Certain aspects of holomorphic function theory on some genus zero arithmetic groups
aa r X i v : . [ m a t h . N T ] M a y CERTAIN ASPECTS OF HOLOMORPHIC FUNCTION THEORY ON SOME GENUS ZEROARITHMETIC GROUPS
JAY JORGENSON, LEJLA SMAJLOVI´C, AND HOLGER THEN
Abstract.
There are a number of fundamental results in the study of holomorphic function theory associated to thediscrete group PSL(2 , Z ) including the following statements: The ring of holomorphic modular forms is generatedby the holomorphic Eisenstein series of weight four and six, denoted by E and E ; the smallest weight cusp form∆ has weight twelve and can be written as a polynomial in E and E ; and the Hauptmodul j can be written asa multiple of E divided by ∆. The goal of the present article is to seek generalizations of these results to someother genus zero arithmetic groups Γ ( N ) + with square-free level N , which are related to ”Monstrous moonshineconjectures”. Certain aspects of our results are generated from extensive computer analysis; as a result, many of thespace-consuming results are made available on a publicly accessible web site. However, we do present in this articlespecific results for certain low level groups. Introduction and statement of results
Consider the discrete group PSL(2 , Z ) which acts on the upper half plane H . The quotient space PSL(2 , Z ) \ H has one cusp which can be taken to be at i ∞ . Let Γ ∞ denote the stabilizer subgroup for the cusp at i ∞ , whichconsists of isometries (cid:18) a bc d (cid:19) ∈ PSL(2 , Z )with c = 0. For every integer k ≥
2, the holomorphic Eisenstein series E k ( z ) is defined by the absolutely convergentsum E k ( z ) := X γ ∈ Γ ∞ \ PSL(2 , Z ) ( cz + d ) − k where γ = (cid:18) ∗ ∗ c d (cid:19) .There is an abundance of important and classical formulae which can be wound back to the holomorphic Eisensteinseries E k . For example, if one defines G k ( z ) := X ( n,m ) ∈ Z \{ (0 , } ( nz + m ) − k , then E k ( z ) = G k ( z ) / ζ (2 k ) where ζ ( s ) is the Riemann zeta function. If we set g = 60 G and g = 140 G , themodular discriminant ∆( z ) := (2 π ) e πiz ∞ Y n =1 (cid:0) − e πinz (cid:1) can be written as ∆( z ) = g ( z ) − g ( z ) = 11728 ( E ( z ) − E ( z )) . (1)The function ∆ is a weight twelve cusp form with respect to PSL(2 , Z ), meaning it vanishes as z approaches i ∞ .It can be shown that no smaller weight cusp form exists. Furthermore, ∆ is related to the algebraic discriminantof the cubic equation y = 4 x − g x − g , in the complex projective coordinates [ x, y, z .All higher weight modular forms associated to PSL(2 , Z ), including Eisenstein series, can be written in terms of E ( z ) and E ( z ). For example, the formulae E ( z ) = E ( z ), E ( z ) = E ( z ) E ( z ) and691 E ( z ) = 441 E ( z ) + 250 E ( z ) , are just the beginning of the never ending list of interesting relations which one can write.Whereas the content of the above discussion is classical, there is a very modern component. The function j ( z ) = 1728 E ( z ) E ( z ) − E ( z )(2) J. J. acknowledges grant support from NSF and PSC-CUNY grants, and H. T. acknowledges support from EPSRC grantEP/H005188/1. is a weight zero modular form on PSL(2 , Z ) \ H which can be viewed as the biholomorphic function that mapsPSL(2 , Z ) \ H onto the Riemann sphere P . If we set q = e πiz , then one can expand j ( z ) as a function of q , namelyone has j ( z ) = 1 q + 744 + 196884 q + 21493760 q + O ( q ) as q → j ( z ).Setting to the side the important formulae themselves, one can summarize the above discussion as the threefollowing points. First, the ring of holomorphic modular forms associated to PSL(2 , Z ) is generated by E and E .Second, the smallest weight cusp form ∆ has weight twelve, hence can be written as a polynomial in E and E .Third, the Hauptmodul j is equal to a multiple of E divided by ∆, hence is a rational function in E and E .The goal of this article is to seek generalizations of the above three statements to certain other arithmetic groupsrelated to the ”Monstrous moonshine conjectures”. Specifically, for any positive integer N , letΓ ( N ) + = (cid:26) e − / (cid:18) a bc d (cid:19) ∈ SL(2 , R ) : ad − bc = e, a, b, c, d, e ∈ Z , e | N, e | a, e | d, N | c (cid:27) (4)and let Γ ( N ) + = Γ ( N ) + / {± Id } , where Id denotes the identity matrix. Observe that PSL(2 , Z ) = Γ (1) + . It hasbeen shown that there are 43 square-free integers N > X N := Γ ( N ) + \ H has genuszero (see [3]). Each group has one cusp, which we can always choose to be at i ∞ . As stated in the title, the aim ofthis paper is to present results in the study of the holomorphic function theory associated to these 43 spaces.Let η ( z ) = e πiz/ ∞ Y n =1 (cid:0) − e πinz (cid:1) denote the Dedekind eta function. For any square-free N assume that N has r prime factors, and let lcm( · , · ) denotethe least common multiple function. Let σ ( N ) equal the sum of divisors of N . It was proven in [7] that the function∆ N ( z ) = Y v | N η ( vz ) ℓ N , where ℓ N = 2 − r lcm (cid:16) , r − , σ ( N )) (cid:17) is a weight k N = 2 r − ℓ N modular form on Γ ( N ) + , vanishing at the cusp i ∞ only. For reasons discussed in [7], werefer to ∆ N as the Kronecker limit function on Γ ( N ) + .The main results of the present paper are the following statements which hold true for each square-free N provided that X N has genus zero. (1) There is an explicitly computed integer M N such that ∆ M N N is equal to a polynomial Q N in holomorphicEisenstein series associated to Γ ( N ) + ;(2) The Hauptmodul j N associated to Γ ( N ) + is equal to a rational function whose numerator is a polynomial P N in holomorphic Eisenstein series and whose denominator is ∆ M N N ;(3) The polynomials P N and Q N are explicitly computed; hence, we determine, for each N , a finite set T ( N ) ofholomorphic Eisenstein series such that any meromorphic form with at most polynomial growth at i ∞ canbe expressed as a rational function involving elements of T ( N ) . Points 1 and 2 are direct generalizations of the formulae (1) and (2). Point 3 is a weak generalization of the resultthat the ring of holomorphic modular forms associated to PSL(2 , Z ) is generated by the holomorphic Eisensteinseries of weight four and six. For certain small levels, we are able to compute generators of the ring of holomorphicforms; however, for general N , and for future investigations we plan to undertake, we are content with point 3 asstated.The present article is organized as follows. In section 2 we establish notation and cite appropriate backgroundmaterial. In particular, we recall the Kronecker limit formula associated to the non-parabolic Eisenstein serieson X N = Γ ( N ) + \ H and an computer algorithm of [7]. In section 3, we prove some basic results regarding lowweight modular forms for any level N >
1. In section 4, we present results regarding the ring of holomorphic forms
ERTAIN ASPECTS OF HOLOMORPHIC FUNCTION THEORY ON SOME GENUS ZERO ARITHMETIC GROUPS 3 for certain small levels. In section 5, we present a variant of the algorithm of [7] from which we prove that forevery square-free N , provided that X N has genus zero, there is an integer M N such that ∆ M N N can be written as apolynomial in holomorphic Eisenstein series. Let j N denote the biholomorphic map from X N to the Riemann sphere P which maps i ∞ to zero. The algorithm described in section 5 allows us to prove that j N ∆ M N N can be writtenas a polynomial in holomorpic Eisenstein series. The data provided by the algorithm is presented in Table 1, as isa comparison of the results of the original algorithm of [7] and the modified variant thereof. From the algorithmdeveloped in this paper, we are able to determine for each level N a set of holomorphic Eisenstein series whichgenerate T ( N ) , the ring of holomorphic modular forms associated to X N ; this information is given in Table 2. Itis important to note that the entries in Table 2 may not be a minimal set of generators, meaning that for each N there may exist further relations amongst the sets listed in Table 2.As N grows, so does the complexity of the formulae for ∆ N and j N . For example, when N = 17, our algorithmshows that the five holomorphic Eisenstein series of weights four through twelve generate T (17) and M = 9,meaning ∆ and j ∆ can be written as a polynomial in these five Eisenstein series. As an indication of thecomplexity of the formulae, we present these two examples in section 5. The formula for ∆ and j each occupyapproximately one page.We note that the Tables 3 and 4a of [2] describe, in their notation, how one can express each Hauptmodul j N interms of holomorphic forms. In Table 3 we translate the aforementioned data from [2], related to 43 groups definedby (4) with square-free N and genus zero, such that we explicitly write these formulae in terms of the Dedekind etafunction and theta function attached to quadratic forms. By combining our formulae for j N and the formulae from[2] one has the prospect of obtaining further identities involving holomorphic Eisenstein series and theta functions.As in [7], the theoretical work developed in this article is supplemented by extensive computer analysis and, quitefrankly, some of the results are not printable. For example, for N = 119, the formula for j from [7] occupiesnearly 60 pages. Nonetheless, in order to disseminate the results obtained by our algorithms, we have posted allformulae to a web site [8]. 2. Background material
Holomorphic modular forms.
Let Γ be a Fuchsian group of the first kind. Following [10], we define aweakly modular form f of weight 2 k for k ≥ f which is meromorphic on H andsatisfies the transformation property f (cid:18) az + bcz + d (cid:19) = ( cz + d ) − k f ( z ) for all (cid:18) a bc d (cid:19) ∈ Γ.Assume that Γ has at least one class of parabolic elements. By transforming coordinates, if necessary, we mayalways assume that the parabolic subgroup of Γ has a fixed point at i ∞ , with identity scaling matrix. In thissituation, any weakly modular form f will satisfy the relation f ( z + 1) = f ( z ), so we can write f ( z ) = ∞ X n = −∞ a n q n where q = e πiz .If a n = 0 for all n <
0, then f is said to be holomorphic in the cusp; f is called a cusp form if a n = 0 for all n ≤ H and in all ofthe cusps of Γ.For Γ = PSL(2 , Z ), the full modular surface, there is no weight 2 holomorphic modular form. Nonetheless, onedefines the function E ( z ) by the q -expansion E ( z ) = 1 − ∞ X n =1 σ ( n ) q n . (5)It can be shown that E ( z ) transforms according to the formula E ( γz ) = ( cz + d ) E ( z ) + 6 πi c ( cz + d ) for (cid:18) ∗ ∗ c d (cid:19) ∈ SL(2 , Z ) . (6)From this, it is elementary to show that for a prime p , the function E ,p ( z ) := pE ( pz ) − E ( z ) p − J. JORGENSON, L. SMAJLOVI´C, AND H. THEN is weight 2 holomorphic form associated to the congruence subgroup Γ ( p ) of SL(2 , Z ). The q -expansion of E ,p is E ,p ( z ) = 1 + 24 p − ∞ X n =1 σ ( n )( q n − pq pn ) . Certain arithmetic groups related to ”moonshine”.
For any square-free integer N , the subset of SL(2 , R )defined by (4) is an arithmetic subgroup of SL(2 , R ). As shown in [3], there are precisely 44 such groups which havegenus zero and which appear in ”Monstrous moonshine conjectures”. In this article we will focus on the 43 genuszero groups for which N > ( N ) + = Γ ( N ) + / {± Id } the corresponding subgroup of PSL(2 , R ). Basic properties of Γ ( N ) + ,for square-free N are derived in [6] and references therein. In particular, we use that the surface X N = Γ ( N ) + \ H has exactly one cusp, which can be taken to be at i ∞ .Let T ( N ) denote the ring of holomorphic modular forms associated to X N , and let T ( N )2 k denote the holomorphicmodular forms of weight 2 k . We will denote the subspace of cusp forms on X N of weight 2 k by S ( N )2 k .2.3. Holomorphic Eisenstein series on Γ ( N ) + . In the case when
N > ( N ) + are defined for k ≥ E ( N )2 k ( z ) := X γ ∈ Γ ∞ ( N ) \ Γ ( N ) + ( cz + d ) − k with γ = (cid:18) ∗ ∗ c d (cid:19) where Γ ∞ ( N ) denotes the stabilizer group of the cusp at i ∞ . In [7] it is proven that E ( N )2 k ( z ) may be expressed as alinear combination of forms E k ( z ), the holomorphic Eisenstein series associated to PSL(2 , Z ). Namely, it is knownthat E ( N )2 k ( z ) = 1 σ k ( N ) X v | N v k E k ( vz ) , (7)where σ α denotes the generalized divisor function σ α ( m ) = X δ | m δ α . Formula (7), together with a well-known q -expansion of classical forms E k yields that the q -expansion of E ( N )2 k isgiven by E ( N )2 k ( z ) = 1 σ k ( N ) X v | N v k − kB k ∞ X j =1 σ k − ( j ) q vj , (8)where B k denotes the k -th Bernoulli number.2.4. Kronecker limit function on Γ ( N ) + . Associated to the cusp of Γ ( N ) + one has a non-holomorphic Eisen-stein series denoted by E par ∞ ( z, s ) which is defined for z ∈ H and Re( s ) > E par ∞ ( z, s ) = X η ∈ Γ ∞ ( N ) \ Γ ( N ) + Im( ηz ) s . In [7] it is proven that, for any square-free N which has r prime factors, the parabolic Eisenstein series E par ∞ ( z, s )admits a Taylor series expansion of the form E par ∞ ( z, s ) = 1 + s · log r sY v | N | η ( vz ) | · Im( z ) + O ( s ) , as s → , where η ( z ) is Dedekind’s eta function associated to PSL(2 , Z ). As stated above, it is proven that the function∆ N ( z ) = Y v | N η ( vz ) ℓ N , (9)where ℓ N = 2 − r lcm (cid:16) , r − , σ ( N )) (cid:17) ERTAIN ASPECTS OF HOLOMORPHIC FUNCTION THEORY ON SOME GENUS ZERO ARITHMETIC GROUPS 5 and lcm( · , · ) denotes the least common multiple function, is weight k N = 2 r − ℓ N modular form on Γ ( N ) + , vanishingat the cusp i ∞ only. We call the function ∆ N ( z ) defined by (9) the Kronecker limit function on Γ ( N ) + .2.5. The algorithm.
Let X N = Γ ( N ) + \ H have genus g . For any positive integer M , the function F b ( z ) = Y ν (cid:16) E ( N ) m ν ( z ) (cid:17) b ν .(cid:0) ∆ N ( z ) (cid:1) M where X ν b ν m ν = M k N and b = ( b , . . . )(10)is a holomorphic modular function on X N , meaning a weight zero modular form with polynomial growth near i ∞ .The q -expansion of F b follows from substituting the q -expansions of E ( N ) k and ∆ N .Let S M denote the set of all possible rational functions defined in (10) for all vectors b = ( b ν ) and m = ( m ν )with fixed M . In [7], we implemented the following algorithm, which we refer to as the JST2 algorithm.Choose a non-negative integer κ . Let M = 1 and set S = S ∪ S .(1) Form the matrix A S of coefficients from the q -expansions of all elements of S , where each element in S isexpanded along a row with each column containing the coefficient of a power, negative, zero or positive, of q . The expansion is recorded out to order q κ .(2) Apply Gauss elimination to A S , thus producing a matrix B S which is in row-reduced echelon form.(3) Implement the following decision to determine if the algorithm is complete: If the g + 1 lowest non-trivialrows at the bottom of B S correspond to q -expansions whose lead terms have precisely g gaps, meaning zerocoefficients, in the set { q − , . . . , q − g } , then the algorithm is completed. If the indicator to stop fails, thenreplace M by M + 1, S by S M ∪ S and reiterate the algorithm.If g = 0, then the algorithm stops if the lowest non-trivial row at the bottom of B S has a q -expansion whichbegins with q − . We also denote by M N the value of M for the group of level N at which Step 3 shows that thealgorithm is completed.As stated in [7], the rationale for the stopping decision in Step 3 above is based on two ideas, one factual andone hopeful. First, the Weierstrass gap theorem states that for any point P on a compact Riemann surface thereare precisely g gaps in the set of possible orders from 1 to 2 g of functions whose only pole is at P . Second, for anygenus, the assumption which is hopeful is that the function field is generated by the set of holomorphic modularfunctions defined in (10), which is related to the question of whether the field of meromorphic modular forms onΓ ( N ) + is generated by holomorphic Eisenstein series and the Kronecker limit function. The latter assumption isnot obvious, and, indeed, the assumption itself is equivalent to the statement that the rational function field on X N is generated by the holomorphic Eisenstein series. As it turned out, for all groups Γ ( N ) + that we have studiedso far, which includes all groups of genus zero, genus one, genus two, and genus three, the algorithm stopped.Therefore, we conclude that, in particular, the rational function field associated to all genus zero groups Γ ( N ) + isgenerated by a finite set of holomorphic Eisenstein series.We described the algorithm with choice of an arbitrary κ and g . For reasons of efficiency, we initially selected κ to be zero, so that all coefficients for q ν for ν ≤ κ are included in A S . In [7], it is shown that for each N , thereis an explicitly computable κ = κ N such that if a modular form associated to Γ ( N ) + has integral coefficients inits q -expansion out to q κ N , then all remaining coefficients are also integral. The list of κ N for square-free levels N provided that Γ ( N ) + has genus zero is given in Table 1 of [7]. In the implementation of the above algorithm, bothin the present article and in [7], the value of κ was finally increased to κ N .In the present article, we implemented a slight variant of the above algorithm, which we refer to as the JST3 algorithm. The difference between the
JST2 and the
JST3 algorithm is the following action should the decisionin Step 3 fail.
Replace M by M + 1 , S by S M , and reiterate the algorithm. In other words, the
JST3 algorithm studies the q -expansions of the space of rational functions of the form (10)with a fixed denominator (cid:0) ∆ N ( z ) (cid:1) M . Should the JST3 algorithm successfully complete, then the row in B S with q -expansion beginning with q − would correspond to a formula for j N with denominator (cid:0) ∆ N ( z ) (cid:1) M and numeratorgiven as a polynomial in Eisenstein series. Furthermore, any lower row in B S would correspond to a q -expansionbeginning with q , which would yield, upon clearing the denominator, a formula for (cid:0) ∆ N ( z ) (cid:1) M in terms of Eisensteinseries.As we will report below, the JST3 algorithm has successfully completed for all genus zero groups Γ ( N ) + withsquare-free level N . J. JORGENSON, L. SMAJLOVI´C, AND H. THEN Modular forms on surfaces X N From Proposition 7, page II-7, of [1], we immediately obtain the following formula which relates the number ofzeros of a modular form, counted with multiplicity, with its weight and volume of X N . Lemma 1.
Let f be a modular form on X N of weight k , not identically zero. Let F N denote the fundamentaldomain of X N and let v z ( f ) denote the order of zero z of f . Then, k Vol( X N )2 π = v i ∞ ( f ) + X e ∈ E N n e v e ( f ) + X z ∈ F N \ E N v z ( f ) , (11) where E N denotes the set of elliptic points in F N and n e is the order of the elliptic point e ∈ E N . Lemma 1 enables us to deduce the following proposition.
Proposition 2.
Let N be a square-free number such that the surface X N has genus zero. Then, there are no weighttwo holomorphic modular forms on X N .Proof. From the tables in [3], one determines that all genus zero groups Γ ( N ) + , for a square-free N have a at mostone elliptic point of order three, four or six and a various number of order two elliptic points. Let e N (2) denote thenumber of order two elliptic points on X N , and let n N ∈ { , , } denote the order of the additional elliptic point on X N . Since all surfaces X N have one cusp and genus zero, the Gauss-Bonnet formula for the volume of the surface X N becomes Vol( X N )2 π = 12 e N (2) + (cid:18) − n N (cid:19) δ ( N ) − , (12)where δ ( N ) is equal to one if X N has an elliptic point of order different from two and zero otherwise.For an arbitrary, square-free N and e | N , the elliptic element of Γ ( N ) + is of the form (cid:18) a √ e b/ √ e ( cN ) / √ e d √ e (cid:19) , (13)where a, b, c, d, ∈ Z are such that | ( a + d ) √ e | < ade − ( bcN ) /e = 1. The first condition implies that either a + d = 0 or | a + d | = 1 and e ∈ { , , } .If | a + d | = 1, then d = ± − a , hence (cid:18) a √ e b/ √ e ( cN ) / √ e d √ e (cid:19) = (cid:18) ± ae − ± b ± cN ± ae − (cid:19) = ± Id , for any choice of a, b, c ∈ Z such that a ( ± − a ) e − ( bcN ) /e = 1. Therefore, there are no order two elements inΓ ( N ) + such that | a + d | = 1.On the other hand, if a + d = 0, then − a e − ( bcN ) /e = 1, hence (cid:18) a √ e b/ √ e ( cN ) / √ e d √ e (cid:19) = (cid:18) − − (cid:19) . In other words, any elliptic element (13) of Γ ( N ) + has order two if and only if a + d = 0. Let η = (cid:18) a √ e b/ √ e ( cN ) / √ e − a √ e (cid:19) denote arbitrary elliptic element of Γ ( N ) + of order two, and let z η be its fixed point. Solving the equation η ( z η ) = z η leads to the conclusion that ( z η cN/ √ e − a √ e ) = − f ,N is a holomorphic modular form on X N of weight 2. By the transformation rule, we have that f ,N ( z η ) = f ,N ( ηz η ) = ( − f ,N ( z η ) , hence z η is vanishing point of f ,N . Since this holds true for any order two elliptic element of Γ ( N ) + , we concludethat all order two elliptic points of X N are vanishing points of f ,N . Applying Lemma 1 to f ,N , we arrive at theinequality Vol( X N )2 π ≥ e N (2) , which contradicts (12). Therefore, there are no weight two holomorphic modular forms on X N . (cid:3) Though there are no weight two holomorphic forms on Γ ( N ) + , we may construct forms that transform almostlike a weight two form, up to an order two character. ERTAIN ASPECTS OF HOLOMORPHIC FUNCTION THEORY ON SOME GENUS ZERO ARITHMETIC GROUPS 7
Proposition 3.
Let N = p · · · p r be a square-free positive integer. Let µ ( ν ) denote the M¨obius function and E the series defined in (5) . Then the holomorphic function E ,N ( z ) := ( − r ϕ ( N ) X v | N µ ( v ) vE ( vz ) satisfies the transformation rule E ,N ( γ e z ) = µ ( e )( c N √ e z + d √ e ) E ,N ( z ) for any γ e = (cid:18) a √ e b/ √ e ( cN ) / √ e d √ e (cid:19) ∈ Γ ( N ) + . Proof.
Choose and fix any e | N . For any v | N , let ( e, v ) denote the greatest common divisor of e and v . Then,using the transformation formula (6) for E , it is easy to deduce that vE ( v ( γ e z )) = ev ( e, v ) ( c N √ e z + d √ e ) E (cid:18) ev ( e, v ) z (cid:19) + 6 πi cN (cid:18) c Ne z + d (cid:19) . Since N is square-free with r prime factors, it is easy to see that X v | N µ ( v ) 6 πi cN (cid:18) c Ne z + d (cid:19) = 6 πi cN (cid:18) c Ne z + d (cid:19) r X j =1 (cid:18) rj (cid:19) ( − j = 0 , hence X v | N µ ( v ) vE ( v ( γ e z )) = X v | N µ ( v ) ev ( e, v ) ( c N √ e z + d √ e ) E (cid:18) ev ( e, v ) z (cid:19) . We claim that { v : v | N } = { ev ( e,v ) : v | N } , which is easily deduced by induction in r . Furthermore, when e has an even number of prime factors, the parity of the number of factors of ev ( e,v ) remains the same as theparity of the number of factors of v , while when e has odd number of factors, the parity changes, meaning that µ ( v ) = µ ( e ) µ ( ev ( e,v ) ). Therefore X v | N µ ( v ) vE ( v ( γ e z )) = µ ( e )( c N √ e z + d √ e ) X v | N µ ( v ) vE ( vz )and the proof is complete. (cid:3) Proposition 4.
The smallest even integer k N such that there exists a weight k N cusp form f N vanishing only atthe cusp i ∞ is given by the formula k N = lcm(4 , r − , σ ( N )) )(14) where lcm denotes the least common multiple and ( · , · ) stands for the greatest common divisor.Proof. From [7], one has that the volume of the surface X N is given byVol( X N ) = πσ ( N )6 · r − , (15)where r is the number of (distinct) prime factors of N . By combining (15) with (11), we have that k N · σ ( N )24 · r − = v i ∞ ( f N ) , hence 2 r − ,σ ( N )) | k N .On the other hand, the cusp form f N does not vanish at order two elliptic points. As proven above, every surface X N for a square-free N has at least one order two elliptic point that is a fixed point of the Atkin-Lehner involution τ N : z
7→ − / ( N z )). Since f N ( τ N ( i/ √ N )) = f N ( i/ √ N ) = ( i ) k N f N ( i/ √ N ) , it folows that 4 | k N . The smallest k N divisible by both 4 and 2 r − ,σ ( N )) is given by (14). Therefore, the proofis complete. (cid:3) The above proposition, together with Theorem 16 form [7] yields the following corollary.
J. JORGENSON, L. SMAJLOVI´C, AND H. THEN
Corollary 5.
Let ℓ N = 2 − r k N , where k N is given by (14) . Then, the function ∆ N ( z ) := Y v | N η ( vz ) ℓ N is the smallest weight cusp form on X N vanishing at the cusp only. Furthermore, the order of vanishing of ∆ N atthe cusp is given by v i ∞ (∆ N ) = σ ( N ) ℓ N
24 = k N · σ ( N )24 · r − The next proposition determines the smallest weight e k N for square-free N such that the space S ( N ) e k N is not empty. Proposition 6.
Let N = p · · · p r be a square-free positive integer where N > . Then, the smallest even integer e k N such that there exist a weight e k N cusp form on a genus zero surface X N is equal to , for N ∈ { , } and equalto for all other N .Proof. When N = 2, it is immediate that k = 8 is the smallest number such that k · Vol( X )4 π ≥
1. Since ∆ is weight8 cusp form, the assertion is proven when N = 2.When N = 3, k = 6 is the smallest number such that k · Vol( X )4 π ≥
1. However, if there exists a weight 6 cuspform on X , this cusp form also vanishes at order two elliptic point e of X . Therefore, the right hand side of theformula (11) is at least 3 /
2, while the left hand side of the same formula with k = 6 is equal to 1, which yields acontradiction. This shows that 8 is the smallest possible weight of cusp form on X . An example of weight eightcusp form on X is E (3)8 − ( E (3)4 ) , so the case when N = 3 is complete.When N ≥ X N , whether or not the genus is zero, as follows.Let E ,N ( z ) := ( E ,N ( z )) . From Proposition 3, it is immediate that E ,N is weight four holomorphic form on Γ ( N ) + . Recall that for asquare-free N with r prime factors we have the formula ϕ ( N ) = ( − r X v | N vµ ( v ) . The q -expansion (5) implies that E ,N ( z ) is normalized so that its q -expansion has a leading coefficient equal toone. Therefore, the difference e ∆ N ( z ) := E ( N )4 ( z ) − E ,N ( z )is a weight four cusp form. By computing the q -expansion of E ,N , we deduce that the term multiplying q in the q -expansion of E ,N ( z ) is ϕ ( N ) , while the term multiplying q in the q -expansion of E ( N )4 ( z ) is equal to N . Inother words, for square-free N / ∈ { , } , we have the expansion e ∆ N ( z ) = 48 (cid:18) ϕ ( N ) − N + 1 (cid:19) q + . . . . The leading coefficient is non-zero whenever N ≥
5, hence e ∆ N ( z ) is a weight four cusp form on X N . (cid:3) Expressing the Hauptmodul in terms of Eisenstein series
In this section we discuss the main results of this article.
Theorem 7.
For any square-free N ≥ such that the surface X N has genus zero, there exist effectively computableintegers M N and m N , and explicitly computable polynomials P N ( x , . . . , x m N − ) and Q N ( x , . . . , x m N − ) in m N − variables with integer coefficients such that the Hauptmodul j N can be written as j N ( z ) = P N ( E ( N )4 , . . . , E ( N )2 m N ) Q N ( E ( N )4 , . . . , E ( N )2 m N ) and the Kronecker limit function can be written as ∆ M N N = Q N ( E ( N )4 , . . . , E ( N )2 m N ) . ERTAIN ASPECTS OF HOLOMORPHIC FUNCTION THEORY ON SOME GENUS ZERO ARITHMETIC GROUPS 9
Proof.
The result follows, because for each square-free level N , provided that X N has genus zero, the JST3 algorithm terminates in finite time. As stated, the computer code as well as the output is available on web site[8]. In the space below, let us describe in further detail the output of the computational algorithm. We remind thereader that the
JST2 and the
JST3 algorithm are described in section 2.5.After Gauss elimination, one of the q -expansions has a pole of order 1. This is the Hauptmodul, see section 5 forexplicit examples. Keeping track of the linear algebra, we have an exact expression for the Hauptmodul as a linearcombination of holomorphic modular functions (10) with rational coefficients. In other words, j N ( z ) = 1(∆ N ( z )) M N X b C b · Y ν (cid:16) E ( N ) m ν ( z ) (cid:17) b ν ! , where the sum is taken over all b = ( b , . . . ) such that P ν b ν m ν = k N M N , where M N is given in the right columnof Table 1 and C b ∈ Q .There is also a q -expansion which is equal to the constant 1. Again, by keeping track of the linear algebra, wehave an exact expression for the constant 1 as1 = 1(∆ N ( z )) M N X b D b · Y ν (cid:16) E ( N ) m ν ( z ) (cid:17) b ν ! , where the sum is taken over the same set of b as above and D B ∈ Q .By the design of the JST3 algorithm, this exact expression can easily be solved for the M N -th power of theKronecker limit function, showing that j N ( z ) = P b C b · (cid:18)Q ν (cid:16) E ( N ) m ν ( z ) (cid:17) b ν (cid:19)P b D b · (cid:18)Q ν (cid:16) E ( N ) m ν ( z ) (cid:17) b ν (cid:19) . After multiplication of both numerator and denominator with the least common multiple of the denominators ofthe numbers C b and D b , we deduce the statement of the theorem. (cid:3) Remark 8.
The polynomials P N and Q N appearing in Theorem 7 are weighted homogeneous in the sense thatthere exists an integer M N such that the coefficient of the term ( x ) α · · · ( x m N − ) α mN − is non-zero only if 4 α +6 α + . . . + 2 m N α m N − = k N M N , where k N is the weight of the Kronecker limit function ∆ N . Remark 9.
Table 1 provides the data regarding the performance of the
JST2 and
JST3 algorithms. Moreprecisely, the first columns of data in Table 1 lists, for each level N provided that X N has genus zero, the weight k N of the Kronecker limit function and the integer κ N . To recall, it is shown in [7] if the q -expansion of a holomorphicmodular form has integer coefficients out to q κ N , then all further coefficients are also integral. The columns of datain Table 1 under the heading JST2 algorithm lists the integer M such that the JST2 algorithm stops, togetherwith the q -expansions which are used in the Gauss elimination algorithm as well as the order of the largest pole at i ∞ amongst the rational functions considered. The columns of data in Table 1 under the heading JST3 algorithmpresent similar information.
Remark 10.
Table 2 provides a list of the holomorphic Eisenstein series E ( N ) m ν which appear in the expression forthe Hauptmodul j N cited in Theorem 7. For all levels, the highest weight Eisenstein series has weight 26. Remark 11.
Expressions that are based on the track record of the linear algebra depend on how the base change ismade through Gauss elimination. In particular, there may be linearly dependent functions, some of which survivethe Gauss elimination while others get annihilated. We sought to express our results in terms of Eisenstein serieswhose weights are as small as possible. In other words, in the Gauss elimination we prioritized the holomorphicmodular functions accordingly.By expressing the Hauptmodul in terms of holomorphic Eisenstein series of smallest possible weights, we wereable to determine a finite list of holomorphic Eisenstein series which generates the rational function field. Let G denote any modular form of weight k and consider the function F ( z ) = G ( z ) (cid:16) E ( N )6 ( z ) (cid:17) n (cid:16) E ( N )4 ( z ) (cid:17) n .(cid:0) ∆ N ( z ) (cid:1) nM N , with non-negative integers n , n , and n such that k + 6 n + 4 n = k N nM N . There is a rational function R in onevariable such that F ( z ) = R ( j N ( z )). Therefore, we conclude that G can be written as a rational function in termsof the holomorphic Eisenstein series that are listed in Table 2. Table 1.
Performance of the
JST2 and the
JST3 algorithm. For all genus zero groups Γ ( N ) + we list the level N , the weight k N of the Kronecker limit function, the value of κ N in the proof ofintegrality [7] (left); the level N , the number of iterations M , the number of equations, and thelargest order of pole for the JST2 algorithm (middle) and similar for the
JST3 algorithm (right).
N k N κ N JST2 algorithm
N M { eqs } pole1 1 5 12 1 3 13 1 5 25 1 2 16 1 2 17 1 5 410 2 10 611 3 8 613 2 26 1414 3 8 615 3 8 617 4 15 1219 3 114 3021 2 26 1622 4 15 1223 5 27 2026 3 31 2129 6 48 3030 4 15 1231 4 434 6433 5 27 2034 3 31 2735 5 27 2038 5 27 2539 3 114 4241 7 82 4942 5 27 2046 6 48 3647 8 137 6451 6 48 3655 6 48 3659 9 225 9062 7 82 5666 6 48 3669 7 82 5670 6 48 3671 10 362 12078 6 48 4287 7 82 7094 8 137 9695 7 82 70105 7 82 56110 7 82 63119 8 137 96 JST3 algorithm
N M N { eqs } pole1 1 4 12 1 2 13 1 4 25 3 4 36 3 4 37 2 21 810 2 7 611 9 88 1813 3 88 2114 6 21 1215 5 12 1017 9 88 2719 4 320 4021 2 21 1622 6 21 1823 15 1039 6026 4 55 2829 15 1039 7530 6 21 1831 5 1039 8033 8 55 3234 4 55 3635 7 34 2838 10 137 5039 3 88 4241 21 8591 14742 7 34 2846 14 708 8447 27 56224 21651 11 210 6655 8 55 4859 33 310962 33062 18 3094 14466 8 55 4869 14 708 11270 8 55 4871 39 1512301 46878 9 88 6387 17 2167 17094 26 41646 31295 11 210 110105 9 88 72110 9 88 81119 10 137 120 ERTAIN ASPECTS OF HOLOMORPHIC FUNCTION THEORY ON SOME GENUS ZERO ARITHMETIC GROUPS 11
Table 2.
Finite sets of Eisenstein series which include the generators of the holomorphic Eisensteinseries on groups Γ ( N ) + of genus zero. Listed are level and finite set. N finite set1 E (1)4 , E (1)6 E (2)4 , E (2)6 , E (2)8 E (3)4 , E (3)6 , E (3)12 E (5)4 , E (5)6 , E (5)8 , E (5)12 E (6)4 , E (6)6 , E (6)8 , E (6)12 E (7)4 , E (7)6 , E (7)8 , E (7)10 , E (7)12 E (10)4 , E (10)6 , E (10)8 , E (10)10 , E (10)12 , E (10)16 E (11)4 , E (11)6 , E (11)8 , E (11)10 , E (11)12 E (13)4 , E (13)6 , E (13)8 , E (13)10 , E (13)12 E (14)4 , E (14)6 , E (14)8 , E (14)10 , E (14)12 E (15)4 , E (15)6 , E (15)8 , E (15)10 , E (15)12 , E (15)14 , E (15)16 E (17)4 , E (17)6 , E (17)8 , E (17)10 , E (17)12 E (19)4 , E (19)6 , E (19)8 , E (19)10 , E (19)12 E (21)4 , E (21)6 , E (21)8 , E (21)10 , E (21)12 , E (21)14 , E (21)16 E (22)4 , E (22)6 , E (22)8 , E (22)10 , E (22)12 , E (22)14 , E (22)16 , E (22)18 E (23)4 , E (23)6 , E (23)8 , E (23)10 , E (23)12 E (26)4 , E (26)6 , E (26)8 , E (26)10 , E (26)12 , E (26)14 E (29)4 , E (29)6 , E (29)8 , E (29)10 , E (29)12 E (30)4 , E (30)6 , E (30)8 , E (30)10 , E (30)12 , E (30)14 , E (30)16 , E (30)18 E (31)4 , E (31)6 , E (31)8 , E (31)10 , E (31)12 E (33)4 , E (33)6 , E (33)8 , E (33)10 , E (33)12 , E (33)14 E (34)4 , E (34)6 , E (34)8 , E (34)10 , E (34)12 , E (34)14 , E (34)16 E (35)4 , E (35)6 , E (35)8 , E (35)10 , E (35)12 , E (35)14 , E (35)16 , E (35)18 E (38)4 , E (38)6 , E (38)8 , E (38)10 , E (38)12 , E (38)14 E (39)4 , E (39)6 , E (39)8 , E (39)10 , E (39)12 , E (39)14 E (41)4 , E (41)6 , E (41)8 , E (41)10 , E (41)12 E (42)4 , E (42)6 , E (42)8 , E (42)10 , E (42)12 , E (42)14 , E (42)16 , E (42)18 E (46)4 , E (46)6 , E (46)8 , E (46)10 , E (46)12 E (47)4 , E (47)6 , E (47)8 , E (47)10 , E (47)12 E (51)4 , E (51)6 , E (51)8 , E (51)10 , E (51)12 , E (51)14 E (55)4 , E (55)6 , E (55)8 , E (55)10 , E (55)12 , E (55)14 , E (55)16 , E (55)18 , E (55)20 , E (55)22 E (59)4 , E (59)6 , E (59)8 , E (59)10 , E (59)12 E (62)4 , E (62)6 , E (62)8 , E (62)10 , E (62)12 E (66)4 , E (66)6 , E (66)8 , E (66)10 , E (66)12 , E (66)14 , E (66)16 , E (66)18 , E (66)20 , E (66)22 E (69)4 , E (69)6 , E (69)8 , E (69)10 , E (69)12 E (70)4 , E (70)6 , E (70)8 , E (70)10 , E (70)12 , E (70)14 , E (70)16 , E (70)18 , E (70)20 , E (70)22 E (71)4 , E (71)6 , E (71)8 , E (71)10 , E (71)12 E (78)4 , E (78)6 , E (78)8 , E (78)10 , E (78)12 , E (78)14 , E (78)16 , E (78)18 E (87)4 , E (87)6 , E (87)8 , E (87)10 , E (87)12 E (94)4 , E (94)6 , E (94)8 , E (94)10 , E (94)12 E (95)4 , E (95)6 , E (95)8 , E (95)10 , E (95)12 , E (95)14 , E (95)16 E (105)4 , E (105)6 , E (105)8 , E (105)10 , E (105)12 , E (105)14 , E (105)16 , E (105)18 , E (105)20 E (110)4 , E (110)6 , E (110)8 , E (110)10 , E (110)12 , E (110)14 , E (110)16 , E (110)18 , E (110)20 , E (110)22 , E (110)24 , E (110)26 E (119)4 , E (119)6 , E (119)8 , E (119)10 , E (119)12 , E (119)14 , E (119)16 , E (119)18 , E (119)20 , E (119)22 , E (119)24 Remark 12.
We note that the sets in Table 2 are not necessarily minimal sets of generators. A specific examplein the case N = 2 is discussed below. As stated in the introduction, our goal was to determine a set of generatorsof the function field. Indeed, it seems to be a difficult problem to determine the structure of the ring of modularforms in any setting when M N >
1, meaning when there is an expression for the M N -th power of the Kroneckerlimit function in terms of holomorphic Eisenstein series yet no apparent expression for any smaller power of theKronecker limit function. 5. Examples
In this section we will present a number of specific formulae for various levels. It seems as if each level has itsown idiosyncratic characteristics, so we choose various examples which, in our opinion, depict some of the mostcomprehensible and quantifiable nuances.5.1. N = We will cite specific results here, referring the reader to the article [9] for additional information andproofs. The Kronecker limit function can be written as∆ ( z ) = (cid:0) E (2)4 ( z ) (cid:1) − E (2)8 ( z ) . (16)In addition, one has that j ( z )∆ ( z ) = − (cid:0) E (2)4 ( z ) (cid:1) + E (2)8 ( z ) . By arguing as in [10], one can prove a dimension formula for the space of automorphic forms of weight 2 k , namelythat dim T (2)2 k = ( ⌊ k ⌋ , if k is congruent to 1 modulo 4, k ≥ ⌊ k ⌋ + 1 , if k is not congruent to 1 modulo 4, k ≥ . (17)The space T (2)2 k is generated by the set of monomials ( E (2)4 ( z )) l ( E (2)6 ( z )) m ( E (2)8 ( z )) n , where l, m, n are non-negativeintegers such that 4 l + 6 m + 8 n = 2 k . The dimension formula (17) yields some interesting number-theoreticalformulae. For example, since dim T (2)10 = 1, we see that E (2)10 ( z ) = E (2)6 ( z ) E (2)4 ( z ). By equating the q -expansions (8)for k ∈ { , , } , one obtains the following summation formula for the generalized sum of divisors: A (2)9 ( n ) = 336 n − X j =1 A (2)3 ( j ) A (2)5 ( n − j ) + 7 A (2)5 ( n ) − A (2)3 ( n ) , where A (2)2 k − ( n ) = σ k − ( n ) + 2 k δ ( n ) σ k − ( n/ k = 1 , , . . . and δ ( n ) = 1 for even positive integers n and δ ( n ) = 0, otherwise.Analogously, using formula (16), the q -expansion (8) and the q -expansion for the delta function, ∆( z ) = P ∞ n =1 τ ( n ) q n , where τ ( n ) is the Ramanujan function, one obtains relations involving τ , σ and σ .5.2. N = As with the case N = 2, we refer the reader to [9] for additional information and proofs. The Kroneckerlimit function vanishes to order 2 at i ∞ and has weight 12. The smallest weight cusp form has weight 8, but itvanishes to order 1 at i ∞ , and, consequently, it vanishes elsewhere. The Kronecker limit function can be written as(18) j ( z )∆ ( z ) = (cid:0) E (3)4 ( z ) (cid:1) + (cid:0) E (3)6 ( z ) (cid:1) − E (3)12 ( z )and the Hauptmodul is given by(19) ∆ ( z ) = − (cid:0) E (3)4 ( z ) (cid:1) − (cid:0) E (3)6 ( z ) (cid:1) + E (3)12 ( z ) . The dimension formula for the space of automorphic forms of weight 2 k isdim T (3)2 k = ( ⌊ k ⌋ , if k is congruent to 1 or 3 modulo 6, k ≥ ⌊ k ⌋ + 1 , if k is not congruent to 1 or 3 modulo 6, k ≥ . We note that the forms E (3)8 ( z ) − ( E (3)4 ( z )) and E (3)10 ( z ) − E (3)4 ( z ) E (3)6 ( z ) are cusp forms which vanish at ellipticpoints on X ; see Appendix B of [9]. In other words, there are cusp forms of weight smaller than the weight ofthe Kronecker limit function, but these forms necessarily vanish at some point in the interior of X , whereas theKronecker limit function vanishes at i ∞ only.Finally, let us explain why E (3)8 does not appear in Table 2. The information in Appendix B of [9] describes thezeros of small weight holomorphic forms. In particular, we conclude from the information provided that E (3)8 ( z ) E (3)8 ( z ) − ( E (3)4 ( z )) = c j ( z ) + c for some explicitly computable constants c and c . From this, we get that(20) E (3)8 ( z ) = ( E (3)4 ( z )) c j ( z ) + c c j ( z ) + c − . When combining (18), (19) and (20), we get a formula which expresses E (3)8 as a rational function involving E (3)4 , E (3)6 and E (3)12 , as asserted by Table 2.5.3. N = In the case N = 5, the surface X has genus zero, three order two elliptic elements e = i/ √ e = 2 / i/ e = 1 / i/ (2 √ hyp ( X ) = π . Its Kronecker limit function has weightfour, which is minimal, and the function vanishes at i ∞ to order one, which is also minimal. As a result, we havethat the mapping f ∆ f is an isometry between the spaces T (5)2 k − and S (5)2 k ; therefore, we arrive at the dimensionformula dim T (2)2 k = ( ⌊ k ⌋ , if k is congruent to 1 modulo 2, k ≥ ⌊ k ⌋ + 1 , if k is not congruent to 0 modulo 2, k ≥ . The space T (5)2 k is generated by the set of monomials ( E (5)4 ( z )) l (∆ ( z )) m ( E (5)6 ( z )) n , where l, m, n are non-negativeintegers such that 4 l + 4 m + 6 n = 2 k . From the output of the JST2 algorithm, we have that j ( z )∆ ( z ) = E (5)4 ( z ) − ∆ ( z ) . The analysis of ∆ differs between the JST2 and
JST3 algorithms. From
JST2 , we have that ∆ is a rationalfunction in the holomorphic Eisenstein series of weights four, six, eight and twelve. From JST3 , we have that ∆ is a polynomial in the holomorphic Eisenstein series of weights four, six, eight and twelve. Namely, from the outputof the JST3 algorithm, we have that j ( z ) (cid:0) ∆ ( z ) (cid:1) = (cid:0) E (5)4 ( z ) (cid:1) + (cid:0) E (5)6 ( z ) (cid:1) − E (5)8 ( z ) E (5)4 ( z ) + E (5)12 ( z )and (cid:0) ∆ ( z ) (cid:1) = − (cid:0) E (5)4 ( z ) (cid:1) − (cid:0) E (5)6 ( z ) (cid:1) + E (5)8 ( z ) E (5)4 ( z ) − E (5)12 ( z ) . N = Topologically, X and X are identical, with the same number of cusps, elliptic points of order two,and consequently, the same hyperbolic volume. The JST2 and
JST3 algorithms performed similarly in both cases,as one can see from Table 1 and Table 2. All comments above regarding the holomorphic function theory for X hold for X . However, as show in [6], the analytic function theory of X and X are different. Specifically, thecounting functions for the analytic Maass forms, when ordered by their Laplacian eigenvalues, are shown to beequal in their lead term but unequal in lower order terms.5.5. N = As we stated in the introduction, as N becomes larger, the formulae become massive. Our lastexample for N = 17. The Kronecker limit function has weight four and vanishes at i ∞ to order four. From the JST3 algorithm, we have the following formulae: j ( z ) (cid:0) ∆ ( z ) (cid:1) = (cid:0) E (17)4 ( z ) (cid:1) − (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) + (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) − (cid:0) E (17)6 ( z ) (cid:1) − E (17)8 ( z ) (cid:0) E (17)4 ( z ) (cid:1) + E (17)8 ( z ) (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) − E (17)8 ( z ) (cid:0) E (17)6 ( z ) (cid:1) E (17)4 ( z ) − (cid:0) E (17)8 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) − (cid:0) E (17)8 ( z ) (cid:1) (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) + (cid:0) E (17)8 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) − (cid:0) E (17)8 ( z ) (cid:1) (cid:0) E (17)6 ( z ) (cid:1) + (cid:0) E (17)8 ( z ) (cid:1) E (17)4 ( z ) − E (17)10 ( z ) E (17)6 ( z ) (cid:0) E (17)4 ( z ) (cid:1) − E (17)10 ( z ) (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) + E (17)10 ( z ) E (17)8 ( z ) E (17)6 ( z ) (cid:0) E (17)4 ( z ) (cid:1) + E (17)10 ( z ) E (17)8 ( z ) (cid:0) E (17)6 ( z ) (cid:1) − E (17)10 ( z ) (cid:0) E (17)8 ( z ) (cid:1) E (17)6 ( z ) E (17)4 ( z )+ (cid:0) E (17)10 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) − (cid:0) E (17)10 ( z ) (cid:1) (cid:0) E (17)6 ( z ) (cid:1) E (17)4 ( z ) − (cid:0) E (17)10 ( z ) (cid:1) E (17)8 ( z ) (cid:0) E (17)4 ( z ) (cid:1) − (cid:0) E (17)10 ( z ) (cid:1) (cid:0) E (17)8 ( z ) (cid:1) + (cid:0) E (17)10 ( z ) (cid:1) E (17)6 ( z )+ E (17)12 ( z ) (cid:0) E (17)4 ( z ) (cid:1) − E (17)12 ( z ) (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) + E (17)12 ( z ) (cid:0) E (17)6 ( z ) (cid:1) + E (17)12 ( z ) E (17)10 ( z ) E (17)6 ( z ) (cid:0) E (17)4 ( z ) (cid:1) − (cid:0) E (17)12 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) + (cid:0) E (17)12 ( z ) (cid:1) and (cid:0) ∆ ( z ) (cid:1) = − (cid:0) E (17)4 ( z ) (cid:1) + (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) − (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) + (cid:0) E (17)6 ( z ) (cid:1) + E (17)8 ( z ) (cid:0) E (17)4 ( z ) (cid:1) − E (17)8 ( z ) (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) + E (17)8 ( z ) (cid:0) E (17)6 ( z ) (cid:1) E (17)4 ( z )+ (cid:0) E (17)8 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) + (cid:0) E (17)8 ( z ) (cid:1) (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) − (cid:0) E (17)8 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) + (cid:0) E (17)8 ( z ) (cid:1) (cid:0) E (17)6 ( z ) (cid:1) − (cid:0) E (17)8 ( z ) (cid:1) E (17)4 ( z )+ E (17)10 ( z ) E (17)6 ( z ) (cid:0) E (17)4 ( z ) (cid:1) + E (17)10 ( z ) (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) − E (17)10 ( z ) E (17)8 ( z ) E (17)6 ( z ) (cid:0) E (17)4 ( z ) (cid:1) − E (17)10 ( z ) E (17)8 ( z ) (cid:0) E (17)6 ( z ) (cid:1) + E (17)10 ( z ) (cid:0) E (17)8 ( z ) (cid:1) E (17)6 ( z ) E (17)4 ( z ) − (cid:0) E (17)10 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) + (cid:0) E (17)10 ( z ) (cid:1) (cid:0) E (17)6 ( z ) (cid:1) E (17)4 ( z )+ (cid:0) E (17)10 ( z ) (cid:1) E (17)8 ( z ) (cid:0) E (17)4 ( z ) (cid:1) + (cid:0) E (17)10 ( z ) (cid:1) (cid:0) E (17)8 ( z ) (cid:1) − (cid:0) E (17)10 ( z ) (cid:1) E (17)6 ( z ) − E (17)12 ( z ) (cid:0) E (17)4 ( z ) (cid:1) + E (17)12 ( z ) (cid:0) E (17)6 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) − E (17)12 ( z ) (cid:0) E (17)6 ( z ) (cid:1) − E (17)12 ( z ) E (17)10 ( z ) E (17)6 ( z ) (cid:0) E (17)4 ( z ) (cid:1) + (cid:0) E (17)12 ( z ) (cid:1) (cid:0) E (17)4 ( z ) (cid:1) − (cid:0) E (17)12 ( z ) (cid:1) . Using the exact identity of the Hauptmodul in terms of Eisenstein series, we can read off that E (17)4 ( z ), E (17)6 ( z ), E (17)8 ( z ), E (17)10 ( z ), and E (17)12 ( z ) generate the holomorphic Eisenstein series E (17) k ( z ) for all even k ≥ Concluding remarks
Known relations for Hauptmoduli.
In [2], the authors computed expressions for j N , up to an additiveconstant. The authors call their function t N . The data from [2] relates to genus zero groups Γ ( N ) + with square-free level N as given in Table 3, using the Dedekind eta function together with θ ( a, b, c ), which is the theta functiondefined by the series θ ( a, b, c ) = X ( x,y ) ∈ Z q ( ax + bxy + cy ) . Table 3.
Known expressions of the Hauptmoduli j N for the genus zero groups Γ ( N ) + . N Formula for t N = j N + const. t = (cid:0) η ( z ) η (2 z ) (cid:1) + 4096 (cid:0) η (2 z ) η ( z ) (cid:1) t = (cid:0) η ( z ) η (3 z ) (cid:1) + 729 (cid:0) η (3 z ) η ( z ) (cid:1) t = (cid:0) η ( z ) η (5 z ) (cid:1) + 125 (cid:0) η (5 z ) η ( z ) (cid:1) t = (cid:0) η ( z ) η (2 z ) η (3 z ) η (6 z ) (cid:1) + 81 (cid:0) η (3 z ) η (6 z ) η ( z ) η (2 z ) (cid:1) = (cid:0) η ( z ) η (3 z ) η (2 z ) η (6 z ) (cid:1) + 64 (cid:0) η (2 z ) η (6 z ) η ( z ) η (3 z ) (cid:1) + c = (cid:0) η (2 z ) η (3 z ) η ( z ) η (6 z ) (cid:1) + (cid:0) η ( z ) η (6 z ) η (2 z ) η (3 z ) (cid:1) + c t = (cid:0) η ( z ) η (7 z ) (cid:1) + 49 (cid:0) η (7 z ) η ( z ) (cid:1) t = (cid:0) η ( z ) η (2 z ) η (5 z ) η (10 z ) (cid:1) + 25 (cid:0) η (5 z ) η (10 z ) η ( z ) η (2 z ) (cid:1) = (cid:0) η ( z ) η (5 z ) η (2 z ) η (10 z ) (cid:1) + 16 (cid:0) η (2 z ) η (10 z ) η ( z ) η (5 z ) (cid:1) + c = (cid:0) η (2 z ) η (5 z ) η ( z ) η (10 z ) (cid:1) + (cid:0) η ( z ) η (10 z ) η (2 z ) η (5 z ) (cid:1) + c t = (cid:0) θ (2 , , η ( z ) η (11 z ) (cid:1) = (cid:0) η ( z ) η (11 z ) η (2 z ) η (22 z ) (cid:1) + 16 (cid:0) η (2 z ) η (22 z ) η ( z ) η (11 z ) (cid:1) + 16 (cid:0) η (2 z ) η (22 z ) η ( z ) η (11 z ) (cid:1) + c t = (cid:0) η ( z ) η (13 z ) (cid:1) + 13 (cid:0) η (13 z ) η ( z ) (cid:1) t = (cid:0) η ( z ) η (7 z ) η (2 z ) η (14 z ) (cid:1) + 8 (cid:0) η (2 z ) η (14 z ) η ( z ) η (7 z ) (cid:1) = (cid:0) η (2 z ) η (7 z ) η ( z ) η (14 z ) (cid:1) + (cid:0) η ( z ) η (14 z ) η (2 z ) η (7 z ) (cid:1) + c t = (cid:0) η ( z ) η (5 z ) η (3 z ) η (15 z ) (cid:1) + 9 (cid:0) η (3 z ) η (15 z ) η ( z ) η (5 z ) (cid:1) = (cid:0) η (3 z ) η (5 z ) η ( z ) η (15 z ) (cid:1) + (cid:0) η ( z ) η (15 z ) η (3 z ) η (5 z ) (cid:1) + c t = (cid:0) θ x ( , , ) − θ y ( , , )2 η ( z ) η (17 z ) (cid:1) t = (cid:0) θ (2 , , θ (1 , , − θ (4 , , (cid:1) t = (cid:0) η ( z ) η (3 z ) η (7 z ) η (21 z ) (cid:1) + 7 (cid:0) η (7 z ) η (21 z ) η ( z ) η (3 z ) (cid:1) = (cid:0) η (3 z ) η (7 z ) η ( z ) η (21 z ) (cid:1) + (cid:0) η ( z ) η (21 z ) η (3 z ) η (7 z ) (cid:1) + c t = (cid:0) η ( z ) η (11 z ) η (2 z ) η (22 z ) (cid:1) + 4 (cid:0) η (2 z ) η (22) η ( z ) η (11 z ) (cid:1) t = (cid:0) θ (2 , , η ( z ) η (23 z ) (cid:1) = (cid:0) η ( z ) η (23 z ) η (2 z ) η (46 z ) (cid:1) + 4 (cid:0) η (2 z ) η (46 z ) η ( z ) η (23 z ) (cid:1) + 4 (cid:0) η (2 z ) η (46 z ) η ( z ) η (23 z ) (cid:1) + c t = (cid:0) η (2 z ) η (13 z ) η ( z ) η (26 z ) (cid:1) + (cid:0) η ( z ) η (26 z ) η (2 z ) η (13 z ) (cid:1) t = θ x ( , , ) − θ y ( , , )2 η ( z ) η (29 z ) t = (cid:0) η ( z ) η (6 z ) η (10 z ) η (15 z ) η (2 z ) η (3 z ) η (5 z ) η (30 z ) (cid:1) + (cid:0) η ( z ) η (6 z ) η (10 z ) η (15 z ) η (2 z ) η (3 z ) η (5 z ) η (30 z ) (cid:1) − = (cid:0) η ( z ) η (3 z ) η (5 z ) η (15 z ) η (2 z ) η (6 z ) η (10 z ) η (30 z ) (cid:1) + 4 (cid:0) η ( z ) η (3 z ) η (5 z ) η (15 z ) η (2 z ) η (6 z ) η (10 z ) η (30 z ) (cid:1) − + c t = (cid:0) η (3 z ) η (5 z ) η (6 z ) η (10 z ) η ( z ) η (2 z ) η (15 z ) η (30 z ) (cid:1) + (cid:0) η (3 z ) η (5 z ) η (6 z ) η (10 z ) η ( z ) η (2 z ) η (15 z ) η (30 z ) (cid:1) − + c = (cid:0) η (2 z ) η (3 z ) η (10 z ) η (15 z ) η ( z ) η (5 z ) η (6 z ) η (30 z ) (cid:1) + (cid:0) η (2 z ) η (3 z ) η (10 z ) η (15 z ) η ( z ) η (5 z ) η (6 z ) η (30 z ) (cid:1) − + c t = (cid:0) θ (2 , , − θ (4 , , η ( z ) η (31 z ) (cid:1) t = (cid:0) η ( z ) η (11 z ) η (3 z ) η (33 z ) (cid:1) + 3 (cid:0) η ( z ) η (11 z ) η (3 z ) η (33 z ) (cid:1) − t is deduced from the formula t ( z ) + t ( z ) − j ( z ) + j (2 z )35 t = (cid:0) η (5 z ) η (7 z ) η ( z ) η (35 z ) (cid:1) − (cid:0) η (5 z ) η (7 z ) η ( z ) η (35 z ) (cid:1) − t is deduced from the formula t ( z ) + t ( z ) − j ( z ) + j (2 z )39 t = (cid:0) η (3 z ) η (13 z ) η ( z ) η (39 z ) (cid:1) + (cid:0) η (3 z ) η (13 z ) η ( z ) η (39 z ) (cid:1) − t = θ x ( , , ) − θ y ( , , )2 η ( z ) η (41 z ) t = (cid:0) η ( z ) η (6 z ) η (14 z ) η (21 z ) η (2 z ) η (3 z ) η (7 z ) η (42 z ) (cid:1) + (cid:0) η ( z ) η (6 z ) η (14 z ) η (21 z ) η (2 z ) η (3 z ) η (7 z ) η (42 z ) (cid:1) − = (cid:0) η (2 z ) η (6 z ) η (7 z ) η (21 z ) η ( z ) η (3 z ) η (14 z ) η (42 z ) (cid:1) + (cid:0) η (2 z ) η (6 z ) η (7 z ) η (21 z ) η ( z ) η (3 z ) η (14 z ) η (42 z ) (cid:1) − + c t = (cid:0) η ( z ) η (23 z ) η (2 z ) η (46 z ) (cid:1) + 2 (cid:0) η ( z ) η (23 z ) η (2 z ) η (46 z ) (cid:1) − t = θ (2 , , − θ (4 , , η ( z ) η (47 z ) t is deduced from the formula t ( z ) − t ( z ) − j ( z ) + j (3 z )55 t is deduced from the formula t ( z ) − t ( z ) − t ( z ) + 16 t ( z ) = j ( z ) + j (5 z )59 t = θ (6 , , θ (2 , , − θ (6 , , t is deduced from the formula t ( z ) + t ( z ) − j ( z ) + j (2 z )66 t = (cid:0) η (2 z ) η (3 z ) η (22 z ) η (33 z ) η ( z ) η (6 z ) η (11 z ) η (66 z ) (cid:1) + (cid:0) η (2 z ) η (3 z ) η (22 z ) η (33 z ) η ( z ) η (6 z ) η (11 z ) η (66 z ) (cid:1) − t is deduced from the formula t ( z ) − t ( z ) − j ( z ) + j (3 z )70 t = (cid:0) η ( z ) η (10 z ) η (14 z ) η (35 z ) η (2 z ) η (5 z ) η (7 z ) η (70 z ) (cid:1) + (cid:0) η ( z ) η (10 z ) η (14 z ) η (35 z ) η (2 z ) η (5 z ) η (7 z ) η (70 z ) (cid:1) − t = θ (4 , , − θ (6 , , η ( z ) η (71 z ) t = (cid:0) η ( z ) η (6 z ) η (26 z ) η (39 z ) η (2 z ) η (3 z ) η (13 z ) η (78 z ) (cid:1) + (cid:0) η ( z ) η (6 z ) η (26 z ) η (39 z ) η (2 z ) η (3 z ) η (13 z ) η (78 z ) (cid:1) − t is deduced from the formula t ( z ) + t ( z ) − j ( z ) + j (3 z )94 t is deduced from the formula t ( z ) + t ( z ) − j ( z ) + j (2 z )95 t is deduced from the formula t ( z ) − t ( z ) + t ( z ) − j ( z ) + j (5 z )105 t is deduced from the formula t ( z ) − t ( z ) − j ( z ) + j (3 z )110 t is deduced from the formula t ( z ) + t ( z ) = j ( z ) + j (2 z )119 t is deduced from the formula t ( z ) − t ( z ) − t ( z ) − t ( z ) − j ( z ) + j (7 z ) ERTAIN ASPECTS OF HOLOMORPHIC FUNCTION THEORY ON SOME GENUS ZERO ARITHMETIC GROUPS 17
Additionally, one has, in the notation of [2], the functions θ x ( a, b, c ) and θ y ( a, b, c ) which are defined by the sameseries which defines θ ( a, b, c ) except one restricts the sum to odd values of x and y , respectively. By combiningour results with the relations for the Hauptmoduli in Table 3, it is possible to deduce many potentially interestingrelations between classical Eisenstein series E k ( z ), eta functions and theta functions.For example, let us take N = 17. In the notation of Theorem 7 one has M = 9 and the Hauptmodul j ( z ) isgiven as a rational function of the form j ( z ) = P ( E (17)4 , E (17)6 , E (17)8 , E (17)10 , E (17)12 ) Q ( E (17)4 , E (17)6 , E (17)8 , E (17)10 , E (17)12 ) , where P and Q denote polynomials of degree 9 in five variables with integer coefficients, where coefficients arenon-zero only if the sum of products of weights and corresponding degrees is equal to 36.In a sense, this result is a direct analogue of formula (2) expressing the classical j -invariant for PSL(2 , Z ) interms of classical holomorphic Eisenstein series.Furthermore, formula (7) implies that the Eisenstein series E (17)2 k , for k = 2 , , , , E k , hence the function (cid:18) θ x ( , , ) − θ y ( , , )2 η ( z ) η (17 z ) (cid:19) is a rational function in the Eisenstein series E ( z ), E (17 z ), E ( z ), E (17 z ), E ( z ), E (17 z ), E ( z ), E (17 z ), E ( z ) and E (17 z ) with integer coefficients.Proceeding in a similar manner, for example when N = 29 or N = 47, we obtain other relations between thetafunctions, eta functions and holomorphic Eisenstein series E k .6.2. Groups Γ ( N ) + of higher genus. There are 38 different square-free levels N such that X N has genus one.Similarly, there are 39 and 31 different square-free N such that X N has genus two and three, respectively. In [7],the authors studied the q -expansions for the corresponding function fields, proving that each function field admitstwo generators with various properties, such as minimal pole at infinity and integer coefficients. In particular, apolynomial relation was computed for each pair of generators, thus giving an algebraic equation for the correspondingprojective curve. In future studies, we plan to investigate the various properties of these elliptic (genus one)and hyperelliptic (genus two) curves. There are a vast number of problems, both arithmetic and analytic, tobe considered given that one knows the uniformizing group, a projective equation, q -expansions, and relations toholomorphic Eisenstein series. References [1]
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