arXiv:2103.02443v2 [math.GM] 16 Apr 2021 ∗ † [email protected],[email protected] [email protected], [email protected] A atraeitiunl iia onnhlmrhcMasf coefficients. Fourier Maass their non-holomorphic by suggested to similar intriguingly are latter L epnec n td t osqecsi em fteMelli the of terms in L consequences its study and respondence epooet soit oamdlrfr a nnt ubro number infinite the (an form on modular functions a to associate to propose We fntosadterMli ul,wihaecnouinpr convolution are which duals, Mellin their and -functions fnto eae otefr.Frhrw ics h aeof case the discuss we Further form. the to related -function 2 p colo hsclSine,Jwhra er University, Nehru Jawaharlal Sciences, Physical of School S:Piay1M9 10;Scnay1F1 11F37 11F11, Secondary 11Z05; 11M99, Primary MSC: 1 atnMdlfrMdlrCusp Modular for Model -arton sts olg,9 haaPaa uhre Road, Mukherjee Prasad Shyama 92 College, Asutosh aisi Dutta Parikshit p ai numbers -adic e eh 107 India 110067, Delhi New okt 006 India 700026, Kolkata Q 1 Forms Abstract ∗ p and o ahprime each for 1 eahsGhoshal Debashis p eeaoaeo h cor- the on elaborate We . rso egtzr as zero weight of orms dcsof oducts rdcso Dirichlet of products rnfr n the and transform n )cmlxvalued complex f) ϑ 2 sre.The -series. † Contents

1 Introduction 2

2 Modular forms and associated L-functions 4

3 A wavelet basis for complex valued functions on Qp 6

4 Vectors for modular (cusp) forms 8

5 Raising, lowering and Hecke operators 11 5.1 InnerproductI ...... 13 5.2 InnerproductII...... 14

6 Products of Dirichlet L-functions and Maass-like forms 16

7 Endnote 22

× A Wavelets on Qp 23

1 Introduction

The modular groups SL(2,Z) and its subgoups, while of interest to mathematicians for their myriad manifestations, also play an important role in models of statistical mechan- ics where they are related to the partition function viewed as a function of complexified temperature and other external parameters. Holomorphic modular forms, with definite transformation properties under a subgroup of the modular group, are essential ingredients in conformally invariant quantum field theories in two dimensions, and more specifically in string theory and black hole physics, where both holomorphic and non-holomorphic mod- ular forms arise (see e.g., [1, 2] and references therein). The usual holomorphic modular forms are defined as q-series expansion in which the coefficients satisfy certain multiplica- tive properties. The coefficients of a modular cusp form define an associated L-series, the two being related by the Mellin transform. Thanks to the multiplicative properties of its coefficients, an L-series admits an Euler product over prime numbers. Another class of L-series is the family of Dirichlet L-series, of which the Riemann zeta function is the most well known member [3], associated to the multiplicative Dirichlet characters. Each of these also admits a Euler product representation in terms of prime numbers. The product form allows one to connect the local factors to the p-adic number

fields Qp. We have recently constructed [4] pseudodifferential operators (as generalisa- tions of the notion of the Vladimirov derivative in [5]) which incorporate the Dirichlet characters and shown that the L-series can be expressed as the trace of an operator in an appropriate vector space. These operators act on the Bruhat-Schwarz class of locally constant complex valued square integrable functions on Qp. The traces of these opera-

2 2 −1 tors are well defined, being restricted to operators that act on the subspace L (p Zp) of functions with compact support in p−1Z Q . In a larger collaboration [6] we have p ⊂ p also initiated a programme to construct a random ensemble of unitary matrix models (UMM) for the prime factors of the Riemann zeta function, and then combine them to get a UMM for the latter. We find that the partition function of the UMM can be writ- 2 −1 ten as a trace of the Vladimirov derivative restricted to L (p Zp). In this approach the Riemann zeta function is essentially a partition function in the sense of statistical mechanics—see also [7–11] for ‘physical models’ of L-functions, a key difference is that we use the orthonormal basis provided by the wavelets [12] in Qp. In [4] we also constructed pseudodifferential operators corresponding to L-series of modular cusp forms. We showed that a family of locally constant functions, called the Kozyrev wavelets [12], known to be the eigenfunctions of the Vladimirov derivative, is the set of common eigenfunctions of all these pseudodifferential operators. The purpose of this article is to propose an association between a modular cusp form 2 and complex valued functions in L (Qp), one function for each prime p. More precisely, the correspondence is between the Fourier expansion of the cusp forms and the functions on Q . In other words, we argue that a cusp form is equivalent to a vector in L2(Q ). p ⊗p p This decomposition reminds us of the parton model of hadrons (which, like primes, begins fortuitously with p). A p-adic Mellin transform of these vectors, when combined for all primes, is shown to be related to the L-series corresponding to the cusp form. We also discuss Hecke operators in terms of the raising-lowering operators on the wavelet basis. This requires the definition of an appropriate inner product at the level of the p-artons. We examine two possibilities and study their properties. In an attempt to elaborate on these, we propose to define a class of toy L-functions (by taking a product of two Dirichlet L-functions) that mimic the Euler product form of L-functions associated with modular forms. As a pleasant surprise, we find that the objects associated to these bear an intriguing resemblance to non-analytic Maass forms [13]. It is likely that these are indeed toy examples of Maass-like forms, however, we have only been able to verify their behaviour under the Im(z) 1/Im(z) transform of the modular group. → − In the following, we review a few relevant facts concerning holomorphic cusp forms of the modular groups related to SL(2,Z) (Section 2) and wavelets on the p-adic numbers (Section 3) that also help us establish the notation used. In Section 4 we propose to associate complex valued functions on Qp, one for each prime p, which we call the p- artons, to a cusp form, and discuss their Mellin transforms. Realisation of the Hecke operators in this description is taken up in Section 5, where we propose possible inner products on the spaces of p-artons and their Mellin duals. Finally, in Section 6 we study the modular objects associated to products of two Dirichlet L-functions and point out their relation to non-analytic Maass forms. We conclude in Section 7 with a brief summary and a comment on the holographic nature of the correspondence proposed in this paper.

3 2 Modular forms and associated L-functions

a b The discrete subgroup1 SL(2, Z) = γ = a,b,c,d Z, ad bc =1 of the c d ∈ −     special linear group SL(2, R), is the symmetry group of lattices Λ in the complex plane

C [14–17]. It is sometimes called the full modular group, and has the following congruence subgroups of finite indices

Γ (N)= γ SL(2, Z) c 0 mod N 0 { ∈ | ≡ } Γ (N)= γ SL(2, Z) a, d 1 and c 0 mod N (1) 1 { ∈ | ≡ ≡ } Γ(N)= γ SL(2, Z) a, d 1 and b, c 0 mod N { ∈ | ≡ ≡ } (Since the conditions are empty for N = 1, Γ(1) is the full modular group SL(2, Z).) They are ordered as follows: Γ(N) Γ (N) Γ (N) Γ(1). Of these, Γ(N), called the ⊂ 1 ⊂ 0 ⊂ principal congruence subgroup of level N, is the kernel of the homomorphism SL(2, Z) → SL(2, Z/NZ). If Γ is a discrete subgroup such that Γ(N) Γ Γ(1), where N is the smallest ⊂ ⊂ such integer, it is referred to as a congruence subgroup of level N. The natural action of the group SL(2, R) on the upper half plane H = z : Im(z) > 0 restricts to that of { } Γ, which partitions it into equivalence classes. A fundamental domain is a subset of F H representing a Γ-equivalence class. For example, the fundamental domain of the full modular group Γ(1) is z H 1 Re(z) 1 , z 1 (ignoring some double counting ∈ | − 2 ≤ ≤ 2 | | ≥ of points on the boundaries).  A modular form [15] f : H C of weight k and level N associated to a Dirichlet 2 → character χN modulo N, is a holomorphic form on the upper half plane H that transforms, under the action of a discrete subgroup Γ(N) Γ Γ(1), as ⊂ ⊂ az + b f(γz) f = χ (d)(cz + d)kf(z), (2) ≡ cz + d N   where k N, N Z, γ Γ SL(2, Z). The modular form vanishes identically unless k ∈ ∈ ∈ ⊂ is an even integer. Using z z+1 in Eq. (2), one sees that a modular form of the full modular group Γ(1) → (i.e., of level 1) is a periodic function in z, hence it has the following Fourier expansion (q-expansion) in q = e2πiz

∞ ∞ f(z)= a(n)qn = a(n)e2πinz (3) n=0 n=0 X X 1More precisely, the relevant groups are the projective special linear groups PSL(2, R) = SL(2,R/ ) and PSL(2,Z) = SL(2,Z)/ 1 . {±} 2 {± } ∗ A Dirichlet character (modulo N) is a group homomorphism χN Hom(G(N), C ) from the multi- ∗ plicative group G(N) = (Z/NZ) of invertible elements of Z/NZ to C∈∗. It is a multiplicative character. It customarily extended to all integers by setting χN (m)=0 for all m which share common factors with N [14].

4 A cusp form is a modular form that vanishes as Im(z) i , or equivalently at q = 0. → ∞ Thus a(0) = 0 for a cusp form. It is conventional, and often convenient, to normalise the first coefficient a(1) of a cusp form to 1, which is what we shall assume in what follows. An equivalent description of modular forms is in terms of scaling functions on C/Λ, where Λ is a lattice left invariant by the action of a subgroup of the modular group.

Modular forms of weight k form a finite dimensional complex vector space Mk (Γ(1)) and of these the subset of cusp forms is a subspace Sk (Γ(1)). Similar notions exist for modular forms of congruence subgroups Γ Γ(1) of level N. However, a cusp form of a ⊂ congruence subgroup Γ is required to vanish as z approaches certain rational points on R = ∂H (equivalently, at certain point on the unit circle q =1), in addition to z i | | → ∞ (q =0). These additional points in the fundamental domain are images of Im(z) i . → ∞ n The Dirichlet series of a cusp form f = n a(n)q is defined by the coefficients in its q-expansion: ∞ P a(n) a(2) a(3) a(4) L(s, f)= =1+ + + + (4) ns 2s 3s 4s ··· n=1 X where we have used the normalisation a(1) = 1. This series converges uniformly to a holomorphic function of s in a half-plane to the right of Re(s) = σ +1 as long as the coefficients a(n) are bounded by some power nσ. The corresponding L-function | | associated to the cusp form f is then defined by an analytic continuation to the complex k+1 s-plane. For a cusp form of weight k, the series above converges in Re(s) > 2 . The series is also related to the cusp form as

(2π)s ∞ L(s, f)= [f](s)= dyys−1f(iy) (5) M Γ(s) Z0 i.e., it is the Mellin transform of f(iy). ∞ The discriminant function ∆(z) =(2π)12 (1 qn) is an example of a cusp form of − n=1 weight 12 (and level 1) of the full modular group.Y Ramanujan noticed that the coefficients in its q-expansion ∞ ∆(z)= τ(n)qn (6) n=1 X satisfy the following properties

τ(mn)= τ(m)τ(n) if gcd (m, n)=1 τ(pm+1)= τ(p)τ(pm) p11τ(pm−1) for p a prime and m> 0 (7) − τ(p) 2p11/2 | | ≤ The function τ : N Z is known as the Ramanujan τ-function. (The first two properties → were proved by Mordell, while the proof for the bound was provided by Deligne.) More

5 generally, the coefficients a(n) of a modular form of weight k and level N satisfy

a(mn)= a(m)a(n) if gcd (m, n)=1 (8) a(pm+1)= a(p) a(pm) χ(p)pk−1a(pm−1) for p a prime and m> 0 − The coefficient function a : N C is said to define a multiplicative character [14]. The → convergence of the series also puts a bound on the growth of the coefficients, which for a cusp form is a(p) Cp(k−1)/2 for C of order 1. Due to the above properties of its | | ≤ coefficients, the L-function of a cusp form f Eq. (4) too admit an Euler product form

∞ a(n) 1 L(s, f)= = (9) ns (1 a(p)p−s + χ(p)pk−1p−2s) n=1 p ∈ primes X Y − This is analogous to the Dirichlet L-functions Eq. (48), however, unlike that case, each local factor L (s, f)= 1 a(p)p−s + χ(p)pk−1p−2s −1 in the denominator of L(s, f) is a p − quadratic in p−s.

3 A wavelet basis for complex valued functions on Qp

The prime factors in the Euler product of Dirichlet L-functions Eq. (48), as well as L- functions associated with cusp forms can be related to complex valued functions on the p-adic space Qp. Using the ultrametric p-adic norm, which itself is an example of a complex valued function : Q C, one may write the prime factor (1 p−s)−1 in |·|p p → − the Riemann zeta function as 1 p ζ (s)= dx x s−1, Re(s) > 0 (10) 1 p−s ≡ p p 1 | |p − − ZZp where the dx is the (complex valued) translation invariant Haar measure on Qp. The integral may also be thought of as

p × s d x x p, Re(s) > 0 (11) p 1 × | | − ZZp in which the integral is over the non-zero elements in Zp with respect to the multiplicative (scale invariant) measure d×x = dx/ x of Q× viewed as a multiplicative group. The | |p p integrand x s, a multiplicative character on Q , is another complex valued function. | |p p Let us introduce two other functions

χ (x)= e2πix, x Q p ∈ p 1 if x x0 p 1 (12) Ωp(x, x0)= | − | ≤ , x, x0 Qp (0 otherwise ∈ where the first one is an additive character and the second is an indicator function for a unit ball centred at x [5,18]. This additive character defines the Fourier transform [f] of 0 F

6 −1 a function f : Qp C as [f](k)= dx χp(kx)f(x) and its inverse . The indicator → F Qp F function Ωp(x) retains its form after Fourier transform, i.e., [Ωp](k) = Ωp(k). In this R 2 F sense, it is an analogue of the Gaussian function e−πx on R. These are the ingredients of a family of orthonormal functions [12] on Qp

(p) − n 2πi jpnx n ψ (x)= p 2 e p Ω (p x m) (13) n,m,j p − (p) (p) dx ψn,m,j(x)ψn′,m′,j′ (x)= δnn′ δmm′ δjj′ (14) ZQp where n Z, m Qp/Zp and j = 1, 2, ,p 1. These functions, which we shall refer ∈ ∈ ··· − 2 to as the Kozyrev wavelets, provide an orthonormal basis for the set L (Qp) of square- integrable Bruhat-Schwarz (locally constant) functions on Qp. These wavelets may be considered to be analogous to the (generalised) Haar wavelets on R. All these functions can be obtained from the mother wavelets ψ0,0,j(x) by the action of the affine group ‘ax+b’ with suitable choices for a and b. The usual definition of the derivative of a function does not work due to the totally dis- connected topology of Qp. However, a pseudodifferential operator, called the generalised Vladimirov derivative [5, 19], is defined by an integral kernel

′ α 1 ′ f(x ) f(x) D(p)f(x)= dx ′ − α+1 Γ(p)( α) x x − Z | − |p (15) dx 2πix −α where, Γ(p)( α)= e x p − × x | | ZQp | |p for α in a suitable half plane of C and elsewhere by analytic continuation when appropriate. The Kozyrev wavelets are eigenfunctions of the Vladimirov derivatives

α (p) α(1−n) (p) D(p)ψn,m,j(x)= p ψn,m,j(x) (16)

α(1−n) corresponding to the eigenvalues p . The operator logp D(p) can be defined in the limit α 0 as α → D(p) 1 logp D(p) = lim − α→0 α ln p such that log D ψ(p) (x) = (1 n)ψ(p) (x). These wavelets are also common eigen- p (p) n,m,j − n,m,j functions of a more general class of Vladimirov derivatives twisted by a multiplicative character [4]. Since our interest will be in the eigenvalues, which depend only on the quantum number n related to scaling (and not m and j related to translation and phase), we shall restrict our attention to the set of eigenfunctions

(p) − n n−1 n ψ (x)= p 2 χ (p x) Ω (p x) 1 n (17) n,0,1 p p ←→ | − i(p) where we have used the alternative ket-vector notation labelled by the eigenvalue of the wavelet.

7 (p) Let us also define the raising/lowering operators a± on the wavelets

a(p)ψ(p) (x)= ψ(p) (x) a(p) n = n 1 (18) ± n,0,1 n±1,0,1 ←→ ± | i(p) | ∓ i(p) that changes the scaling quantum number by one. With these operators, together with (p) (p) (p) logp D(p), one can define J± = a± logp D(p) and J3 = logp D(p) the commutator algebra of which generate an sl(2, R) symmetry of the wavelets [20]. We shall be more interested in the subset spanned by the wavelets

m m =0, 1, 2, n=1−m ψ(p) (x) n =1, 0, 1, 2, (19) | i(p) | ··· ←→ n,0,1 | − − ···  n o (p) 2 −1 to define the subspace − L (p Zp). It consists of wavelets supported on the com- −1 H ⊂ pact subset p Zp and the corresponding eigenvalues of the operator D(p) on these basis functions are 1,p,p2, , i.e., positive integer powers of p. When restricted to this { ···} (p) subspace, we shall demand that the lowering operator a+ annihilates the mother wavelet corresponding to the lowest eigenvalue of the Vladimirov derivative3

a(p) 0 =0 a(p)ψ(p) (x)=0 (20) + | i(p) ←→ + 1,0,1 (p) Thus the wavelet ψ1,0,1(x) is like a ‘ground state’ or ‘lowest weight state’ in this subspace, (p) and the other wavelets arise from repeated applications of the raising operator a− on it, namely n = a(p) n 0 . | i − | i(p)
4 Vectors for modular (cusp) forms

We are now ready to propose an association between a modular cusp form and functions on Q , more precisely on (p) , i.e., those spanned by the wavelets in the subspace ⊗p p ⊗H− Eq. (19). For definiteness, we shall discuss the association for cusp forms of the principal congruent subgroup Γ(N) of the modular group. First we use prime factorisation of a natural number n N to relate it to a wavelet ∈ in (p) Q as follows ⊗H− ⊂ ⊗p p n = pnp Primes 7−→ p∈Y n = n n n n (21) | pi(p) | 2i(2) ⊗| 3i(3) ⊗| 5i(5) ⊗| 7i(7) ⊗··· p O

Since all but a finite number of nps are zero, the associated wavelets are the lowest weight or ground states 0 corresponding to the wavelets ψ(p) (x) for the prime p. Only | i(p) 0,0,1 a finite number of wavelets are, therefore, non-trivial in this correspondence. Similar association between natural numbers as vectors in the Hilbert space of fictitious quantum

3By an abuse of notation, we shall continue to use the same symbol for the restrictions of the raising- (p) lowering to − . However, since it is the subspace that is of primary interest to us, this should hopefully not be a causeH of confusion.

8 systems (dubbed arithmetic gas or primon gas) using the prime factorisation has been made in Refs. [7–10]. The present proposal relates to a mathematically well-defined space of functions, however, without any apparent physical meaning. On the other hand, the types of modular forms and L-functions for which we shall apply this correspondence are much larger. Recall the expansion of a cusp form in Eqs. (3) and (4) in which the leading coefficient a(1) has been normalised to unity. To this cusp form, we associate the following vector in (p) ⊗pH− ∞ ∞ np f = a(n)qn = a(pnp ) qQ p

n=1 np=0 p ! X X∞ Y f = a(pnp ) n 7−→ | i | pi(p) np=0 p X∞O n2 n3 n5 = a(2 ) n2 (2) a(3 ) n3 (3) a(5 ) n5 (5) n2,n3,n5, | i ⊗ | i ⊗ | i ⊗··· X···=0 ∞ = a(pnp )anp 0 (22) − | i(p) n =0 p Xp O where we have used the multiplicative property of the coefficients Eq. (8) for arguments which are coprime to each other. It is understood that for np =0, the coefficients a(1) = 1 for all p. We note that in the above too, most of the terms are ‘trivial’, i.e., most vectors correspond to the ground state 0 = n = 0 (p). In the last line, we use the fact | i(p) | p i ∈ H− that ket vectors of higher occupation number n 1 can be obtained from the ground p ≥ state by the action of the raising operator Eq. (18), i.e., n = anp 0 . | pi − | i(p) The vector f (p) appears to be a complicated one which is really entangled. | i ∈ ⊗pH− However, thanks to the multiplicative property of the coefficients Eq. (8), it simplifies to a product form

∞ −1 f = a(pnp )anp 0 = 1 a(p)a + pk−1χ(p)a2 0 f (23) | i − | i(p) − − − | i(p) ≡ (p) np=0 p p X O
O as can be seen by expanding the right hand side and comparing it with the left hand side. We would like to think of the vector f (p) as the p-th p-arton, i.e., the ‘part’ of (p) ∈ H− the cusp form f at the prime p. It is interesting to note that the operator that acts on the ‘ground state’ to generate the p-arton resembles the form of the local L-function at a prime p in Eq. (9). Indeed, that was a clue to find the factorisation above. All this can equivalently be expressed in terms of the complex valued wavelet func- tions, if we introduce the coordinate basis, familiar in quantum mechanics, consisting of generalised kets x , x Q which satisfy the orthogonormality condition (p) (p) ∈ p  ′ ′ x x ′ = δ ′ δ x x (p) (p ) pp (p) − (p)

9 involving the Dirac δ-functions. In this basis the components x f (also called the h (p)| (p)i wavefunction) of f are | (p)i ∞ ∞ f (x ) x f = a(pnp ) x n = a(pnp )ψ(p) (x ) (24) (p) (p) ≡ (p) (p) (p) p 1−np,0,1 (p) np=0 np=0 X X i.e., a linear combination of wavelets compactly supported on p−1Z Q . p ⊂ p

In summary, according to this correspondence, a cusp form f : H C is → equivalent to the infinite set of functions f : Q C, one function for (p) p → each prime p. The two are equivalent in the sense that from f we can get f , f , f , and vice versa. (2) (3) (5) ···
We would like to show that the association proposed in Eqs. (22) to (24) has other ramifications. To that end, we shall consider a Mellin transformation of the wavelets. There are different proposals for p-adic Mellin transforms [18, 21], however, all involve 4 integrands with multiplicative characters on Qp. The following definition that we shall employ is similar to the one in Ref. [21]

2πiℓ × p x|x|p s g˜ω(s) (p,ω)[g](s)= d x e x p g(x), s C and ℓ =0, 1, ,p 1 (25) ≡M × | | ∈ ··· − ZQp × where d x = dx/ x p is the scale invariant measure and the kernel contains the unitary 2πiℓ | | x|x|p character ωℓ(x) = e p . The character, which determines a phase depending on the leading p-adic ‘digit’ of x, has been mentioned in [5]. The transformation satisfies the scaling property [g(αx)](s)= α −s [g(x)](s) M(p,ω) | |p M(p,ω) similar to usual Mellin transforms, with ωℓ ωℓα|α|p , which is also a primitive p-th →n root of unity, if ωℓ is. In particular, for α = p the unitary character does not change, consequently [g(pnx)](s) = pns [g(x)](s) may be used to relate the Mellin M(p,ω) M(p,ω) transform of the wavelets. The inverse Mellin transform

p−1 2π/ ln p 2πiℓ ln p −1 − p x|x|p −it (p,ω)[˜gω](x)= e dt x p g˜ω(it) (26) M 2π 0 | | Xℓ=0 Z involves a discrete Fourier transform of the character ω and the path of integration is from 0 to 2π/ ln p along t, the imaginary axis in the complex s-plane.

With this definition, the Mellin transform of the Kozyrev wavelet ψn,0,1(x) is

1 1 n s− 1 [ψ ](s)= δ δ p ( 2 ) (27) M(p,ω) n,1,0 − p(1 p−s) − ps 1 ℓ,0 − ℓ,p−1  − −  4In [18] the multiplicative character in the kernel is taken to be unitary, which is satisfied if s in Eq. (25) is purely imaginary. The definition in [21] (see definitions 2.8.4 and 2.8.5) uses a normalised unitary character ω, which has a conductor pN , which for this case is N =1.

10 One can verify that the inverse transform of the above is the Kozyrev wavelet by an explicit calculation. It is relevant to point out that even though the contribution of the last term to the inverse transform adds to zero, while the contributions from the first two terms reproduce the wavelet function, the discrete Fourier transform involving the character ω is absolutely crucial for the inverse transform to work. As an aside, it is interesting to note that the Mellin transform of a Kozyrev wavelet is related to the eigenvalue of the generalized Vladimirov derivative D−s. From Eq. (24) and Eq. (27), we get

∞ 1 n (1−np) s− f (x ) (s)= c (ℓ,s) a(p p ) p ( 2 ) (28) M(p,ω) (p) (p) p np=0   X where

p−sΓ (s), ℓ =0 − p 1 δℓ,0 −1 cp(ℓ,s)= + + δℓ,p−1 = p ζp(s), ℓ =1, 2, ,p 2 (29) −p(1 p−s) ps 1  ··· − − − p−1ζ (s) 1, ℓ = p 1 p − −  is the np-independent factor in the parenthesis in Eq. (27). Now we can combine the results for all primes: the Mellin transform of the ‘wavefunction’ of Eq. (24) is

(ξ ,ξ ,ξ , ) f (s)= f (ξ ) (s) M(p,ω) (2) (3) (5) ··· M(p,ω) (p) (p) p   Y  ∞  1 np (1−np)(s− ) = cp(ℓ,s) a(p )p 2 p n =0 Y Xp s− 1 1 = c (ℓ,s)p 2 L s , f (30) p − 2 p ! Y   Curiously the L-function we get this way has its argument shifted from s to s 1 . The − 2 infinite product in the prefactor also depends on ℓ, which in turn depends on p.

5 Raising, lowering and Hecke operators

It would be instructive to understand the proposed decomposition of a modular form in terms of p-adic wavelets in the context of the profound Hecke theory of modular forms. However, this is outside the scope of our present understanding. We shall need to know how to extend the definition of the Kozyrev wavelets, which are defined on Qp, to the projective space P(Q ) Q so that their transformation under the full group p ∼ p ∪ {∞} GL(2,Qp) can be addressed. For now, we shall set a more modest goal of an operational understanding of the Hecke operators T (m), m N and their algebraic properties. ∈ Recall that the Hecke operators T (m), m N, are a set of commuting operators ∈ whose action on the modular form f(z)= a(n)e2πinz is to return the coefficients in the P 11 q-expansion as eigenvalues [14–17, 21]

T (m)f(z)= a(m)f(z) (31)

In other words a modular form is an eigenvector of the Hecke operators with the eigen- values as the coefficients in its q-expansion. They satisfy

T (m)T (n)= T (mn) for m ∤ n (32) T (p)T (pℓ)= T (pℓ+1)+ χ(p)pk−1T (pℓ−1)

Alternatively, the action of the Hecke operator T (n) on the underlying lattice can be understood as a sum of sublattices of index n: T (n)Λ = Λ′. ′ [Λ:ΛX]=n The Hecke operator T (m) can be written as a sum of two operators [14, 15], the first of which, V (m) gives a new series by replacing each qn in f by qmn

∞ ∞ (V (m)f)(z)= a(n)qmn = a(n)e2πimnz = f(mz) (33) n=1 n=1 X X while the action of the second, U(m), is to define a new series by keeping only those terms qn that are divisible by m

∞ m−1 n 1 z + j (U(m)f)(z)= a(n)q m = f (34) m m n=1 j=0   (Xm|n) X Note that U(m)V (m)= 1, but since V (m)U(m) deletes terms in Eq. (3) not divisible by m, V (m)U(m) = 1. A Hecke operator for prime argument can be written as 6 T (p)= U(p)+ χ(p)pk−1V (p) (35)

Thus, if we denote (T (p)f)(z)= b(n)qn, then b(n)= a(pn)+ χ(p)pk−1a(n/p). The actions of the U and V operatorsP remind us of the raising and lowering operators Eq. (18) on the wavelets. Consequently

∞ ∞ ∞ a a(pn) n − a(pn) n +1 = a(pn−1) n | i −→ | i | i n=0 n=0 n=1 X∞ X∞ X∞ a a(pn) n + a(pn) n 1 = a(pn+1) n | i −→ | − i | i n=0 n=1 n=0 X X ∞ X ∞ = a(p) a(pn) n χ(p)pk−1 a(pn−1) n | i − | i n=0 n=1 X X Hence, we propose to define the following Hecke operator for a prime argument

k−1 T (p)= a+ + χ(p)p a− (36)

12 to obtain

T (p) f(p) = a(p) f(p) It is easy to see that this will lead to the correct eigenvalue s Eq. (31).

5.1 Inner product I

The vector space of modular (cusp) forms of weight k is equipped with the Petersson inner product, defined as

f g = d2z (Im(z))k−2 f ∗(z)g(z) (37) h | i ZF which is invariant under the action of the modular group. The Hecke operators, in general, are not hermitian, rather they satisfy

T †(p)= χ∗(p)T (p) where, hermitian conjugation is defined in terms of inner product Eq. (37). In other ∗ 1 words, χ 2 T is a hermitian operator.

How does T (p) in Eq. (36) acting on the space of vectors f(p) behave under con- jugation? First, one needs to define an inner product. At this point we do not have an  understanding of the transformation properties of the wavelets under the local modular group GL(2,Qp), therefore, we do not know how to define a GL(2,Qp) invariant inner product analogous to the one in Eq. (37). We shall examine two choices, both seemingly natural, in turn. The first is defined in analogy with the Petersson inner product above, however, using only the subgroup of scaling transformations, as follows:

× k−1 ∗ f(p) g(p) = d x x f(p)(x) g(p)(x) (38) × | | ZQp

±1 Recall that the effects of the operators a± on the wavelets is to scale x by a factor of p (as well as a change in the overall normalisation such the raised/lowered wavelet functions remain orthonormal). By a change of variable of integration in the inner product defined above, it is easy to check, that

† k−1 † 1−k a+ = p a− and a− = p a+

Hence, it follows that T †(p)= χ∗(p)T (p), as well as that

a (p) χ∗(p)a f g =0 g − f(p) (p) (p)
Therefore, either f(p) = g(p), in which case the expression in the parenthesis vanishes establishing the hermiticity of T (p), or the two vectors associated to distinct cusp forms are orthogonal f g =0. h (p)| (p)i We should hasten to add that the arguments (and hence the conclusions) above are

13 formal. This is because the manipulation by scaling and redefinition of the variable of 2 integration work only for functions in L (Qp). We do, however, need to restrict to a 2 −1 subspace L (p Zp), in which scaling is not a symmetry as it may take us out of the subspace. Consequently, the relevant operators have to be projected back to the subspace of our interest. It is not obvious that the process of conjugation will commute with the projection. The arguments in Section 5.2 based on another proposal for an inner product does not rely on scaling. We would like to present an interesting observation, the analysis of the implication of which we shall leave for future. In the inner product Eq. (38), let us write the function g(p)(x) as the inverse (generalised) Mellin transform of its (generalised) Mellin transform, as defined in [21], to get

× (k−1)/2 ∗ −1 (k−1)/2 f(p) g(p) = d x x f(p)(x) ω ω x g(p) (x) × | | M M | | ZQp h 2πh ii k−1 ln p ln p k−1 × 2 ∗ −it ∗ 2 = d x x f(p)(x) dt x p ω (x) ω x g(p) (it) × Qp | | ω 2π 0 | | M | | Z (modXpN ) Z h i 2π ∗ ln p ln p k 1 k 1 = dt ω f(p) − + it ω g(p) − + it (39) ω 2π 0 M 2 M 2 mod N Z      ( Xp )     namely, a Parseval type It is interesting to note that the argument of the Mellin transform k 1 in the last line is 2 2 + it, which is shifted by half from the position of the conjectured − k zeroes of the L-function Eq. (4) on the line Re (s)= 2 +it. However, the poles of the local k−1 L-function in Eq. (9) are likely to be on the line Re (s)= 2 + it. This can be verified for those cases where the character χ is trivial, for example, the L(s, ∆) the coefficients of which are the Ramanujan τ-function.

5.2 Inner product II

In order to discuss the other inner product and its applications, it turns out to be more convenient to modify the Kozyrev wavelets Eq. (14) to define a set that is orthonormal × × with respect to the scale invariant measure d x on Qp . This modification is described in the Appendix A. We would also like to redefine the p-artonic modular forms Eq. (24) by rescaling the coefficients with a weight dependent factor as follows

∞ k−1 (p) − 2 np np f(p)(x(p))= p a(p )ψ1−np,0,1(x(p)) (40) n =0 Xp The rescaling is motivated by the bound on the growth of the coefficients of cusp forms. The inner product we define is the simple overlap integral

× ∗ f(p) g(p) = d x f(p)(x) g(p)(x) (41) × ZQp

14 of the modified p-artonic wavefunctions. We use the orthonormality conditions Appendix A to express the action of the raising and lowering operators Eqs. (18) and (20) on the function Eq. (40) to write

∞ × (p) (p)∗ a+f(p)(x)= d y ψ2−np,0,1(x)ψ1−np,0,1(y)f(p)(y) Q× np=1 Z p X∞ × (p) (p)∗ a−f(p)(x)= d y ψ−np,0,1(x)ψ1−np,0,1(y)f(p)(y) (42) × n =0 Qp Xp Z † From this one may check that a± = a∓ with respect to the inner product Eq. (41) above. Moreover, using the same set of equations, we get

∞ k−1 (p) k−1 k−1 (p) − 2 − 2 − 2 np np+1 a+f(p)(x(p))= p a(p)ψ1,0,1(x(p))+ p p a(p )ψ1−np,0,1(x(p)) np=1 ∞ X k−1 k−1 (p) 2 − 2 np np−1 a−f(p)(x(p))= p p a(p )ψ1−np,0,1(x(p)) n =1 Xp Hence the following combination

(p) (p) − k−1 † ∗ T(p)f a + χ(p)a f = p 2 a(p)f and T (p)= χ (p)T(p) (43) (p) ≡ + − (p) (p)   behaves like the Hecke operator T (p) (cf. Eq. (31) and Section 5.1). Thanks to these equations × ∗ ∗ † d x g(p)(x) T(p) χ (p)T (p) f(p)(x)=0 × − ZQp
implies that k−1 − 2 ∗ × ∗ p af (p) χ(p)ag(p) d x g(p)(x)f(p)(x)=0 (44) − × ZQp
from which we conclude that f(p) and g(p) are orthogonal (with respect to the inner ∗ 1 product Eq. (41)) and that χ 2 (p)af (p) are real. Following the steps leading to the equality in Eq. (39) we find the corresponding Parseval type identity for this inner product

× ∗ −1 f(p) g(p) = d x f(p)(x) ω ω g(p) (x) × M M ZQp
2π    ln p ln p ∗ = dt ω f(p) (it) ω g(p) (it) (45) ω 2π 0 M M mod N Z ( Xp )  
  2π ∗ ln p ln p k 1 k 1 = dt Lf(p) − + it Lg(p) − + it 2π 0 2 2 Z       In arriving at the last step above we have used the following. In the Mellin transform it k−1 f (it) = c (ℓ, it)p Lf + it , only the prefactor depends on ω(ℓ), therefore, Mω (p) p (p) 2     15 the sum in the discrete Fourier transform yields 1 for the argument s = it (see Appendix A) and that the t-integral leads to a Kronecker delta with which one of the sums is evaluated −(k−1)np ∗ n n trivially. Thus, both the left and right hand sides reduce to np p af (p )ag(p ). The left hand side vanishes if f and g arise from distinct modular forms. Then the above P is an orthogonality condition for the corresponding modular L-functions. It should be instructive to check this by a direct computation, which we have unfortu- nately not been able to verify. Instead, we offer an indirect argument. Let us parametrise the ‘roots’ of the denominator of the local function Lp(s, f) Eq. (9), which is a quadratic −s (k−1)/2 iα1(p) (k−1)/2 −iα2(p) in p , as a1(p)= p e and a2(p)= p e so that

k−1 iα (p) iα (p) 1 k−1 i (α −α ) a(p)= p 2 e 1 + e 2 = 2 cos (α + α ) p 2 e 2 1 2 2 1 2 χ(p)= ei(α1(p)−α2(p))

Notice that this is consistent with the condition on the growth of the coefficients a(p).

The function Lp(s, f) is the generating function

1 ∞ = U (cos θ)tn (46) 1 2t cos θ + t2 n n=0 − X of the family of orthogonal Chebyshev polynomials of type II, denoted by Un(x), with k−1 i 1 2 −s 2 (α1−α2) θ = 2 (α1 + α2) and t = p e . The Chebyshev polynomials of type II satisfy the three-term recursion relation

U (ξ)=2ξ U (ξ) U (ξ), U (ξ)=1 and U (ξ)=2ξ (47) n+1 n − n−1 0 1

In terms of trigonometric functions Un(cos θ) = sin((n + 1)θ)/sin θ. Furthermore, thanks to these recursion relation, they satisfy the multiplicative property Eq. (8). The orthogo- nality of the local functions Lp(s, f), and hence of the product functions L(s, f) will follow as a consequence of the orthogonality of the Chebyshev polynomials. The appearance of the Chebyshev polynomials of type II in this context has been noticed5 in Refs. [22, 23]. In the next section, we study a simpler class of functions, for which these properties will be manifest by construction.

6 Products of Dirichlet L-functions and Maass-like forms

Since the modular forms and the associated L-functions are rather complicated objects, we shall instead study a family of functions related to the Dirichlet L-function (corresponding to the Dirichlet character ν)

∞ ν(n) 1 L(s, ν)= = (48) ns (1 ν(p)p−s) n=1 p ∈ primes X Y − 5We became aware of these results after submitting a version of this article to the arXiv.

16 which, we shall see, is simpler. We cannot use these L-functions directly because, unlike the L-functions associated with a modular cusp form Eq. (9), where the local factors at prime p, 1 a(p)p−s + χ(p)pk−1p−2s −1 are quadratic functions of p−s, the local factors − L (s, ν)= (1 ν(p)p−s)−1 above are linear. Therefore, in order to mimic the properties p −
of a modular L-function, we consider a product of two Dirichlet L-functions to define the function 2L(s, ν) as follows

1 L(s, ν)= L(s, ν)L(s, ν∗)= 2 (1 ν(p)p−s)(1 ν∗(p)p−s) p Y − − ∞ a(n) 1 = (49) ns (1 2 cos(arg ν )p−s + p−2s) n=1 p p X Y − where ν∗ is the complex conjugate of the character ν (it is further assumed that ν is not a n principal character) and a(n) = ν(d) ν∗ is the convolution of the characters [24]. d Xd|n   Notice that the function 2L(s, ν) is still meromorphic as a function of s. Formally it has ∗ the same form as Eq. (9) with a(p)= ν(p)+ ν (p) = 2cos(arg νp), k =1, and χ =1 (the trivial character).

The local factor 2Lp(s, ν) at a prime p can be recognised as the generating function Eq. (46), hence

1 ∞ L (s, ν)= = U (cos ξ)p−snp (50) 2 p (1 2 cos(arg ν )p−s + p−2s) np p n =0 − Xp where Unp (ξ) are the Chebyshev polynomials of type II of degree np in ξ = arg νp. The property (the second condition in Eq. (8)) of the coefficients in the L-series Eq. (49) is the recursions relation Eq. (47) satisfied by the Chebyshev polynomials. Finally, from the two equations above, we find that the coefficients are n a(n)= ν(d) ν∗ = U (cos(arg ν )) (51) d np p d|n p such that X   n=YQ pnp

Before we proceed to discuss 2L(s, ν) further, let us note that one may define a slightly more general product

−1 ∗ ∗ i (α −α ) α1 + α2 −s −2s L(s, ν , ν )= L(s, ν )L(s, ν )= 1 2e 2 1 2 cos p + χ(p)p 2 1 2 1 2 − 2 p Y  

∗ i(α1−α2) where α1 = arg ν1(p), α2 = arg ν2(p), χ(p) = ν1(p)ν2 (p) = e and the coeffi- i 2 (α1−α2) 1 r cients a(p)= e U1 cos 2 (α1 + α2) satisfy the multiplicative property a(p)a(p )= a(pr+1) + χ(p)pk−1a(pr−1) with k = 1. This is a special case of the parametrisation
discussed at the end of Section 5.2.

17 The Dirichlet L-function in Eq. (48) is related to the ϑ-series

2 ϑ(z, ν)= ν(n)nǫeiπn z/N (52) n∈Z X where ν is a primitive Dirichlet character modulo N and ǫ = 1 (1 ν( 1)) takes the value 2 − − 0 or 1 depending on whether ν is even or odd, respectively. The series above defines a 1 2 −1 ǫ modular form of weight 2 + ǫ, level 4N and character ν(d) d , where the Legendre symbol gives a phase. The Mellin transform of the above is the L-function Eq. (48)

s+ǫ ∞ (π/N) 2 dy s+ǫ L(s, ν)= y 2 ϑ(iy, ν) (53) 2Γ s+ǫ y 2 Z0 We may now use this in Eq. (49) to write

s+ǫ ∞ ∞ (π/N) s+ǫ −1 ∗ 2L(s, ν)= dy1 dy2 (y1y2) 2 ϑ(iy1, ν)ϑ(iy2, ν ) s+ǫ 2 4 Γ 0 0 2 Z Z s+ǫ ∞ ∞ ∞ ∞ ′ 2 ′ 2 2(π/N) dy dy πn1yy πn2y

ǫ ∗ ǫ s+ǫ − N − ′ = ν(n1)n ν (n2)n y e Ny s+ǫ 2 1 2 y y′ Γ n =1 n =1 0 0 2 X1 X2 Z Z s+ǫ ∞ ∞ 4( π/N )

ǫ dy s+ǫ 2πn = a(n)n y K0 y (54) s+ǫ 2 y N Γ n=1 0 2 X Z   The integrals from the

L-functions have been reorganised to be a convolution of the integrals of the two ϑ-series. We have redefined the dummy variables of integrations and 2 ′2 summations to y = y1y2, y = y1/y2, n = n1n2, d = n1 and used Eq. (51) for the coefficients. The integral in y′ can be evaluated from standard tables (e.g., from [25]), or identified as the integral representation of the modified Bessel function K0. Finally, the y-integral in Eq. (54), being a Mellin transform of the modified Bessel function, is known analytically [25]. Performing the integral one of course gets back Eq. (49) as expected.

The expression above means that the function 2L(s, ν) is the Mellin transform of the convolution of two ϑ-series

s+ǫ (π/N) 2 ∞ dy′ iy (ϑ(ν) ⋆ ϑ(ν∗))(iy)= ϑ(iyy′, ν) ϑ , ν∗ 2Γ s+ǫ y′ y′ 2 Z0   s+ǫ (π/N ) 2
∞ dy′ iy = ϑ , ν ϑ(iyy′, ν∗) (55) 2Γ s+ǫ y′ y′ 2 Z0   where the last equality follows from an exchange
n n along with a redefinition of the 1 ↔ 2 variable of integration y′ 1/y′. The convolution is that of the Dirichlet character and → its complex conjugate (denoted by a star above), as well as a Mellin convolution in the imaginary part of their arguments that is shown explicitly. We know that the ϑ-series has modular property, in particular under y 1/y (the S-transformation z 1/z of the → → −

18 modular group restricted to the imaginary part)

1 +ǫ i y 2 ϑ , ν = τ(ν) ϑ(iy, ν∗) (56) y ǫ√   i N N−1 where τ(ν)= ν(m)e2πim/N is the Gauss sum. Hence we find that m=0 X ∞ dy′ i iy′ ∞ dy′ iy ϑ , ν ⋆ ϑ , ν∗ = y1+2ǫ ϑ , ν ⋆ ϑ(iyy′, ν∗) (57) y′ yy′ y y′ y′ Z0     Z0   where we have used τ(ν)τ(ν∗) = ν( 1)N. Thus the Mellin inverse of L(s, ν), namely − 2 the convolution Eq. (55) of the ϑ-series, is (quasi-)modular with weight 1+2ǫ under the transformation y 1/y. → It is interesting to note that this property, as well as the explicit expression on the RHS in Eq. (54), are reminiscent of the harmonic Maass waveform [13] of weight λ =0

∞ 2πn M (x + iy)= a (ny)ǫ √yK y e2πinx/N (58) λ,N n λ N n=1 X   restricted to purely imaginary argument. A Maass waveform M : H C, is a non- λ,N → holomorphic ‘modular function’ on the upper half-plane that is a square-integrable eigen- function of the hyperbolic Γ(1)-invariant Laplacian corresponding to the eigenvalue 1 λ2. 4 − Since a0 =0 in the above, it is actually a Maass cusp form. It may seem that there is a puzzle since the Maass waveform M0,N has zero modular weight, while the ϑ-series related 1 to each of the L-functions Eq. (53) are of weight 2 + ǫ, hence their (convolution) product 1 +ǫ should be of weight 1+2ǫ. This is true, however, it is compensated by y 2 , which is a non-holomorphic form of weight 1 2ǫ. The factor of √y was introduced by shifting − − the argument in the Mellin transform. It may also be noted that one loses complex ana- lyticity by performing a Mellin transform in only the imaginary part of the argument of the ϑ-series.

We propose to identify the modular object related to the series 2L(s, ν) to be a Maass form of the type above

∞ (ν) ǫ 2πny 2πinx √y f(x + iy, ν) M (x + iy)= a(n)(ny) √yK e N (59) → 0,N 0 N n=1 X   where a(n) is related to the Dirichlet character ν by Eq. (51). We must caution that this identification is tentative since we have not been able to show the transformation property of Eq. (55) under the full modular group. Even though this is only a special case of weight λ =0 and is moreover ‘reducible’ by construction, this toy construction of (quasi-)Maass waveforms through a product of Dirichlet L-series could be useful to understand aspects of the proposed correspondence. The following analysis is in that spirit. Henceforth we shall restrict to the case of even characters ν(1) = ν( 1) for definiteness. −

19 The case of the odd characters may be studied in an analogous fashion. Let us now consider the p-artonic modular forms corresponding to Eqs. (49) to (51)

∞ ∞ np (p) (p) f(p)(ν, x)= a(p )ψ1−np,0,1(x)= Unp (cos(arg νp)) ψ1−np,0,1(x) (60) n =0 n =0 Xp Xp While these vectors in (p) are well-defined, the correspondence of the tensor product H− f (ν, x ) to a modular object on the upper half-plane H proposed above would re- ⊗p (p) (p) quire further justification. This is because the Maass waveform Eq. (58), not being a meromorphic function, does not admit a q-series expansion. The latter form is what we had used for the correspondence Eq. (22) in Section 4. However, the dependence of both the holomorphic and non-holomorphic modular forms Eq. (3) and Eq. (59) on the variable x is of the same form. Therefore, the association proposed should correctly be thought of as that between the Fourier coefficients in the Fourier series expansion (in x) of the corresponding modular objects. The L-functions related to the modular objects are also defined with the help of these Fourier coefficients. Nevertheless, it would be desirable to make a distinction between the holomorphic and non-holomorphic forms at the level of the proposed p-artons. 6 Be that as it may, the inner product Eq. (41) of two such functions f(p)(νf ) and g(p)(νg) satisfies the orthogonality condition

× ∗ f(p)(νf ) g(p)(νg) = d x f(p)(νf , x) g(p)(νg, x) × ZQp
∞ ∗ = Unp cos(arg νf,p) Unp (cos(arg νg,p)) = δνf ,νg (61) np=0 X
as a consequence of the completeness of the Chebyshev polynomials. Hence the inner product of their tensor products f(νf ) and g(νg)

f(νf ) g(νg) = f(p)(νf ) g(p)(νg) p
Y
∞ ∗ np np = af (p ) ag(p ) p np=0 Y∞ X ∗ = af (n) ag(n) (62) n=1 X will also be orthogonal. Let us consider the inner product Eq. (37) of two Maass forms Mf = √y f(x + iy, νf ) and Mg = √y g(x + iy, νg)

dxdy ∗ f(νf ) g(νg) = (√y f(x + iy, νf )) √y g(x + iy, νg) N < x N 2 − 2 | |≤ 2 y Z |x+iy|>0

6For modular forms of weight k =1, the two inner products Eq. (38) and Eq. (41) are the same.

20 Unfortunately, we are not able to perform the integrals with the restrictions imposed by the fundamental domain. However, we can compute the above in the rectangular region N < x N ,y> 0 , which contains an infinite number of copies of the fundamental {− 2 | | ≤ 2 } domain . By the modular properties of the integrand and the measure, each of these F copies are identical, therefore, we expect to get the original integral multiplied by an infinite factor. That is exactly what we find. The x-integral in

∞ N/2 ∞ dy 2πmy 2πny f(ν ) g(ν ) = a∗(m)a (n) dx e2πi(n−m)x/N K K f g f g y 0 N 0 N m,n=1 Z−N/2 Z0     X vanishes unless m = n, giving δmn, implying that two different Maass forms are orthogonal. For the y-integral we use the expression [25]

∞ aµ−1−βbβ 1 µ + α + β 1 µ α + β dyy−µK (ay) K (by)= Γ − Γ − − α β 2µ+2Γ(1 µ) 2 2 Z0 −     1 µ + α β 1 µ α β Γ − − Γ − − − × 2 2     1 µ + α + β 1 µ α + β b2 F − , − − , 1 µ;1 × 2 1 2 2 − − a2   for Re(a + b) > 0, Re µ< 1 Re α Re β −| |−| | The integral we need is exactly at the border of the condition on µ, and indeed the arguments of all the Γ-functions are zero, giving a divergent factor. Thus, upto an in- finite normalisation factor arising out of the divergent Γ-functions, the modular objects related to the product L-functions 2L(s, ν) are orthogonal non-analytic Maass waveforms of modular weight zero. This corresponds to the orthogonality Eq. (62) obtained from the wavelet expansion. The expression Eq. (62) above can be related to the inner product of the corresponding product L-functions Eq. (49) by using the identity

1 T eit(ln m−ln n) δ (ln n ln m) lim dt = − = δmn T →∞ 2T √mn √mn Z−T α 1−α It should be mentioned that while the identity above for δmn works for m n for any 1 0 <α< 1 in the denominator, only for α = 2 does it lead to the properties required of an inner product in the following. Thus

∞ ∞ ∗ f(νf ) g(νg) = af (n) ag(m) n=1 m=1 X X ∞ ∞ 1 T a∗(n) a (n) = lim dt f g 1 −it 1 +it T →∞ 2T −T m 2 n 2 Z n=1 n=1 T X X 1 L∗ 1 L 1 = lim dt 2 f it, νf 2 g + it, νg (63) T →∞ 2T 2 − 2 Z−T    

21 where in the last line we have used the analytical continuation of the power series. The fi- nal expression allows for an interpretation as an inner product 2Lf 2Lg of the L-functions related to the Maass forms. This is a Parseval type identity, like that in Eqs. (39) and(45).

Hence, the family of product L-functions, defined in Eq. (50), seems to form an orthogonal set of functions (for a fixed prime value of N). Although the L-functions are known ex- plicitly, it is not straightforward to verify their orthogonality. It may be seen (by plotting the functions in Mathematica) that the imaginary part of the integrand is odd, therefore, its integral vanishes for f = g, however, its real part oscillates with increasing amplitude 6 making it difficult to perform the integral numerically.

7 Endnote

2 In this paper we have used the orthonormal bases provided by the wavelets [12] in L (Qp) to associate complex valued functions, one function for each prime number p to a cusp form of a congruence subgroup of the modular group SL(2,Z). We have studied the local functions (which may appropriately be called p-artons, borrowing a term from the physics of strong interaction) obtained through this decomposition, their Mellin transforms and the related L-functions. We leave the study of the action of the group GL(2,Qp) on these p-artons to the future. By taking a product of two Dirichlet L-functions, associated to a Dirichlet character and its complex conjugate, we have also defined functions that are similar to these modular L-functions. Rather surprisingly these turn out to behave like non-analytic Maass waveforms of weight zero. Although being reducible by construction, these may not be of intrinsic interest mathematically, however, being much simpler to work with, they may be useful toy objects to further the correspondence. Let us close with the following remark. The one to one relation between a cusp form (a complex function on the upper half of the complex plane) and a vector in (a subspace of) L2(Q ) is reminiscent of the holographic correspondence, which has dominated the ⊗p p landscape of research in theoretical physics in the recent decades. The conformal boundary of the upper half plane (H = SL(2, R)/U(1)) is the real line R. The latter is closely related to Qp, which can be thought of as the conformal boundary of the Bruhat-Tits tree, which in turn is the coset of GL(2,Qp) by its maximal compact subgroup GL(2,Zp). The association between a modular form f : H C and the products of functions f , → ⊗p (p) where f : Q C for a prime p is a complex valued function on Q , is suggestive of a (p) p → p bulk-boundary correspondence in holography. It is so, since the data related to a function in the bulk of the upper half of the complex plane is in the ‘boundary’ Q , which in ⊗p p turn is related to R = ∂H. In this sense, the proposed relation may even be called a holographic p-arton model of modular forms. It would be interesting to explore if there are conformal field theories (and their bulk duals) related to the p-atronic objects as well as the non-analytic Maass waveforms.

Acknowledgments: DG is supported in part by the Mathematical Research Impact Centric Support (MATRICS) grant no. MTR/2020/000481 of the Science & Engineering Research Board, Department of Science & Technology, Government of India. One of us

22 (DG) presented a preliminary version of some of these results (as well as those in [4]) in the National String Meeting 2019 held at IISER Bhopal, India during Dec 22–27, 2019. We would like to thank Chandan Singh Dalawat, and Vijay Patankar for useful discussions. We are particularly grateful to Krishnan Rajkumar for many patient explanations and valuable comments on the manuscript.

A Wavelets on Qp×

Let us form modify the Kozyrev wavelets Eq. (17) to define

1 (p) 2 (p) ψ (x)= x p ψ (x) (64) n,m,j | | n,m,j × which are naturally defined on Qp since they are orthonormal with respect to the scale invariant multiplicative Haar measure d×x:

dx (p) (p) (p) (p) ψn,0,1(x)ψn′,0,1(x)= dx ψn,0,1(x)ψn′,0,1(x)= δnn′ × x ZQp | |p ZQp Notice that the wavelets above, being different from the Kozyrev wavelets Eqs. (14) and (17) by a coordinate dependent factor, is not the constant p−n/2 for x < pn, | |p although it is still a locally constant function on Qp. The raising and lowering operators (p) (p) (p) (p) (p) Eqs. (18)and (20) act as before: a± ψn,0,1(x)= ψn±1,0,1(x) and a+ ψ1,0,1(x)=0, however, we choose to use a different notation to emphasise the fact that they act on a different space of functions. The Mellin transform Eq. (25) of these modified wavelets

(p) ns 1 1 ns (p,ω)[ψn,0,1](s)= cp(ℓ,s) p = 1 δℓ,0 δℓ,p−1 p (65) −s− 1 s+ M − p 1 p 2 − p 2 1 −  − −     2 1−|ps|2 likewise differ somewhat from Eq. (27). Let us note that c (ℓ,s) =1+ 2 , ℓ p s 1 | | p + 2 −1 thus, if the argument s is purely imaginary, the sum is 1. P

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