2.1 Level, Weight, Nominator and Denominator of an Eta Product. By an eta product we understand any finite product of functions. f(z) = m.

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1 Ea Producs.1 Level, Weigh, Noinaor and Denoinaor of an Ea Produc By an ea produc we undersand any finie produc of funcions f(z = η(z a where runs hrough a finie se of posiive inegers and he exponens a ay ake any values fro Z, posiive or negaive or 0. (Of course, an exponen 0 conribues a rivial facor 1 o he produc, and herefore we ay as well assue ha a 0forall. Since he produc is finie, he lowes coon uliple N =lc{} exiss, and every divides N. We wrie f(z = η(z a, (.1 and we call f an ea produc of level N. Here, forally, runs hrough all posiive divisors of he posiive ineger N, and soe of he exponens a igh be 0. We will use his noaion also in cases when N is bigger han lc{}; henn is a uliple of he level of he ea produc. Soe auhors use he er ea quoien for funcions as in (.1, and hey reserve he er ea produc for he case when a 0 for all. Ofen we will use he noaion [1 a1, a, 3 a3,...] as an abbreviaion for he ea produc η(z a1 η(z a η(3z a3... This noaion is adoped fro [4]. The er in square brackes will ofen be wrien as a fracion wih posiive exponens in is nueraor and denoinaor. G. Köhler, Ea Producs and Thea Series Ideniies, Springer Monographs in Maheaics, DOI / , c Springer-Verlag Berlin Heidelberg

2 3. Ea Producs An ea produc (.1 ransfors like a odular for of weigh k = 1 a wih soe uliplier syse on he congruence group Γ 0 (N. This eans ha for every L = ( a b c d Γ0 (N wehave ( az + b f(lz =f = v f (L(cz + d k f(z cz + d where v f (L is soe 4h roo of uniy which can be copued fro he uliplier syse v η of he ea funcion. We will rarely need o know he values v f (L of he uliplier syse of f explicily. We have ( a b v f (L =v f = ( ( a a b v c d η c/ d where he values of v η are given explicily in Theore 1.7. Highly iporan for us, however, is he value v f (T for he ranslaion T = ( We wrie 1 a = s (. 4 in lowes ers, i.e., wih gcd(s, = 1. Then i is a rivial consequence fro η(z +1=e ( 1 4 η(z hawehavevf (T =e ( s, ( s f(tz=f(z +1=e f(z. I follows ha f has a Fourier expansion of he for ( nz f(z = c n e n s(od, n s (.3 s wih coefficiens c n Z, c s = 1. In paricular, is he order of f a he cusp. We call s he nueraor and he denoinaor of he ea produc (.1. The denoinaor is a divisor of 4. An explici forula for v f (L is given in [105], Theore 1.64 in he case when he weigh k and he nuber (. are inegers (whence = 1 and when also a N/ is an ineger; in his case v f (L is a funcion of d only. 1 4 For a Fourier series (.3, he sign ransfor is f ( z + 1 = e ( s n s(od,n s ( 1 (n s/ c n e ( nz.

3 .. Ea Producs on he Fricke Group 33 Modifying our concep fro Sec. 1., we will also call he series for e( s f(z + 1 hesign ransfor of he series for f(z. An ea produc f of level N as in (.1 will be called old if here is an ineger d 1, a proper divisor N 1 of N and an ea produc g of level N 1 such ha f(z =g(dz. Oherwise f will be called a new ea produc. Since f and g have idenical Fourier coefficiens, i ofen suffices o sudy new ea producs. Neverheless, soeies i is advanageous o consider old ones. For exaple, g(z =η(zη(z andf(z =η(8zη(16z boh are old ea producs of level 16, while g is new of level. Bu f has period 1, and hence is Fourier expansion is a power series in he variable q = e(z, which igh be nicer han he expansion of g wih fracional powers of q. We ephasize ha our concep of a new ea produc has lile o do wih he concep of a newfor in he heory of Hecke operaors as explained in Sec Only occasionally i will happen ha a new ea produc is also a Hecke eigenfor. (Incidenally, η(zη(z is such an exaple; see Sec Ea Producs on he Fricke Group For he oen, le us pu f (z =η(z, where is a posiive ineger. Fro η( 1/z = iz η(z i follows ha ( f (W N z=f 1 ( 1 = η = ( N (in/zη Nz (N/z z. Thus, for an ea produc f of level N as in (.1, we obain f(w N z = ( ( a N ( i(n/z 1/ η z = ( ( iz 1/ η(z a N/ ( 1/ = ( iz k a N/ η(z a N/. The ea produc f ransfors like a odular for of weigh k for he Fricke group Γ (N if and only if f(w N z= ( i Nz k f(z. We see ha his holds if and only if he condiion a N/ = a for all (.4 is saisfied. An ea produc wih his propery will be called an ea produc on he Fricke group of level N.

4 34. Ea Producs We observe ha an ea produc of level N is deerined by is syse of τ(n exponens a, whereas roughly half of hese paraeers exacly τ(n/ of he suffice o deerine an ea produc on he Fricke group. Here, τ(n = σ 0 (N is he nuber of posiive divisors of N, as inroduced in Sec Expansion and Order a Cusps The produc for η(z ells us ha his funcion is nowhere 0. Therefore, ea producs (.1 are holoorphic on he upper half plane regardless of heir syse of exponens a. However, we will resric our sudy o ea producs which are holoorphic a all cusps, oo. In paricular, he order a he cusp should be non-negaive, i.e., s 0. We need condiions for an ea produc o be holoorphic a he oher cusps r Q. For his purpose we give a forula for he order of funcions η(z a an arbirary cusp and, soewha ore general, for he Fourier expansion of η(z a cusps. This expansion will evenually be useful when we wan o decide wheher a linear cobinaion of ea producs is a cusp for, where he ea producs are holoorphic a all cusps, bu no cusp fors heselves. Proposiion.1 Le f (z =η(z wih N, andler = d c Q be a reduced fracion wih c 0. Le a, b be chosen such ha A = ( a b SL (Z. Then we have: c d (1 The expansion of f a he cusp r is ( gcd(c, f (A 1 z = v η (L n=1 ( 1 n e ( n 1/ ( cz + a 4 ( (gcd(c, z + ν gcd(c, where L = ( x u SL (Z, x = d gcd(c,, u = c gcd(c,,andν is soe ineger. ( The order of f a he cusp r is ord(f,r= 1 4 (gcd(c,.

5 .3. Expansion and Order a Cusps 35 Proof. Since c, d are relaively prie, we can choose a, b Z such ha A = ( a b c d SL (Z. We ge A 1 ( = ( d b c a ( = d c = r and ( dz b f (A 1 z=η = η(αz cz + d where α = ( d b c a,de(α =. The expansion of f a r is given by he expansion of f (A 1 za. In order o find i, we need soe arix L = ( x y u v SL (Z such ha he lower lef enry in L 1 α vanishes. We have ( ( ( L 1 v y d b α = =. u x c a du cx Therefore we need ha du + cx = 0. Thus for he firs colun of L we can choose he relaively prie inegers x = d gcd(c, d = d g, u = c, wih g = gcd(c,. g Fro de(l 1 α=de(α = we infer ha ( ( L 1 g ν α = = 0 /g 0 /g wih soe ν Z. (Observe ha we can copue ν = bv ya explicily, depending on and r. Now we ge f (A 1 z = η(αz =η(ll 1 αz ( = v η (L u gz + ν 1/ /g + v η(l 1 αz ( 1/ ( cz cν/g gz + ν = v η (L + v η /g /g ( ( 1/ g cν v g = v η (L cz η( g z + νg ( 1/ g g = v η (L ( cz + a η( z + νg ( 1/ g ( ( 1 n = v η (L ( cz + a e n 4 (g z + νg. This proves our firs asserion. The firs non-vanishing er in ( cz+a 1/ f (A 1 z is a consan uliple of e(g z/4. Thus, by our definiion of he order, we obain ord(f,r=g /4, which is he second asserion. n=1 We noe an iediae consequence of he second asserion:

6 36. Ea Producs Corollary. Le f be an ea produc as in (.1, andler = d c Q, gcd(c, d =1. Then he order of f a he cusp r is ord(f,r = 1 (gcd(c, a. 4 An ea produc f will be called a holoorphic ea produc if is orders a all cusps are non-negaive, ord(f,r 0 for all r Q. Holoorphic ea producs (.1 are (enire odular fors for Γ 0 (N. They are cusp fors if and only if all he orders are posiive, ord(f,r > 0 for all r Q. In his case we will call he cuspidal ea producs, andnon-cuspidal oherwise..4 Condiions for Holoorphic Ea Producs Fro Corollary. we ge condiions for an ea produc o be holoorphic or a cusp for. These are condiions for infiniely any cusps. Of course, i suffices o check hese condiions for a finie syse of represenaives of inequivalen cusps of Γ 0 (N, i.e., for he orbis of his group on Q.The nuber of inequivalen cusps of Γ 0 (N is ϕ(gcd(, N/, where ϕ is he Euler funcion; his is known fro several exbooks; see [15], p. 10, for exaple. A se of represenaives of inequivalen cusps is given in [9], forula (. Using his, i would be possible o characerize holoorphic and cuspidal ea producs by syses of finiely any inequaliies. In fac, one can find such a characerizaion using nohing else bu Corollary.: We observe ha he order of f a a cusp does only depend on he denoinaor c of ha cusp. If is any divisor of N hen for all c Z we have gcd(c, = gcd(gcd(c, N,, and gcd(c, N isadivisorofn. Therefore he condiions ord(f,r 0are saisfied for all r Q if and only if ord(f,1/c 0 for all c N, and siilarly for sric inequaliies. This proves he following resul:

7 .5. The Cones and Siplices of Holoorphic Ea 37 Corollary.3 An ea produc f as in (.1 is holoorphic if and only if he inequaliies (gcd(c, a 0 hold for all posiive divisors c of N. I is a cuspidal ea produc if and only if all hese inequaliies hold sricly..5 The Cones and Siplices of Holoorphic Ea Producs According o Corollary.3, we inroduce raional nubers α c, a arix A and a colun vecor X by (gcd(c, α c =, A = A(N =(α c c,, X =(a R τ(n, (.5 where he posiive divisors, c of N are aken in soe arbirary, bu fixed order. (Usually he divisors will be in heir naural order. Then he condiion for holoorphic ea producs of level N reads A(N X 0, (.6 and cuspidal ea producs are characerized by A(N X>0. The syse of linear inequaliies in (.6 defines an inersecion of τ(n closed halfspaces in R τ(n whose bounding hyperplanes all pass hrough he origin. So his syse defines a closed siplicial cone wih is verex a he origin. We denoe his cone by K(N, i.e. K(N ={X R τ(n A(NX 0}. (.7 We can reforulae Corollary.3 as follows: Corollary.4 An ea produc (.1 is holoorphic if and only if is vecor of exponens X =(a is a laice poin in he cone K(N. I is cuspidal if and only if X is an inerior poin of K(N.

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Chapter 2. First Order Scalar Equations

Chapter 2. First Order Scalar Equations Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.