρ ) E A , ( 2 π i / υ ρ ) E B , ( 2 π i / υ ρ ) E C , so that u A = ˙ A / A , u B = ˙ B / B , u C = ˙ C / C . Then a bit of calculus applied to Eq s. (7.11abcd) yields Corollary 7.2. The w eight - 2 , d epth- 6 1 quasi-mod ular forms u A , u B , u C satisfy ˙ u A = u 2 A − ( 1 + ρ α )( u A − u B )( u A − u C ) , ˙ u B = u 2 B − ( 1 + ρ β )( u B − u C )( u B − u A ) , ˙ u C = u 2 C − ( 1 + ρ γ )( u C − u A )( u C − u B ) . This is a so-called gen eralized Darbo ux–Halph en (gDH) system o f ODEs [1, 10, 22]. It is evident that the gDH system with ( α , β , γ ) = ( 1 / n A , 1 / n B , 1 / n C ) an d ρ = 2 1 − α − β − γ arises naturally fro m the unique (up to conju gacy) triangle subgroup of PS L ( 2 , R ) with sig nature ( n A , n B , n C ) . Exam ples of g DH systems coming from modular subgroup s ha ve appeared in the literature; e.g., the ones coming from the six (up to con jugacy) trian gle sub group s of Γ ( 1 ) = PS L ( 2 , Z ) , w hich are inci- dentally the o nly g DH systems for which some linea r com bination of u A , u B , u C satisﬁes the classical Chazy equation [10]. Ho wever , the general statement is ne w . 7.2. T riangle gr oups commensurable with Γ ( 1 ) The triang le subg roups o f PS L ( 2 , R ) co mmensur able with Γ ( 1 ) = PSL ( 2 , Z ) are well kn own. (Subgro ups Γ 1 , Γ 2 are said to be commen surable if Γ 1 ∩ Γ 2 is of ﬁnite index in both.) Up to conjugacy the re are e xactly nine [43], lis ted in T able 3. Each is h yperb olic with at least one cu sp. T hey are of th ree type s, and it will be shown that to eac h ty pe there is an associa ted d ifferential system, parame trized by ρ , which is satisﬁed by we ight-1 form s A , B , C . For the ﬁrst typ e the system will 30 Robert S. Maier be that of Theorem 2.3(2), in which ρ equals the signature of R amanu jan’ s elliptic theories (i.e., ρ = r = 4 , 3 , 2). T yp e I com prises Γ 0 ( N ) , N = 2 , 3 , 4 , and T ype II comprises Γ ( 1 ) an d the Fricke extensions Γ + 0 ( N ) , N = 2 , 3 , which are not sub grou ps o f Γ ( 1 ) . T y pe III comprises three grou ps that will be called 2a ′ , 4a ′ , 6a ′ . The group 2a ′ is the index-2 subgroup Γ 2 < Γ ( 1 ) , but the latter two are not subgr oups of Γ ( 1 ) . These n ames ar e taken from Harn ad and McKay [22] 1 . It is known th at the intersections of the grou ps 4a ′ , 6a ′ with Γ ( 1 ) are Γ 0 ( 4 ) ∩ Γ ( 2 ) , Γ 0 ( 3 ) ∩ Γ ( 2 ) , which are conju gates of Γ 0 ( 8 ) , Γ 0 ( 12 ) under τ 7→ 2 τ . For each triang le grou p Γ , the expone nts ( α , β , γ ) = ( 1 / n A , 1 / n B , 1 / n C ) are giv en; a s is ρ = 2 / ( 1 − α − β − γ ) , which subsumes the signatur e of Ramanu- jan’ s theories. For concreteness, ge nerators a , b , c ∈ Γ of c orrespon ding stabi- lizing sub group s are given as well. ( What are given are homog eneous version s ¯ a , ¯ b , ¯ c ∈ GL ( 2 , R ) . T o co n vert th em to a , b , c ∈ PS L ( 2 , R ) , satisfyin g cab = ± I , divide ea ch b y its deter minant and prepend ± .) These generato rs are adap ted from [22, T able 1]. By convention, n C = ∞ and γ = 0 , and the co rrespond ing class C of ﬁxed points ( cusps) inclu des the inﬁn ite cusp. The width of the inﬁnite cusp is denoted υ , as above. The expressions for th e fo rmal (i.e., potentially multivalued) weight- 1 m odu- lar fo rms A , B , C , which in e ach c ase satisfy A ρ = B ρ + C ρ , come as fo llows. Starting with the on es fo r Γ of T ype I and for Γ = Γ ( 1 ) , which have a lready been discussed, they are obtaine d as con sequences o f the in dex-2 subg roup r ela- tions [44] Γ 0 ( 2 ) < 2 Γ + 0 ( 2 ) , Γ 0 ( 3 ) < 2 Γ + 0 ( 3 ) ; (7.16) 2a ′ < 2 Γ ( 1 ) , 4 a ′ < 2 Γ + 0 ( 2 ) , 6 a ′ < 2 Γ + 0 ( 3 ) . (7.17) For instance, suppo se th at ˜ Γ < 2 Γ and that the resp ectiv e signatures satisfy 2 ˜ ρ = ρ . The goal is to relate the associated triples ˜ A , ˜ B , ˜ C and A , B , C . Let the corre - sponding classes of ﬁx ed points be ˜ A , ˜ B , ˜ C and A , B , C. Suppo se that ˜ B , ˜ C are cusp classes und er ˜ Γ , which merge into a sing le class C under Γ , b ut that A = ˜ A. This is precisely what happen s when ( ˜ Γ , Γ ) = ( Γ 0 ( N ) , Γ + 0 ( N )) for N = 2 , 3. Under these assumption s, o ne will have A = ˜ A . Expressions for B , C in terms of ˜ B , ˜ C com e from a Haup tmodu l relation . Hauptmod uls fo r ˜ Γ , Γ , i.e., ratio nal pa- rameters on the quotien t cu rves H ∗ \ ˜ Γ , H ∗ \ Γ , will be ˜ λ = ˜ C ˜ ρ / ˜ A ˜ ρ , resp. λ = C ρ / A ρ . T he ind ex-2 relation ˜ Γ < 2 Γ induces a double covering H ∗ \ ˜ Γ → H ∗ \ Γ , i.e., P 1 ( C ) ˜ λ → P 1 ( C ) λ , i.e. , a quadratic rational map ˜ λ 7→ λ . Since ˜ λ = 1 , 0 (cor- respond ing to classes ˜ B , ˜ C) must be map ped to λ = 0 (cor respondin g to class C), and ˜ λ = ∞ (cor respond ing to class ˜ A) mu st be ma pped to λ = ∞ ( correspo nding to class A), the map must be λ = 4 ˜ λ ( 1 − ˜ λ ) = 1 − ( 1 − 2 ˜ λ ) 2 . (7.18) 1 The primes on 2a ′ , 4a ′ , 6a ′ indicate a relation to t he groups labeled 2a , 4a , 6a in the ex- tended Conway–Norton classiﬁcation. In the Conway–Norton notation used in [22], t he T ype I groups Γ 0 ( 2 ) , Γ 0 ( 3 ) , Γ 0 ( 4 ) are referred to as 2B , 3B , 4C , and the T ype II groups Γ ( 1 ) , Γ + 0 ( 2 ) , Γ + 0 ( 3 ) as 1A , 2A , 3A. Nonlinear dif ferential equations for classical modular forms 31 T able 3. For each triangle su bgroup Γ < PSL ( 2 , R ) commensurab le with Γ ( 1 ) = PSL ( 2 , Z ) , the basic data and the triple A , B , C of (p ossibly mu ltiv alued) weight-1 modular f orms, satisfying A ρ = B ρ + C ρ . The nine subgro ups are partitioned into T ypes I,II,III. If n A < ∞ , resp. n B < ∞ , then the minimum po wer of A , resp. B , which is single-valued, equals the numerator of ρ α , resp. ρ β , expressed in lowest terms. The forms A , B on 2a ′ = Γ 2 can alternativ ely be written as ( B 2 2 − ¯ ζ 3 C 2 2 ) 1 / 2 ( q 1 / 2 ) and ( B 2 2 − ζ 3 C 2 2 ) 1 / 2 ( q 1 / 2 ) , where ζ 3 = exp ( 2 π i / 3 ) ; and the forms A , B on 4a ′ as A 2 ± i C 2 . Γ ( n A , n A , n A ) ( α , β , γ ) ρ ¯ a ¯ b ¯ c υ A B C Γ 0 ( 2 ) ( 2 , ∞ , ∞ ) ( 1 2 , 0 , 0 ) 4  1 − 1 2 − 1   − 1 0 2 − 1   1 1 0 1  1 A 4 B 4 C 4 Γ 0 ( 3 ) ( 3 , ∞ , ∞ ) ( 1 3 , 0 , 0 ) 3  1 − 1 3 − 2   − 1 0 3 − 1   1 1 0 1  1 A 3 B 3 C 3 Γ 0 ( 4 ) ( ∞ , ∞ , ∞ ) ( 0 , 0 , 0 ) 2  1 − 1 4 − 3   − 1 0 4 − 1   1 1 0 1  1 A 2 B 2 C 2 Γ ( 1 ) ( 3 , 2 , ∞ ) ( 1 3 , 1 2 , 0 ) 12  0 − 1 1 − 1   0 − 1 1 0   1 1 0 1  1 E 4 1 / 4 E 6 1 / 6 ( 12 3 ∆ ) 1 / 12 Γ + 0 ( 2 ) ( 4 , 2 , ∞ ) ( 1 4 , 1 2 , 0 ) 8  0 − 1 2 − 2   0 − 1 2 0   1 1 0 1  1 A 4 h B 4 4 − C 4 4 i 1 / 4 2 1 / 4 √ B 4 C 4 Γ + 0 ( 3 ) ( 6 , 2 , ∞ ) ( 1 6 , 1 2 , 0 ) 6  0 − 1 3 − 3   0 − 1 3 0   1 1 0 1  1 A 3 h B 3 3 − C 3 3 i 1 / 3 2 1 / 3 √ B 3 C 3 2a ′ = Γ 2 ( 3 , 3 , ∞ ) ( 1 3 , 1 3 , 0 ) 6  1 − 3 1 − 2   0 − 1 1 − 1   1 2 0 1  2 h E 6 + i √ 12 3 ∆ i 1 / 6 h E 6 − i √ 12 3 ∆ i 1 / 6 ( 2i ) 1 / 6 ( 12 3 ∆ ) 1 / 12 4a ′ ( 4 , 4 , ∞ ) ( 1 4 , 1 4 , 0 ) 4  2 − 5 2 − 4   0 − 1 2 − 2   1 2 0 1  2 h B 4 2 + i C 4 2 i 1 / 2 h B 4 2 − i C 4 2 i 1 / 2 ( 4i ) 1 / 4 √ B 4 C 4 6a ′ ( 6 , 6 , ∞ ) ( 1 6 , 1 6 , 0 ) 3  3 − 7 3 − 6   0 − 1 3 − 3   1 2 0 1  2 h B 3 3 / 2 + i C 3 3 / 2 i 2 / 3 h B 3 3 / 2 − i C 3 3 / 2 i 2 / 3 ( 4i ) 1 / 3 √ B 3 C 3 32 Robert S. Maier Here, the propor tionality constant (i.e., 4) is determined by th e con dition th at λ = 1 , correspo nding to the class B of ﬁxed p oints un der Γ , should b e a criti- cal value of th e map . If it were n ot, then B would also be such a class under ˜ Γ ; which would violate the assumption that ˜ Γ is a triangle gr oup, with only three s uch classes. Using the above expressions for ˜ λ , λ , an d also the identities ˜ A ˜ ρ = ˜ B ˜ ρ + ˜ C ˜ ρ , A ρ = B ρ + C ρ , with ρ = 2 ˜ ρ , o ne immediately obtains A = ˜ A , (7.19a ) B = ˜ ρ p ˜ B ˜ ρ − ˜ C ˜ ρ , (7.19b) C = 2 1 / ˜ ρ p ˜ B ˜ C . (7.19c ) Applied to the pair s ( ˜ Γ , Γ ) of (7.16), these yield the triples A , B , C for Γ = Γ + 0 ( 2 ) , Γ + 0 ( 3 ) that are shown in T ab le 3 . A similar but ‘reversed’ proced ure, ap- plied to the ( ˜ Γ , Γ ) of (7. 17), allows th e triples A , B , C for the T ype-II I gr oups 2a ′ , 4a ′ , 6a ′ , to b e computed in terms of those for th e c orrespon ding T ype-I I gro ups Γ ( 1 ) , Γ + 0 ( 2 ) , Γ + 0 ( 3 ) . The resulting triples are giv en in the table. Alternative repr esentations for the fo rms A , B on the group 2a ′ = Γ 2 are supplied in the c aption, and are d erived as follows. Althoug h these forms are not single-valued, th eir squares A 2 , B 2 are single-valued, by the test for single- valuedness mention ed immediately befor e T heorem 7.1 (and r eprod uced in th e caption). Each of A 2 , B 2 has a { 1 , ζ 3 , ζ 2 3 } -valued mu ltiplier system, and sinc e Γ 2 has as in dex-3 subgro up th e prin cipal m odular subg roup Γ ( 2 ) , e ach of them lies in M 2 ( Γ ( 2 )) . Bu t Γ ( 2 ) is is conju gated to Γ 0 ( 4 ) b y τ 7→ 2 τ . Since B 2 2 , C 2 2 span M 2 ( Γ 0 ( 4 )) , the f orms A 2 , B 2 must be co mbination s of B 2 2 ( q 1 / 2 ) , C 2 2 ( q 1 / 2 ) , i.e., of ϑ 4 4 ( q 1 / 2 ) , ϑ 2 4 ( q 1 / 2 ) . The com bination s are easily worked out by linear al- gebra, if one expands to second order in q 2 = q 1 / 2 . 7.3. Explicit systems and Chazy equatio ns For each o f t he nine (conjugacy classes of) tr iangle groups Γ commensu rable with Γ ( 1 ) = PSL ( 2 , Z ) , the associated dif feren tial system and generalized Chazy equa- tion are computed belo w . They come respecti vely from Theorems 7.1 and 6.4. F or each Γ , a hypergeo metric (i.e., elliptic-integral) representatio n of the corre spond- ing weight-1 form A , comin g from Eq. (7.9a), is gi ven as well. As was e xplained in § 7.2, th ese triangle subgroups are of three types, denoted I,II,II I. Fro m a c lassical-analytic rather than a modular po int of vie w , the y dif fer in the depende nce of the exponent dif ferences ( α , β , γ ) on the signature ρ . • T yp e I, for which ( α , β , γ ) = ( 1 − 2 ρ , 0 , 0 ) . It compr ises Γ = Γ 0 ( 2 ) , Γ 0 ( 3 ) , Γ 0 ( 4 ) , for which ρ = 4 , 3 , 2; in each case the inﬁnite cusp has width υ = 1. F or each o f these group s the associated trip le A ρ , B ρ , C ρ of weight-1 forms equals A ρ , B ρ , C ρ , and th e weight-2 quasi-mo dular fo rm E ρ : = υ ( C ρ ρ ) ′ / ( C ρ ρ ) equals E ρ . The system satisﬁed by A ρ , B ρ , C ρ ; E ρ and the generalized Chazy equation satisﬁed by u = ( 2 π i / ρ ) E ρ were giv en in Theorem 2.3. Nonlinear dif ferential equations for classical modular forms 33 By Eq. (7 .9a), the hyp ergeometric repre sentation for A ρ = A ρ is A ρ = ˆ K I ρ ( λ ρ ) , where λ ρ : = C ρ ρ / A ρ ρ is a Hauptmo dul for Γ and ˆ K I ρ ( λ ρ ) : = 2 F 1 ( 1 ρ , 1 − 1 ρ ; 1; λ ρ ) (7.20) = sin ( π / ρ ) π Z 1 0 x − 1 / ρ ( 1 − x ) − 1 + 1 / ρ ( 1 − λ ρ x ) − 1 / ρ d x is the (normalized) T ype-I complete elliptic integral. These cases of T ype I cor- respond to Ramanujan’ s elliptic theories of signature ρ , for ρ = 4 , 3 , 2 (see [3, 7]). Th e classical (Jaco bi) case is ρ = 2 , and ˆ K I 2 is the (norm alized) complete integral ˆ K , which was introd uced in § 5. • T yp e II, fo r which ( α , β , γ ) = ( 1 2 − 2 ρ , 1 2 , 0 ) . It comp rises Γ = Γ ( 1 ) , Γ + 0 ( 2 ) , Γ + 0 ( 3 ) , for wh ich ρ = 12 , 8 , 6; in each case the inﬁnite cusp has width υ = 1. The associated triples A ρ , B ρ , C ρ of weigh t-1 for ms are in T ab le 3 . By direct computatio n, the weight-2 quasi-modular forms E ρ : = υ ( C ρ ρ ) ′ / ( C ρ ρ ) are E 12 ( q ) = E 2 ( q ) (7.21) = 1 − 24 ∞ ∑ n = 1 σ 1 ( n ; 1 ) q n = 1 − 24 ∞ ∑ n = 1 σ c 1 ( n ; 1 ) q n , E 8 ( q ) = 1 3  2 E 2 ( q 2 ) + E 2 ( q )  (7.22) = 1 − 8 ∞ ∑ n = 1 σ 1 ( n ; 2 , 1 ) q n = 1 − 8 ∞ ∑ n = 1 σ c 1 ( n ; 3 , 1 ) q n , E 6 ( q ) = 1 4  3 E 2 ( q 3 ) + E 2 ( q )  (7.23) = 1 − 6 ∞ ∑ n = 1 σ 1 ( n ; 2 , 1 , 1 ) q n = 1 − 6 ∞ ∑ n = 1 σ c 1 ( n ; 4 , 1 , 1 ) q n . They lie resp ectiv ely in M 6 1 2 ( Γ ( 1 )) , M 6 1 2 ( Γ + 0 ( 2 )) , M 6 1 2 ( Γ + 0 ( 3 )) . By Theo - rem 7.1, the differential system parametrized by ρ is ( A ρ ρ ) ′ = E ρ · A ρ ρ − A ρ ρ / 2 + 2 B ρ ρ / 2 , (7.24a) ( B ρ ρ ) ′ = E ρ · B ρ ρ − A ρ ρ / 2 + 2 B ρ ρ / 2 , (7.24b ) ( C ρ ρ ) ′ = E ρ · C ρ ρ , (7.24c ) ρ E ′ ρ = E ρ · E ρ − A ρ 4 . (7.24d ) This is an extension o f Ramanuja n’ s P – Q – R system ( 2.5abcd), to which it reduces when ρ = 12 and Γ = Γ ( 1 ) . The gener alized Ch azy eq uation satisﬁed by u = ( 2 π i / υ ρ ) E ρ = ˙ C ρ / C ρ , ac- cording to Theore m 6.4, is the non linear third-ord er ODE C p ρ , i.e. , p ρ = 0 , in which the polynomial p ρ ∈ C [ u 4 , u 6 , u 8 ] is de ﬁned by p 12 = u 8 + 24 u 2 4 , (7.25) p 8 = 2 u 4 u 8 − u 2 6 + 32 u 3 4 , (7.26) p 6 = 4 u 4 u 8 − 3 u 2 6 + 48 u 3 4 , (7.27) 34 Robert S. Maier and u 4 , u 6 , u 8 were given in Deﬁnition 2.1. T he differential eq uation C p 12 as- sociated to Γ ( 1 ) , c oming fr om ( 7.25), is th e c lassical Chazy equation ( 2.3a) that is satisﬁed by u = ( 2 π i / 12 ) E 2 . The po lynom ials (7.26),(7. 27) yield th e generalized Chazy equations associated to Γ + 0 ( 2 ) , Γ + 0 ( 3 ) , wh ich are ne w . By Eq . (7 .9a), th e hypergeo metric representation for the weight-1 form A ρ is A ρ = ˆ K II ρ ( λ ρ ) , in which λ ρ : = C ρ ρ / A ρ ρ is a Hauptmo dul for Γ and ˆ K II ρ ( λ ρ ) : = 2 F 1 ( 1 ρ , 1 2 − 1 ρ ; 1; λ ρ ) ( 7.28a) = cos ( π / ρ ) π Z 1 0 x − 1 / 2 − 1 / ρ ( 1 − x ) − 1 / 2 + 1 / ρ ( 1 − λ ρ x ) − 1 / ρ d x is the (normalized ) T y pe-II complete elliptic integral. Equi valently , h ˆ K II ρ ( λ ρ ) i 2 = 3 F 2 ( 2 ρ , 1 2 , 1 − 2 ρ ; 1 , 1 ; λ ρ ) . (7.28b ) Such representatio ns, wh en ρ = 12 and Γ = Γ ( 1 ) , are fairly well known. By T able 3, the ρ = 12 versions of (7.28ab) are E 4 1 / 4 = 2 F 1 ( 1 12 , 5 12 ; 1; 12 3 / j ) , (7.29a ) E 4 1 / 2 = 3 F 2 ( 1 6 , 1 2 , 5 6 ; 1 , 1 ; 12 3 / j ) , (7.29 b) where j = E 4 3 / ∆ = 12 3 E 4 3 / ( E 4 3 − E 6 2 ) is th e Klein –W eber inv ariant, the canonical Hauptm odul for Γ ( 1 ) , so that 12 3 / j = ( E 4 3 − E 6 2 ) / E 4 3 . Equ ation (7.29a) was kn own to Dede kind and was rediscovered by Stiller [42]. These identities hold in a neighb orho od of th e inﬁn ite cusp , at which j = ∞ an d 12 3 / j = 0. In the same way , Eq . (7.9b) yields E 6 1 / 6 = 2 F 1  1 12 , 7 12 ; 1; 1 2 3 / ( 12 3 − j )  . (7.30) From (7.10a) one also has ∆ 1 / 12 ∝ 2 F 1 ( 1 12 , 1 12 ; 2 3 ; j / 12 3 ) , (7.31a ) ∆ 1 / 6 ∝ 3 F 2 ( 1 6 , 1 6 , 1 6 ; 1 3 , 2 3 ; j / 1 2 3 ) , (7.31b ) which hold n ear any cu bic elliptic ﬁxed point, wh ere j / 12 3 = 0. (E.g ., near τ = ζ 3 = exp ( 2 π i / 3 ) .) The co nstants of pr oportio nality depend o n the choice of ﬁxed point. The ρ = 8 , 6 rep resentations, for Γ = Γ + 0 ( 2 ) , Γ + 0 ( 3 ) , were derived by Zudilin [47, Eqs. (23bc)]. The corresponding differential systems that he obtained [47, Props. 6,7] are equ i valent to the ρ = 8 , 6 ca ses of the system ( 7.24abcd), but are more complicated as they are not e xpressed in terms of weight-1 forms. • T yp e III, for wh ich ( α , β , γ ) = ( 1 2 − 1 ρ , 1 2 − 1 ρ , 0 ) . It co mprises Γ = ( 2a ′ = Γ 2 ) , 4a ′ , 6a ′ , for which ρ = 6 , 4 , 3; in each case th e inﬁnite c usp has width υ = 2. The associated tr iples A ρ , B ρ , C ρ of weight-1 f orms ar e in T able 3. T he weight-2 quasi-modular f orms E ρ : = υ ( C ρ ρ ) ′ / ( C ρ ρ ) , ρ = 6 , 4 , 3 , are identical to the T ype- II forms E 12 , E 8 , E 6 , given in Eqs. (7.21),(7.22),(7.23). Nonlinear dif ferential equations for classical modular forms 35 By Theorem 7.1, the differential system parametrized by ρ is 2 ( A ρ ρ ) ′ = E ρ · A ρ ρ − A ρ ρ / 2 + 1 B ρ ρ / 2 + 1 , (7.32a ) 2 ( B ρ ρ ) ′ = E ρ · B ρ ρ − A ρ ρ / 2 + 1 B ρ ρ / 2 + 1 , (7.32b) 2 ( C ρ ρ ) ′ = E ρ · C ρ ρ , (7.32c ) 2 ρ E ′ ρ = E ρ · E ρ − A ρ 2 B ρ 2 . ( 7.32d ) Although this system is sign iﬁcantly different fr om (7.24abcd), the system o f T yp e II, the resulting gen eralized Chazy equations C p ρ , ρ = 6 , 4 , 3 , are id enti- cal to the T ype -II equations for ρ = 12 , 8 , 6 (see (7.25),(7.26),(7.27)). By Eq . (7 .9a), th e hypergeo metric representation for the weight-1 form A ρ is A ρ = ˆ K III ρ ( λ ρ ) , in which λ ρ : = C ρ ρ / A ρ ρ is a Hauptmo dul for Γ and ˆ K III ρ ( λ ρ ) : = 2 F 1 ( 1 ρ , 1 2 ; 1; λ ρ ) (7.33) = 1 π Z 1 0 x − 1 / 2 ( 1 − x ) − 1 / 2 ( 1 − λ ρ x ) − 1 / ρ d x is the (no rmalized) T y pe-III complete elliptic integral. These represen tations are new . The case ρ = 4 , i.e., Γ = 4a ′ , is espec ially noteworthy . It follows from the formulas A 4 , B 4 = A 2 ± i C 2 = ϑ 3 2 ± i ϑ 2 2 (7.34) A 4 4 = B 4 4 + C 4 4 (7.35) that when ρ = 4 , the equ ation A ρ = ˆ K III ρ ( λ ρ ) specialize s to ϑ 3 2 ± i ϑ 2 2 = 2 F 1  1 4 , 1 2 ; 1; ( ϑ 3 2 ± i ϑ 2 2 ) 4 − ( ϑ 3 2 ∓ i ϑ 2 2 ) 4 ( ϑ 3 2 ± i ϑ 2 2 ) 4  . (7 .36) It is unclear whether this remarkable theta identity has a non-mo dular proof. 7.4. Discussion The results of § 7 .3 suggest that elliptic integrals of T yp es II and III (parametrize d by the sign ature ρ ) deserve fu rther study , much like th e elliptic integrals of T ype I, which are those of Raman ujan’ s alternative theor ies [3]. His th eories ﬁt into a larger framew ork: one that is lar ger by a factor of three, at least. The ne w generalized Chazy equations C p are especially interesting, since they open a ‘mod ular window’ into the space of no nlinear th ird-ord er OD Es. Each of the gener alized Chazy equ ations derived in this article can be integrated in closed form in terms of mod ular function s. This has taxon omic ramiﬁcatio ns. Th e clas- sical Chazy eq uation u 8 + 24 u 2 4 = 0 has the P ainlev ´ e pr operty , in that its so lu- tions have no ‘m ovable bra nch points’ [2, § 7.1 .5 a nd Ex. 6 .5.14] . The no nlinear third-or der ODEs in u which have this prop erty , an d in which ... u is po lynomial in ¨ u , ˙ u , u and rational in x , were classiﬁed by Chazy [11] into classes number ed 36 Robert S. Maier I thro ugh XIII. But to date, there has been no extension of his sch eme to third-order ODEs with the property , in which ... u is non-p olynom ial but r ationa l in ¨ u , ˙ u , u . The equations C p of this article provide examples. (F or the deﬁning polynomi- als p ∈ C [ u 4 , u 6 , u 8 ] , see Eqs. (2.6)–(2.8) and (7.25)–(7.2 7 ).) Each equation p = 0 is a no nlinear third-o rder ODE satisﬁed by u = ˙ C / C , wher e C is the weight-1 modular form that vanishes on t he third class o f ﬁxed poin ts o f a triangle su bgrou p Γ < PSL ( 2 , R ) with sign ature ( n A , n B ; n C ) . Oth er than C p 3 , th e rather comp li- cated weig ht-20 ODE comin g from Γ = Γ 0 ( 3 ) , these n onlinear ODEs lie in a single new clas s. W ith one seeming exception, each is of the form ( M − 2 ) u 4 u 8 − ( M − 3 ) u 2 6 + 8 M u 3 4 = 0 . (7.3 7) Equation (7.37) is of weight 12 unless M = 3 , in which case the u 2 6 term drops out and it red uces to the classical Ch azy equ ation, of weight 8 . By Th eorem 6.4, Eq. ( 7.37) comes from the triang le gro ups with sign atures ( M , 2; ∞ ) and ( M , M ; ∞ ) , i.e., with vertex ang les ( α , β ; γ ) (expressed in terms o f π radians) equ al to ( 1 M , 1 2 ; 0 ) or ( 1 M , 1 M ; 0 ) . T he a bovementioned seeming exception is the gen eralized Chazy equation attached to Γ 0 ( 2 ) and Γ 0 ( 4 ) , which must be obtaine d from Eq . (7 .37) b y taking a f ormal M → ∞ limit. But such limits are familiar from Ch azy’ s ana lysis. For in stance, the classical Chazy equ ation, wh ich is a ttached to the g roups Γ ( 1 ) and Γ 2 , with respecti ve sign atures ( 3 , 2; ∞ ) and ( 3 , 3 ; ∞ ) , is also the fo rmal N → ∞ limit of the Chazy-XII equation ( N 2 − 36 ) u 8 + 24 N 2 u 2 4 = 0 , (7.38) which is of weight 8 and comes fro m the triangle g roup s with signature s ( 3 , 2; N ) and ( 3 , 3; N / 2 ) . T o date, the Chazy -XII class is the only on e that has been given a modular interpretation , e.g., expressed in terms of the forms u 4 , u 6 , u 8 . One expe cts that wh en a n extend ed Chazy classiﬁcation is ﬁnally con structed by non -modu lar , classical-analytic tech niques, the equatio n (7 .37), p arametrized by in teger M , will b elong to an add itional ‘mo dular’ class. Inter estingly , it is the limiting ( N → ∞ ) case of a two-parameter generalized Chazy equatio n,  ( M − 2 ) 2 N 2 − 4 M 2  ·  ( M − 2 ) u 4 u 8 − ( M − 3 ) u 2 6  + 8 M ( M − 2 ) 2 N 2 u 3 4 = 0 , (7.39) which is also of weight 12 (g enerically) . By Theorem 6.4, Eq. (7. 39) comes fr om the triangle group s with signatur es ( M , 2; N ) and ( M , M ; N / 2 ) , i.e., with ( α , β ; γ ) equal to ( 1 M , 1 2 ; 1 N ) o r ( 1 M , 1 M ; 2 N ) . The fund amental do mains of the latter tria ngle group s are hy perbolic isosceles trian gles. Equatio n (7 .39) red uces to Eq. (7.38), the weight-8 Chazy–XII equation , when M = 3. In a similar way , one can derive a two-p arameter g eneralized Chazy equ ation extending C p 3 , the weight- 20 ODE that comes from Γ 0 ( 3 ) . Th e extension , also of weight 20 (gene rically), is  ( 2 M − 3 ) 2 N 2 − 9 M 2  2 ·  ( M − 2 ) u 4 u 8 − ( M − 3 ) u 2 6  u 8 + 12 M N 2  ( 2 M − 3 ) 2 N 2 − 9 M 2  · u 2 4  4 ( M − 2 )( 2 M − 3 ) u 4 u 8 − ( M − 3 )( 5 M − 9 ) u 2 6  + 576 M 2 ( M − 2 )( 2 M − 3 ) 2 N 4 · u 5 4 = 0 . (7.40) Nonlinear dif ferential equations for classical modular forms 37 By Theorem 6.4, this no nlinear ODE comes f rom th e triangle gro up with signature ( 3 , M ; N ) , i.e ., with ( α , β ; γ ) equal to ( 1 3 , 1 M ; 1 N ) . I t r educes to C p 3 when M → ∞ and N → ∞ , to th e Chazy–XII equation when M = 2 , an d to th e classical Chazy equation when M = 2 and N → ∞ , or when M = 3 and N → ∞ . Writing u 8 , u 6 , u 4 in terms of ... u , ¨ u , ˙ u , u , one sees that like C p 3 itself, the ODE (7.40), when M 6 = 2 , expresses ... u as a degree-2 algebraic function of ¨ u , ˙ u , u , rath er than a rational function . It is of a more general type than the ODE (7.39). Ap pendix: Theta representations and A GM identities The functions A r , B r , C r , r = 2 , 3 , 4 , satisfying A r r = B r r + C r r on th e d isk | q | < 1 , were origin ally deﬁned b y the Borwein s [ 7] as the sum s of theta series of certain quadra tic form s, o ccurring in Ramanujan’ s theories of elliptic functions to altern a- ti ve bases [3]. T o facilitate com parison, several of their r esults are restated b elow in the no tation of the pr esent article. They deﬁned ( A 2 , B 2 , C 2 ) = ( ϑ 3 2 , ϑ 4 2 , ϑ 2 2 ) , ( A 4 2 , B 4 2 , C 4 2 ) = ( ϑ 2 4 + ϑ 3 4 , ϑ 4 4 , 2 ϑ 2 2 ϑ 3 2 ) , an d also A 3 ( q ) = ∑ n , m ∈ Z q n 2 + nm + m 2 , (A.1a) B 3 ( q ) = ∑ n , m ∈ Z ζ n − m 3 q n 2 + nm + m 2 , (A.1b) C 3 ( q ) = ∑ n , m ∈ Z q ( n + 1 3 ) 2 +( n + 1 3 )( m + 1 3 )+( m + 1 3 ) 2 , (A.1c) where ζ 3 is a pr imitiv e third root of unity . Their A GM identities include the quadratic signature-2 identities A 2 ( q 2 ) = [( A 2 + B 2 ) / 2 ] ( q ) , (A.2a) B 2 ( q 2 ) = p A 2 B 2 ( q ) , (A.2b) C 2 ( q 2 ) = [( A 2 − B 2 ) / 2 ] ( q ) , (A.2c) which origina ted with Jacobi, the quartic signature-2 identities ϑ 3 ( q 4 ) = [( ϑ 3 + ϑ 4 ) / 2 ] ( q ) , (A.3a) ϑ 4 ( q 4 ) = 4 q ( ϑ 3 2 + ϑ 4 2 )( ϑ 3 ϑ 4 ) / 2 ( q ) , (A.3b) ϑ 2 ( q 4 ) = [( ϑ 3 − ϑ 4 ) / 2 ] ( q ) , (A.3c) the cubic signature- 3 identities A 3 ( q 3 ) = [( A 3 + 2 B 3 ) / 3 ] ( q ) , (A.4a) B 3 ( q 3 ) = 3 q ( A 3 2 + A 3 B 3 + B 3 2 ) B 3 / 3 ( q ) , (A.4b) C 3 ( q 3 ) = [( A 3 − B 3 ) / 3 ] ( q ) , (A.4c) 38 Robert S. Maier and the quadr atic si gnatu re-4 identities A 4 2 ( q 2 ) =  ( A 4 2 + 3 B 4 2 ) / 4  ( q ) , (A .5a) B 4 2 ( q 2 ) = q ( A 4 2 + B 4 2 ) B 4 2 / 2 ( q ) , (A.5b) C 4 2 ( q 2 ) =  ( A 4 2 − B 4 2 ) / 4  ( q ) . (A.5c) References [1] Ablowitz, M.J., Chakrav arty , S., Hahn, H.: Integrable systems and modular forms of lev el 2. J. 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Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

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