INFINITE FAMILIES OF SIMPLE HOLOMORPHIC ETA QUOTIENTS SOUMYA BHATTACHARYA Abstract. We address the problem of constructing a simple holomor- 7 phic eta quotient of a given level N. Such constructions are known 1 for all cubefree N. Here, we provide such constructions for arbitrarily 0 large prime power levels. As a consequence, we obtain an irreducibility 2 criterion for holomorphic eta quotients in general. n a 1. Introduction J 1 The Dedekind eta function is defined by the infinite product: ] ∞ T (1.1) η(z) := q214 (cid:89)(1−qn) for all z ∈ H, N n=1 . h where qr = qr(z) := e2πirz for all r and H := {τ ∈ C|Im(τ) > 0}. Eta is a t a holomorphic function on H with no zeros. This function has its significance m in Number Theory. For example, 1/η is the generating function for the [ ordinary partition function p : N → N (see [1]) and the constant term in the Laurentexpansionat1oftheEpsteinzetafunctionζ attachedtoapositive 1 Q v definite quadratic form Q is related via the Kronecker limit formula to the 8 value of η at the root of the associated quadratic polynomial in H (see [8]). 7 The value of η at such a quadratic irrationality of discriminant −D is also 2 0 related via the Lerch/Chowla-Selberg formula to the values of the Gamma 0 function with arguments in D−1N (see [24]). Further, eta quotients appear . 1 in denominator formula for Kac-Moody algebras, (see [14]), in ”Moonshine“ 0 of finite groups (see [12]), in Probability Theory, e. g. in the distribution of 7 the distance travelled in a uniform four-step random walk (see [5]) and in 1 : the distribution of crossing probability in two-dimensional percolation (see v [16]). In fact, the eta function comes up naturally in many other areas of i X Mathematics (see the Introduction in [2] for a brief overview of them). r The function η is a modular form of weight 1/2 with a multiplier system a on SL (Z) (see [17]). An eta quotient f is a finite product of the form 2 (cid:89) (1.2) ηXd, d where d ∈ N, η is the rescaling of η by d, defined by d (1.3) η (z) := η(dz) for all z ∈ H d and X ∈ Z. Eta quotients naturally inherit modularity from η: The eta d quotient f in (1.2) transforms like a modular form of weight 1 (cid:80) X with a 2 d d 2010 Mathematics Subject Classification. Primary 11F20, 11F37, 11F11; Secondary 11Y05, 11Y16, 11G16, 11F12. 1 2 SOUMYABHATTACHARYA multiplier system on suitable congruence subgroups of SL (Z): The largest 2 among these subgroups is (cid:110)(cid:18)a b(cid:19) (cid:12) (cid:111) (1.4) Γ (N) := ∈ SL (Z)(cid:12)c ≡ 0(mod N) , 0 c d 2 (cid:12) where (1.5) N := lcm{d ∈ N|X (cid:54)= 0}. d We call N the level of f. Since η is non-zero on H, the eta quotient f is holomorphic if and only if f does not have any pole at the cusps of Γ (N). 0 Aneta quotient on Γ (M)isanetaquotientwhoseleveldividesM. Letf, 0 g and h be nonconstant holomorphic eta quotientson Γ (M) such that f = 0 g×h. Then we say that f is factorizable on Γ (M). We call a holomorphic 0 eta quotient f of level N quasi-irreducible (resp. irreducible), if it is not factorizable on Γ (N) (resp. on Γ (M) for all multiples M of N). Here, it is 0 0 worth mentioning that the notions of irreducibility and quasi-irreducibility of holomorphic eta quotients are conjecturally equivalent (see [2]). We say that a holomorphic eta quotient is simple if it is both primitive and quasi- irreducible. 2. The main result and conjecture For a prime p and an integer n > 3, we define the eta quotient f by p,n (2.1)  n/2−1  ηppηp(pn−−11)2 (cid:89) ηpp22s−−31p+1ηpp22s−2p+2   s=1 if n is even.  (ηη )p−1  pn f := p,n  n−1  (η η )p (cid:89)ηp2−3p+2  p pn−1 ps   s=1 if n is odd and p (cid:54)= 2,  (ηη )p−1 pn Clearly, f is invariant under the Fricke involution W . We shall show p,n pn that: Theorem 1. For any integer n > 3, f is a simple holomorphic eta quo- p,n tient of level pn. FromTheorem2in[2],werecallthatanysimpleholomorphicetaquotient of a prime power level is irreducible. So, in particular, the above theorem implies: Corollary 1. For any integer n > 3, the eta quotient f is irreducible. p,n Also, from Corollary 1 in [2], we recall that given an irreducibile holomor- phic eta quotient f of a prime power level, all the rescalings of f by positive integers are irreducible. Thus we obtain: Corollary 2 (Irreducibility criterion for holomorphic eta quotients). Given a holomorphic eta quotient g, if there exist n,d ∈ N and a prime p, such that g is the rescaling of f by d, then g is irreducible. Here f is p,n p,n as defined in (2.1). INFINITE FAMILIES OF SIMPLE HOLOMORPHIC ETA QUOTIENTS 3 With a good amount of numerical evidence, we conjecture that Conjecture 1. For any integer n > 3 and for any odd prime p, there are no simple holomorphic eta quotients of level pn and of weight greater that of f . p,n 3. Notations and the basic facts By N we denote the set of positive integers. For N ∈ N, by D we denote N the set of divisors of N. For X ∈ ZDN, we define the eta quotient ηX by (cid:89) (3.1) ηX := ηXd, d d∈DN where X is the value of X at d ∈ D whereas η denotes the rescaling of η d N d by d. Clearly, the level of ηX divides N. In other words, ηX transforms like a modular form on Γ0(N). We define the summatory function σ : ZDN → Z by (cid:88) (3.2) σ(X) := X . d d∈DN Since η is of weight 1/2, the weight of ηX is σ(X)/2 for all X ∈ ZDN. Recall that an eta quotient f on Γ (N) is holomorphic if it does not have 0 any poles at the cusps of Γ (N). Under the action of Γ (N) on P1(Q) by 0 0 M¨obius transformation, for a,b ∈ Z with gcd(a,b) = 1, we have (3.3) [a : b]∼ [a(cid:48) : gcd(N,b)] Γ0(N) for some a(cid:48) ∈ Z which is coprime to gcd(N,b) (see [10]). We identify P1(Q) with Q∪{∞} via the canonical bijection that maps [α : λ] to α/λ if λ (cid:54)= 0 and to ∞ if λ = 0. For s ∈ Q∪{∞} and a weakly holomorphic modular form f on Γ (N), the order of f at the cusp s of Γ (N) is the exponent of 0 0 q1/ws occurring with the first nonzero coefficient in the q-expansion of f at the cusp s, where w is the width of the cusp s (see [10], [23]). The following s is a minimal set of representatives of the cusps of Γ (N) (see [10], [20]): 0 (3.4) SN := (cid:110)a ∈ Q (cid:12)(cid:12) t ∈ DN, a ∈ Z, gcd(a,t) = 1(cid:111)/ ∼, t a b where ∼ if and only if a ≡ b (mod gcd(t,N/t)). For d ∈ D and for N t t a s = ∈ S with gcd(a,t) = 1, we have N t N ·gcd(d,t)2 1 (3.5) ord (η ;Γ (N)) = ∈ N s d 0 24·d·gcd(t2,N) 24 (see [20]). It is easy to check the above inclusion when N is a prime power. Thegeneralcasefollowsbymultiplicativity(see(3.13)and(3.16)). Itfollows that for all X ∈ ZDN, we have 1 (cid:88) N ·gcd(d,t)2 (3.6) ord (ηX;Γ (N)) = X . s 0 24 d·gcd(t2,N) d d∈DN 4 SOUMYABHATTACHARYA In particular, that implies (3.7) ord (ηX;Γ (N)) = ord (ηX;Γ (N)) a/t 0 1/t 0 for all t ∈ D and for all the ϕ(gcd(t,N/t)) inequivalent cusps of Γ (N) N 0 a represented by rational numbers of the form ∈ S with gcd(a,t) = 1. N t The index of Γ (N) in SL (Z) is given by 0 2 (cid:18) (cid:19) (cid:89) 1 (3.8) ψ(N) := N · 1+ p p|N pprime (see [10]). The valence formula for Γ (N) (see [23]) states: 0 (cid:88) 1 (cid:88) k·ψ(N) (3.9) ·ord (f) + ord (f;Γ (N)) = , P s 0 n 24 P P∈Γ0(N)\H s∈SN where k ∈ Z, f : H → C is a meromorphic function that transforms like a modular forms of weight k/2 on Γ (N) which is also meromorphic at the 0 cusps of Γ (N) and n is the number of elements in the stabilizer of P in 0 P the group Γ (N)/{±I}, where I ∈ SL (Z) denotes the identity matrix. In 0 2 particular, if f is an eta quotient, then from (3.9) we obtain (cid:88) k·ψ(N) (3.10) ord (f;Γ (N)) = , s 0 24 s∈SN because eta quotients do not have poles or zeros on H. it follows from (3.10) andfrom(3.7)thatforanetaquotientf ofweightk/2onΓ (N),thevalence 0 formula further reduces to (cid:88) k·ψ(N) (3.11) ϕ(gcd(t,N/t))·ord (f;Γ (N)) = . 1/t 0 24 t|N Since ord (f;Γ (N)) ∈ 1 Z (see (3.5)), from (3.11) it follows that of any 1/t 0 24 particular weight, there are only finitely many holomorphic eta quotients on Γ (N). More precisely, the number of holomorphic eta quotients of weight 0 k/2 on Γ (N) is at most the number of solutions of the following equation 0 (cid:88) (3.12) ϕ(gcd(t,N/t))·x = k·ψ(N) t t∈DN in nonnegative integers x . t Wedefinetheorder map ON : ZDN → 214ZDN oflevelN asthemapwhich sends X ∈ ZDN to the ordered set of orders of the eta quotient ηX at the cusps {1/t}t∈DN of Γ0(N). Also, we define the order matrix AN ∈ ZDN×DN of level N by (3.13) A (t,d) := 24·ord (η ;Γ (N)) N 1/t d 0 INFINITE FAMILIES OF SIMPLE HOLOMORPHIC ETA QUOTIENTS 5 for all t,d ∈ D . For example, for a prime power pn, we have N  pn pn−1 pn−2 ··· p 1  pn−2 pn−1 pn−2 ··· p 1      pn−4 pn−3 pn−2 ··· p 1    (3.14) A =  . pn  . . . . .  . . . . .  . . . ··· . .       1 p p2 ··· pn−1 pn−2   1 p p2 ··· pn−1 pn By linearity of the order map, we have 1 (3.15) O (X) = ·A X. N N 24 For r ∈ N, if Y,Y(cid:48) ∈ ZDNr is such that Y −Y(cid:48) is nonnegative at each element of Dr , then we write Y ≥ Y(cid:48). In particular, for X ∈ ZDN, the eta quotient N ηX is holomorphic if and only if A X ≥ 0. N From (3.13) and (3.5), we note that A (t,d) is multiplicative in N,t and N d. Hence, it follows that (cid:79) (3.16) A = A , N pn pn(cid:107)N pprime where by ⊗, we denote the Kronecker product of matrices.∗ It is easy to verify that for a prime power pn, the matrix A is invertible pn with the tridiagonal inverse:   p −p 0 −1 p2+1 −p2     −p p·(p2+1) −p3  1   (3.17) A−1 =  , pn pn·(p− p1)  ... ... ...   0     −p2 p2+1 −1   −p p where for each positive integer j < n, the nonzero entries of the column A−1( ,pj) are the same as those of the column A−1( ,p) shifted down by pn pn ∗Kroneckerproductofmatricesisnotcommutative. However,sinceanygivenordering of the primes dividing N induces a lexicographic ordering on D with which the entries N of A are indexed, Equation (3.16) makes sense for all possible orderings of the primes N dividing N. 6 SOUMYABHATTACHARYA j −1 entries and multiplied with pmin{j−1,n−j−1}. More precisely, 1 pn·(p− )·A−1(pi,pj) = p pn  p if i = j = 0 or i = j = n   −pmin{j,n−j} if |i−j| = 1 (3.18) pmin{j−1,n−j−1}·(p2+1) if 0 < i = j < n     0 otherwise. For general N, the invertibility of the matrix A now follows by (3.16). N Hence,anyetaquotientonΓ (N)isuniquelydeterminedbyitsordersatthe 0 setofthecusps{1/t}t∈DN ofΓ0(N). Inparticular,fordistinctX,X(cid:48) ∈ ZDN, we have ηX (cid:54)= ηX(cid:48). The last statement is also implied by the uniqueness of q-series expansion: Let ηX(cid:98) and ηX(cid:98)(cid:48) be the eta products (i. e. X(cid:98),X(cid:98)(cid:48) ≥ 0) obtainedbymultiplyingηX andηX(cid:48) withacommondenominator. Theclaim follows by induction on the weight of ηX(cid:98) (or equivalently, the weight of ηX(cid:98)(cid:48)) when we compare the corresponding first two exponents of q occurring in the q-series expansions of ηX(cid:98) and ηX(cid:98)(cid:48). 4. Proof of Theorem 1 We shall only prove the theorem for the case where n is even. The proof for the case where n is odd is quite similar. Let A be the order matrix N of level N (see (3.13)). Since all the entries of A−1 are rational (see (3.16) N and (3.17)), for each t ∈ D , there exists a smallest positive integer m N t,N such that m ·A−1( ,t) has integer entries, where A−1( ,t) denotes the t,N N N column of AN indexed by t ∈ DN. We define BN ∈ ZDN×DN by (4.1) B ( ,t) := m ·A−1( ,t) for all t ∈ D . N t,N N N From the multiplicativity of A−1(d,t) in N, d and t (see (3.16)), it follows N that B (d,t) (see (4.1)) is also multiplicative in N, d and t. That implies: N (cid:79) (4.2) B = B , N pn p∈℘N pn(cid:107)N where ℘ denotes the set of prime divisors of N. For a prime p, from (4.1) N and (3.17), we have   p −p 0 −1 p2+1 −p     −p p2+1 −p    (4.3) B =  . pn  ... ... ...   0     −p p2+1 −1   −p p INFINITE FAMILIES OF SIMPLE HOLOMORPHIC ETA QUOTIENTS 7 From [3], we recall that if ηX is an irreducible holomorphic eta quotient on Γ0(N), then X ∈ ZN ∩ZDN, where (cid:26) (cid:12) (cid:27) (cid:88) (cid:12) (4.4) ZN = Cdvd(cid:12)Cd ∈ [0,1] for all d|N (cid:12) d|N and for all d ∈ D , v = m u , where u is the column of B indexed by d N d d d d N andmd isthesmallestpositiveintegersuchthatvd ∈ ZDN. Letv := (cid:80)d|N vd and let F = ηv. Again, from [3], we recall that given a holomorphic eta N quotient g on Γ (N), the eta quotient F /g on Γ (N) is holomorphic if and 0 N 0 only if g corresponds to some point in ZN ∩ZDN. In particular, for N = p2m, the eta quotient F = ηv is given by p2m (4.5)  η(pz)p2−1 if m = 1,  F (z) = p2m  (η(pz)η(p2m−1z))p(p−1) (cid:81)2m−2η(prz)(p−1)2 if m > 1. r=2 Let f be the eta quotient defined in (2.1). Then we have p,2m (4.6)  η(z)p−1η(pz)p−2η(p2z)p−1 if m = 1,  F (z)  p2m = fp,2m(z)  η(z)p−1η(pz)p−2η(p2mz)p−1 (cid:81)m−1 η(p2r−1z)η(p2r+1z)p−1 if m > 1. r=1 η(p2rz) Since η(z)η(p2z)p−1 is a holomorphic eta quotient of level p2, it follows that η(pz) Fp2m(z) is a holomorphic eta quotient of level p2m for all m ∈ N. Let X ∈ fp,2m(z) ZDN be such that fp,2m = ηX. From (4.6), we conclude that X ∈ ZN. In other words, Y := A(cid:101)NX has all its entries in the interval [0,1]. From (2.1), it easily follows that ord (f ) = 1/24. Since f is invariant under the ∞ 2m,p 2m,p Fricke involution on Γ (p2m), we also have ord (f ) = 1/24, since the 0 0 2m,p Fricke involution interchanges the cusps 0 and ∞ of Γ (p2m). Since 0 and 0 1 (resp. ∞ and 1/p2m) represent the same cusp of Γ (p2m), from (3.17) we 0 get that both the first and the last entries of Y are equal to 1 . p2m−1(p2−1) There exists U ,V ∈ GL (Z) and a diagonal matrix D such that N N σ0(N) N D = U ×B ×V . We shall see in the next section that if N = pn for N N N N some prime p and some integer n > 2, then (4.7) D = diag(1,1,...,1,pn−1,pn−1(p2−1)) N and the last two columns of V are respectively N  (1,0)t if n = 1,      (4.8) C := (−1,0,1)t if n = 2, n,1     (1,1,p,p2,...,pn−3,pn−2,0)t if n > 2 8 SOUMYABHATTACHARYA and  (p,1)t if n = 1,      (4.9) C := (p2,1,1)t if n = 2, n,2     (pn,pn−2,pn−3...,p,1,1)t if n > 2. Next we briefly recall an useful tool from Linear Algebra: By elementary row and column operations [13], one can reduce any matrix B ∈ GL (Z) to a diagonal matrix D. In other words, there exists U,V ∈ GL (Z) and n n D = diag(d ,d ,...,d ) ∈ GL (Z) such that D = U ·B ·V. Since U,V ∈ GL (Z), 1 2 n n n we have U−1·Zn = Zn and V ·Zn = Zn. Therefore, n (cid:77) Zn/(B·Zn) = U−1·Zn/(B·V ·Zn) (cid:39) Zn/(U·B·V ·Zn) = Zn/(D·Zn) = Zn/d Zn. i i=1 The above isomorphism maps the element (cid:96) := ((cid:96) ...(cid:96) )t of (cid:76)n Zn/d Zn to the 1 n i=1 i element U−1·(cid:96) (mod B·Zn) of Zn/(B ·Zn). Since B is invertible, there is a bijection between Zn/(B ·Zn) and [0,1)n∩B−1·Zn, given by X (mod B·Zn) (cid:55)→ B−1·X (mod Zn). Composing this bijection with the isomorphism above, we get a bijection between (cid:76)n Zn/d Zn and [0,1)n∩B−1·Zn, given by i=1 i (cid:96) (cid:55)→ B−1·U−1·(cid:96) (mod Zn) = V ·D−1·(cid:96) (mod Zn). Now multiplication by B maps [0,1)n∩B−1·Zn bijectively to B·[0,1)n∩Zn. Let N = p2m and suppose, ηX = f is reducible. Let Y = A X. Since 2m,p N ηX isreduciblethereexistsY(cid:48),Y(cid:48)(cid:48) ∈ ZDN(cid:114){0}withY(cid:48) ≥ 0andY(cid:48)(cid:48) ≥ 0such thatY = Y(cid:48)+Y(cid:48)(cid:48) andbothB Y(cid:48) andB Y(cid:48)(cid:48) haveintegerentries. SinceB N N N is an integer matrix with determinant d := p2m−1(p2−1), we see that 1 N dN is the least possible entry for Y(cid:48) and Y(cid:48)(cid:48). Since Y(cid:48) +Y(cid:48)(cid:48) = Y has 1 as its dN first entry, either the first entry of Y(cid:48) or that of Y(cid:48)(cid:48) is zero. Similarly, either the last entry of Y(cid:48) or that of Y(cid:48)(cid:48) is zero. But it is easy to show∗ that if both the first and the last entries of Y(cid:48) (resp. Y(cid:48)(cid:48)) is zero, then Y(cid:48) (resp. Y(cid:48)(cid:48)) is entirely zero. So, without loss of generality, we may assume that the first entry of Y(cid:48) is 1 and the last entry of Y(cid:48) is 0. From the previous section dN and from the entries of the diagonal matrix D , we know that there exists N (cid:96) ∈ {0,1,...,p2m−1−1} and (cid:96) ∈ {0,1,...,p2m−1(p2−1)−1} such that 1 2 (cid:96) (cid:96) (4.10) 1 ·C + 2 ·C ≡ Y(cid:48) (mod Zn). p2m−1 2m,1 p2m−1(p2−1) 2m,2 Case 1. (m = 1) ∗From the congruence relation (4.10) (resp. replacing Y(cid:48) with Y(cid:48)(cid:48) in (4.10)). INFINITE FAMILIES OF SIMPLE HOLOMORPHIC ETA QUOTIENTS 9 Equating only the first and the last entries from both sides of (4.10), we obtain (cid:96) p(cid:96) 1 (cid:96) (cid:96) 1 + 2 ≡ (mod Z) and 1 + 2 ≡ 0 (mod Z), p p2−1 d p p(p2−1) N which together implies that (cid:96) 1 1 ≡ (mod Z). p d N But this modular equation has no solution in (cid:96) ∈ {0,1,...,p−1}. Thus 1 we get a contradiction! Case 2. (m > 1) SincethelastentriesofY(cid:48) andC are0, whereasthelastentryofC 2m,1 2m,2 is 1, it follows that (cid:96) = 0. Since the first entry of C is 1, we get 2 2m,1 (cid:96) 1 1 ≡ (mod Z) p2m−1 d N as in the previous case. Since as before, this has no solution in (cid:96) ∈ 1 {0,1,...,p2m−1−1}, we get a contradiction. Hence, f = ηX is irreducible. (cid:3) 2m,p 5. The matrix identities We continue to prove that the matrix identities B = UDV, UU(cid:48) = 1 and VV(cid:48) = 1 with B = B as defined in (4.3) and D = D as defined in pn pn (4.7) holds if we define U = U , V = V , U(cid:48) = U(cid:48) and V(cid:48) = V(cid:48) as follows pn pn pn pn for n = 1,2,3 or n ≥ 4: For n = 1, we define (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 0 −1 1 p p 1 1 −p U := , V := , U(cid:48) := and V(cid:48) := . 1 p 0 1 −1 0 0 1 For n = 2, we define 0 1 0  0 −1 p2  0 −1 1 U := 0 p 1, V :=  0 0 1, U(cid:48) =  1 0 0 and 1 p 1 −1 1 1 −p 1 0 −1 p2+1 −1 V(cid:48) = −1 p2 0 . 0 1 0 For n = 3, we define 10 SOUMYABHATTACHARYA 0 −1 −p −p2  1 0 1 p3 0 0 −1 −p  0 0 1 p U :=  , V :=  , 0 0 −p −(p2+1) 0 −1 p 1 1 p p2 p3 0 0 0 1  p 0 0 1 1 −1 0 −p(p2−1) U(cid:48) := −1 p 0 0 and V(cid:48) := 0 p −1 −(p2−1).  0 −(p2+1) p 0 0 1 0 −p  0 p −1 0 0 0 0 1 For n > 3: We define U = (U ) by i,j 0≤i,j≤n j = 0 j > 0 (cid:26) −pj−i−1 if j > i, i < n−1 0 (5.1) 0 otherwise. . i = n−1 0 −pn−j · p2(j−1)−1 p2−1 i = n 1 pj We define V = (V ) by i,j 0≤i,j≤n j = 0 0 < j < n−1 j = n−1 j = n i = 0 1 0 1 pn (5.2) (cid:26) −pi−j−1 if i > j, . 0 < i < n 0 pi−1 pn−i−1 0 otherwise. i = n 0 0 0 1 We define U(cid:48) = (U(cid:48) ) by i,j 0≤i,j≤n j = 0 0 < j < n−2 j = n−2 j = n−1 j = n i = 0 p 0 0 0 1 −1 0  0  p if i = j, .. . 0 < i < n−1 . −1 if i = j +1, 0 0 . . .  0 otherwise. 0 0 p i = n−1 0 −pn−j −(p2+1) p 0 i = n 0 pn−j−1 p −1 0 (5.3) We define V(cid:48) = (V(cid:48) ) by i,j 0≤i,j≤n j = 0 0 < j < n j = n i = 0 1 −1 0 ··· 0 −pn−2(p2−1)  p if i = j,  (5.4) 0 < i < n−1 0 −1 if i = j −1, −pn−i−2(p2−1) .  0 otherwise. i = n−1 0 1 0 ··· 0 −pn−2 i = n 0 0 1