A new type of functional equations of Euler products

A new type of functional equations of Euler products Bernhard Heim Abstract: We investigate interplay between generating series and infinite products including partition numbers, functional equations and modular forms. We apply multiplicative Hecke operators on periodic functions and introduce a new type of functional equations characterizing infinite products of Euler type. 2010 Mathematics Subject Classification. 05A19, 11F, 11N, 11P, 11F11, 11F30, 11F50, 11G18 Keywords: Partition function, Fourier coefficients, Infinite Products, Hecke operators, Borcherds products

The paper was written during a research stay July-August 2015 at the RWTH Aachen: Graduiertenkolleg Experimentelle und konstruktive Algebra, supported by the DFG. The author thanks Prof. Dr. Nebe and Prof. Dr. Krieg for the invitation and for providing a stimulating research environment at the RWTH Aachen University. Adress: German University of Technology in Oman, Halban Campus, PO Box 1816, Athaibah PC 130, Muscat, Sultanate of Oman e-mail: [email protected]

1

1

Introduction

The interplay between additive and multiplicative structures is one of the most fascinating topics in mathematics. Especially in algebra, combinatorics and number theory many beautiful and important results have been discovered and proven, some are still unproven (we refer to Ono [12] for a good source). For example, the famous Goldbach conjecture (1742), that every even number larger than 3 should be the sum of two prime numbers, is still open. In this paper we focus on structures related to generating functions, introduced by Euler. We concentrate on the question: How does the additive structure (generating functions) correspond to infinite products (multiplicative structure). This is closely related to the question, how can one deduce the divisor, the zeros and poles from the Fourier expansion of a meromorphic function. We contribute to this very classical topic with a new viewpoint. We show that infinite products of Euler type can be determined by a new type of functional equations. Functional equations of a more symmetric flavor recently appeared in [5]. Borcherds lifts [1, 2, 9] (certain automorphic forms) on the orthogonal group O(2, 2 + n) of signature (2, 2 + n) are characterized by functional equations. In this paper we deal with the case O(2, 1), which is a somehow degenerate case and not included. Moreover we work in a quite general setting. It is interesting to note that these functional equations are related to properties of cyclotomic polynomials. It is well-known that roots of unity and cyclotomic polynomials Φm (X) play a fundamental role in algebra and number theory (see Leutbecher [10] page 218). Nevertheless is seems to us the first time in the literature that they are also closely related to infinite products of Euler type. Let p be a prime number. Then Φp (X) :=

Y 

a

X − e2 π i p



=

0 1 2 2  [ 1 τ ∈ H 0 ≤ Re(τ ) ≤ and |τ | = 1 . 2 e be a subgroup of SL2 (R) which is commensurable with the Definition 3.1. Let Γ modular group Γ, and let k be a real number. A function f : H −→ C is called a (weakly) e if f is holomorphic on H and has modular form of weight k and multiplier system v for Γ the following two properties: e (i) The following equation holds for all γ = ( ac db ) ∈ Γ.   aτ + b = v(γ) (cτ + d)k f (τ ). f (γ(τ )) := f cτ + d

(3.1)

All |v(γ)| = 1. (ii) The function f is (meromorphic) at all cusps. If f vanishes at all cusps, then f is a cusp form. Let z ∈ C and r ∈ R. Then we define z r as given in [7], but not as in [8]. The reader not familiar with the concept of multipliers, commensurability, and cusps is adviced to consult e = Γ and (half-)integral weight, we also [8], section 1.4. Since we are mainly dealing with Γ just state the facts on the attached Fourier expansions (for a comprehensive treatment on the topic, we refer again to [8], section 1.4). One of the most important examples of this paper is given by the Dedekind eta function η (see also [8]): ∞ Y 1 24 η(τ ) := gb1 (τ ) = q (1 − q n ) . (3.2) n=1

The normal convergence of the product implies that gba is a holomorphic function for all a ∈ Z, which also implies that gba (τ ) 6= 0. Note that   2  ∞  X 12 nτ η(τ ) = e , (3.3) n 24 n=1
where ∗∗ denotes the Legendre-Jacobi-Kronecker symbol and e(z) := e2πiz (z ∈ C). Remark. It is well known that the Dedekind eta function satisfies a transformation law for all elements γ the full modular group Γ: 1

η(γ(τ )) = vη (γ) (cτ + d) 2 η(τ ). 7

(3.4)

Here vη is the induced multiplier system. Actually it is known that vη generates the group of multipliers of Γ of order 24. In particular one has η(τ + 1) = eπ i/12 η(τ ). Please note that gba is in general a weakly modular forms of weight a2 . The multiplier system is trivial if 24|a. Then gba (τ + 1) = gba (τ ) Let k be an nonnegative integer. Then we denote by Mk (Γ) and Sk (Γ) the space of modular forms and cusp forms of weight k. These are all modular forms with trivial multiplier with respect to Γ of weight k and with Fourier expansion ∞ X f (τ ) = an q n , q = e(τ ), τ ∈ H (3.5) n=n0

at infinity. Here n0 ≥ 0 and n0 ≥ 1 if f is a cusp form. Moreover Mk! (Γ) denotes the space of weakly modular forms (trivial multiplier system, but we allow poles at infinity). Let γ ∈ GL+ 2 (R) and k ∈ Z then we denote by j(γ, τ ) := cτ + d the usual cocycle and by (f |k γ) (τ ) := j(γ, τ )−k f (γ(τ )). (3.6) the Petersson slash operator. It is well known that Mk (Γ) and Sk (Γ) are finite dimensional vector spaces over C. Before we give several further examples, we just recall some background facts, making this exposition more natural. k k Let k ≥ 4 even. Then dim Mk (Γ) = [ 12 ] for k − 2 divisible by 12 and [ 12 ] + 1 if not divisible by 12. Further we have dim Sk (Γ) = dim Mk (Γ) − 1. Here [x] denotes the Gauss bracket. A meromorphic function f on H is called meromorphic modular form with integer weight k on Γ if f satisfies (f |k γ) (τ ) = f (τ )

(3.7)

for Γ and has at most poles at infinity, i.e. a Fourier expansion of type (3.5). Let √ ρ := 1+ 2 −3 and let F ∗ := {i∞}∪F be the compactified fundamental domain. To consider the zero/pole order ordω f of f is well-defined for all ω ∈ F ∗ . Let further ord ω := 2, 3, 1 if ω = i, ρ, otherwise. Let f be a meromorphic modular form of weight k. The famous valence formula (see for example Koecher and Krieg [7], page 169) states that X 1 k ordω f = . (3.8) ord ω 12 ∗ ω∈F The valence formula implies the dimension formula.

8

Examples. Eisenstein series Ek : Let k ∈ N0 be even. We denote by Bk the k-th Bernoulli number. Let k ≥ 2 then ∞ Bk X σk−1 (n) q n . Ek (τ ) := 1 − 2k n=1

(3.9)

P is a holomorphic function on H. Moreover Ek ∈ Mk (Γ) for k ≥ 4. Here σs (n) := d|n ds . Note that E4 has exactly one zero at ρ and E6 at i in F ∗ . The order for each zero is 1. Ramanujan Delta Function ∆: The dimension formula implies that there exists exactly one normalized cusp forms. This is given by E4 (τ )3 − E6 (τ )2 123 = q − 24q + 252q 2 − . . . .

∆(τ ) :=

(3.10)

It also follows from our observations that η 24 ∈ S12 (Γ). Hence ∆(τ ) = q

∞ Y

(1 − q n )24

(τ ∈ H).

n=1

This implies that ∆ has no zeros.

4

Hecke Theory for Periodic Functions

In this section we recall some standard notation and well-known properties of (additive) Hecke operators (for example, see [14]). Actually we generalize the theory slidely as given by Koecher and Krieg in chapter IV in [7]. At the same time we introduce multiplicative Hecke operators and indicate some useful properties used in this paper. Let V (H) be the C-vector space of all meromorphic functions f with f (τ ) = f (τ + 1) on the upper half space H. We assume that we have at most a pole at infinity. Hence for every f ∈ V (H) and Im(τ ) large enough one has X f (τ ) = af (m)q m (m0 ∈ Z). (4.1) m≥m0

9

Definition 4.1. Let k, n be integers and let n be positive. Let f ∈ V (H). Then we define the additive and multiplicative Hecke operators. ! X a b (k) TΣ (n)(f ) := nk−1 f |k , (4.2) 0 d a·d=n b ( mod n)

TΠ (n)(f ) :=

Y

f |0

a·d=n b ( mod n)

! a b . 0 d

(4.3)

(k)

We denote k the weight of the Hecke operator. Let TΣ (n)(f ) = λf (n) f for n ∈ N, then we call 0 6= f Hecke eigenform of weight k with eigenvalues λf (n). Note that   (k) (k) (k) TΣ (n m)(f ) = TΣ (n) TΣ (m)(f ) (k)

(k)

for all coprime n and m. Moreover let p a prime number. Then TΣ (pn ) ∈ Q[TΣ (p) with suitable interpretation. All the additive Hecke operators build up a commutative algebra (see IV, section 2 [7] for more details). With some more effort one can deduce similar properties for the multiplicative Hecke operators. TΠ (n m)(f ) = TΠ (n) (TΠ (m)(f ))

(4.4)

for all coprime n and m. Moreover let p a prime number. Then the action of TΠ (pn ) is deduced from TΠ (p). For example TΠ (p2 )(f ) = TΠ (p) (TΠ (p)(f )) · f −p TΠ (pl )(f ) = TΠ (pl−1 ) (TΠ (p)(f )) · TΠ (pl−2 )f


−p

(l ≥ 2).

All the operators commute and everything is determined by the action of TΠ (p). Hence it is easy to prove by induction that TΠ (n)(f ) = f σ(n) for all n ⇔ TΠ (p)(f ) = f p+1 for all p prime.

(4.5)

The properties of additive Hecke operators acting on periodic functions are almost the same as the original Hecke operators, since most of the properties are already reflected in the underlying abstract Hecke algebra. In the following we note some of them. Let f be a modular form, then (4.2) coincides with definition of the classical normalized (k) Hecke operators (for example, see chapter two [12]). Let g = TΣ (n)(f ) and let m0 be the possible order of f . Then    m0 ≥ 1  1  mn  X for m ≥ 0 (4.6) ag (m) = dk−1 af m0 = 0 .  d2  d|(m,n)  nm0 m0 < 0 10

Hence we obtain: ag (0) = σk−1 (n)af (0) and ag (1) = af (n). Corollary 4.2. Let 0 6= f ∈ V (H). Then f is a Hecke eigenform of weight k if and only if for all m, n ∈ N  mn  X k−1 λf (n)af (m) = d af . (4.7) d2 d|(m,n)

We have λf (n)af (1) = af (n). If af (1) = 1, we say f is normalized, which implies λf (n) = af (n). It is perhaps also interesting to see how the additive Hecke operators transform with respect to differentiation. Lemma 4.3. Let f ∈ V (H) then we have: 0  (k) (k+2) (n)(f 0 ). n TΣ (n)(f ) = TΣ

(4.8)

In the following we indicate how the multiplicative and the additive structures can be related and transformed vice versa. Please note that we omit questions on convergence to directly see the algebraic structure. To complement them is straightforward. The following result is very useful and gives more or less a tool to translate all formulas for additive Hecke operators into the corresponding formulas for multiplicative Hecke operators. It is also quite amusing how this is naturally related with logarithmic differentiation. Proposition 4.4. Let F, G ∈ V (H). Let F be of the form 1 + O(q) and G of the form q + O(q 2 ) Suppose (formally) that F = exp(G). Then we have the following correspondence between multiplicative and additive Hecke operators   (0) TΠ (n)(F ) = exp nTΣ (n)(G) . (4.9) Moreover the logarithmic derivative on the left side leads to (ln (TΠ (n)(F )))0 = TΣ (n)(G0 ). (2)

(4.10)

The proof mainly depends on the following observation. Let a, d ∈ N and let f ∈ V (H) be an arbitrary element for a moment. Then we decompose the Hecke operator in the following way. Let   X aτ + b (Ta,d (f )) (τ ) := f . (4.11) d b ( mod d)

11

The Fourier expansion of the action is given by X (Ta,d (f )) (τ ) = af (md)q ma . m≥

m0 d

This implies that 



(k) TΣ (n)(f )

= nk−1

X

d−k Ta,d (f ).

ad=n

P∞

P m Now let F (τ ) = 1 + m=1 aF (m)q and G(τ ) = ∞ m=1 aG (m)q . Then aF (1) = aG (1) holds. Finally we mention the following application, which is almost characterizing the Eisenstein series E2 of weight 2 (see (3.9)). The proof is straightforward. P m Lemma 4.5. Let G(τ ) = ∞ m=0 a(m)q ∈ V (H) be normalized with a(1) = 1. Then   (0) n TΣ (n)(G) = σ(n)(G) for all n ∈ N (4.12) m

if and only if G = a0 +

∞ X

σ−1 (m)q m .

(4.13)

m=1 1 1 This is implied by Corollary(4.2). If a0 := − 24 , then G = − 24 E2 .

5

Final Proofs of the Results

Before we give the proofs of the main results of this paper, we note the following useful 1 observation. Let f (τ ) be given as in Theorem 2.1. Let f1 (τ ) := q l +m0 am0 . Since f1 is not period f1 (τ + 1) 6= f1 (τ ), we cannot apply the Hecke operators introduced in the last section, since they are not well-defined. Nevertheless we can proceed in two ways. By abuse of notation, up to the value of εp the action is well defined. Or we fix representatives of the involved matrices. Actually in this way the functional equation (∗n ) is stated. Then f1 (τ ) satisfies the functional equation TΠ! (p)(f1 )

 p−1  Y τ +b = εp f1p+1 . := f (pτ ) f p b=0

Since f (τ ) = f1 (τ ) f2 (τ ) with ∞ X am0 +m m f2 (τ ) := 1 + q , am 0 m=1

(5.1)

we have that f satisfies a functional equation of our particular type if and only if f2 satisfies a functional equation. Here the nontrivial factor εp is induced by f1 . 12

5.1

Proof of Theorem 2.1

In view of the observation above, it is sufficient to prove the Theorem in the following P m variation. Let f (τ ) = 1 + ∞ satisfy m=1 am q TΠ (n)(f ) = f σ(n) for all n ∈ N,

(5.2)

then f is uniquely determined by a(1). There are several ways to do this. We first give a direct way, which reveals a recursion property of the Fourier coefficients am . ! !σ(n) ∞ ∞ X Y X aτ +b = 1+ am q m 1+ am e2πim d (5.3)

Y

1+

a·d=n b ( mod n)

m=1

m=1

a·d=n b ( mod n) ∞ X

! ma2 τ + mab n

am e2πi(

)

=

1+

m=1

∞ X

!σ(n) am q nm

(5.4)

m=1

Comparing the coefficients of q n on both sides leads to the recursion formula nan + Qn (a1 , . . . , an−1 ) = σ(n)a1 .

(5.5)

Here Qn is a polynomial with integer coefficients in n − 1 variables. A second proof (sketch) can be given in the following way. Let f be as before and f = exp(g), with g = q + O(q 2 ). Then the functional equations lead to the situation studied in Lemma 4.5, where a1 is more general. Finally we obtain g = a241 (E2 − 1).

5.2

Proof of Theorem 2.2 and Corollary 2.3

We start with the proof of the Theorem. We have to show that the function f (τ ) =

∞ Y

(1 − q n )a

(a ∈ Z)

(5.6)

n=1

is satisfying the functional equations for all n ∈ N. Applying (4.5) and the fact that the functional equations are compatible with taking powers, we are finally left with the claim: TΠ (p)(h) = hp+1 for all prime numbers p, where h(τ ) :=

Q∞

m=1 (1

(5.7)

− q m ). We have

 p−1  Y τ +b TΠ (p)(h)(τ ) = h(pτ ) h p b=0 =

∞ Y

(1 − q pm )

m=1

p−1 ∞  Y Y b=0 m=1

13

(5.8)  m 1 − q p ξpmb .

(5.9)

Q Q∞ Here ξp := e2πi/p . We interchange the order of the products p−1 b=0 m=1 . Then we consider the two cases p|m and p 6 |m. The first case leads to the expression hp . The second case is related to the p-th cyclotomic polynomial as defined in the introduction and leads to the expression ∞ Y (1 − q m ). (5.10) (m,p)=1

This term matches perfectly with the first term. Putting everything together proves the Theorem since f (τ ) = 1 − aq + O(q 2 ) is the unique solution of the functional equations for all a ∈ Z. As a byproduct we have also proven the Corollary.

5.3

Proofs of Theorem 2.4 and Corollary 2.5

The Dedekind eta function satisfies the functional equation (∗n ) for all n ∈ N. Let f be an automorpic form on H with respect to SL2 (Z) satisfying TΠ (p)(f ) = εp f p+1

(5.11)

for at least one prime p. We show that this already implies that f has no zeros and hence has to be proportional to η(τ )a (a ∈ N). Let f (τ0 ) = 0. Let τ0 ∈ F. Note that |pτ0 |>1 for all prime numbers p. Let τ1 := pτ0 . Then it follows from  p−1  Y τ1 + b = 0, (5.12) TΠ (p)(f )(τ1 ) = f (pτ1 ) f p b=0 that also f (τ1 ) = 0. This procedure can be repeated and shows that pn τ0 are infinitely many nonequivalent zeros of f . Note that for every n ∈ N an s ∈ Z exists that pn τ0 +s ∈ F. But since every automorphic form has only finite many zeros in F we have a contraction, which implies that that f has no zero. Hence f has to be of the prescribed type. If f ∈ Mk of integer weight k and trivial multiplier, and normalized. Then k is divisible by k 12 and f = ∆ 12 .

Acknowledgements To author would like to thank Aloys Krieg and Atsushi Murase for useful conversations related to the topic.

14

References [1] R. E. Borcherds, Automorphic forms on Os+2,2 (R) and infinite products, Invent. Math. 120 (1995), 161–213. [2] J. H. Bruinier, Borcherds Products on O(2, l) and Chern Classes of Heegner Divisors, Lecture Notes in Math. 1780 (2002), Springer Verlag. [3] D. A. Cox, Primes of the form x2 + ny 2 . Fermat, class field theory and complex multiplication, A Wiley-Interscience Publication, New York, 1989. [4] A. Dabholkar, S. Murthy, and D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074v1 [hep-th] 20 Aug 2012. [5] B. Heim and A. Murase, A characterization of Holomorphic Borcherds Lifts by Symmetries, International Mathematics Research Notices (2014). doi: 10.1093/imm/rnv021 [6] B. Heim, C. Kaiser, and A. Murase, Borcherds lifts, Symmetries and the AndreOort conjecture, manuscript 23.02.2015. [7] M. Koecher and A. Krieg, Elliptische Funktionen und Modulformen, 2nd edition, Springer, Berlin-Heidelberg-New York (2007). [8] G. K¨ohler, Eta Products and Theta Series Identities, Springer Monographs in Mathematics, Springer, Berlin-Heidelberg-New York (2011). [9] M. Kontsevich, Product formulas for modular forms on O(2, n), Seminaire Bourbaki 821 (1996). [10] A. Leutbecher, Zahlentheorie, Eine Einf¨ urung in die Algebra, Springer Verlag, Springer, Berlin-Heidelberg-New York (1996). [11] E. Neher, Jacobis Tripleprodukt-Identit¨at und η-Identit¨aten in der Theorie affiner Lie-Algebren, Jahresbericht d. Dt. Math.-Verein 87 (1985), 164–181. [12] K. Ono, The web of modulariy: arithmetic of the coefficients of modular forms and q-series, CBMS Regional Conference Series in Mathematics 102 Amer. Math. Soc., Providence, RI, (2004). [13] R. Remmert, G. Schumacher, Funktionentheorie 2, Dritte Auflage. Springer Berlin-Heidelberg-New York (2007).

15

[14] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Reprint of the 1971 original. Publications of the Math. Soc. of Japan, 11. Kan Memorial Lectures, 1. Princeton Univ. Press, Princeton, NJ, 1994.

16