Results in Mathematics , Vol. 8 (1985) 0378-6218/851020146-07$01.50 + 0.2010

© 1985 Birkhauser Verlag, Basel

On the growth of entire automorphic integrals

MARVIN I. KNopp

1. Introduction

Inspired largely by the Eichler cohomology theory, the development of a theory of automorphic integrals - in a more or less general sense - has taken place during the past decade [1, 2, 3, 5, 6, 7, 8, 9, 10]. Adopting the most general definition likely to be of interest, we here prove that an entire automorphic integral (in a sense generalizing Heeke's notion of "entire automorphic form") has no greater than polynomial growth in the upper half-plane H and at the boundary of H. A slightly restrictive version of this result is proved by Cavaliere [1], using somewhat different methods.

2. Preliminaries

To make this statement precise, we define P to be the space of functions g which are holomorphic in H and satisfy the growth condition

y=Imz>O, (1)

for some positive constants K, p and (T. P is a ring and an infinite dimensional vector space. It contains all polynomials and all rational functions holomorphic in H. P contains as well all entire automorphic forms with respect to any H -group [4]. (An H -group is a discrete group of real linear functional transformations which is finitely generated, contains translations and has the entire real line as its limit set.)

Suppose r is an H-group. Without loss of generality, we may assume that

V = (; !) E r has determinant 1. Let k be a real number and v( V), V E r, a

multiplier system on r for the weight k . That is, Iv(V)1 = 1 and for any function F

defined on H,

(2)

146

On the growth of entire automorphic integrals

where the stroke operator FI~ V, V E r, is defined by

(FI~ V)(z) = iJ(V)(cz + d)-kF(Vz).

147

(3)

Equation (2) is the defining consistency condition of a multiplier system on r. In terms of the stroke operator, the characteristic functional equation of an au­tomorphic form of weight k and multiplier system v on r can be written simply as

FI~ V=F, for VEr. (4)

3. Automorphic integrals

The notion of automorphic integral arises by generalizing the functional equation (4) to

t{lv E P, for V E r. (5)

An automorphic integral of weight k and multiplier system v on r is a function F meromorphic in H, satisfying (5) and certain growth conditions at the parabolic cusps-finite in number-of a fundamental region R of the H -group r. The functions t{lv, V E r, are called the period functions of F; they satisfy the consis­tency (or cocycle) condition

(6)

which follows directly from (5). In order to formulate the growth conditions at the parabolic cusps, we assume that {t{lv} is a parabolic cocycle, that is to say, for l:5j:5u,

(7)

where 01> ... , Ou E r form a complete set of parabolic representatives for r. The 01> ... , Q u are the generators of the parabolic cyclic subgroups leaving invariant the parabolic cusps q1' ... ,qu, respectively, of R. (If there exists a single t{I E P such that t{lv = t{li V - t{I, for all V E r, then the cocycle {t{lv} is called a coboun­dary.) Suppose q1 = i oo, so that 0 1 is a translation.

Assume now that F is holomorphic in H and satisfies (5), with {t{lv} a parabolic cocycle. It then follows from (5), (7) and the analyticity of F in R n H (finitely many poles in R nH would do for this purpose) that F has the Fourier (actually, Laurent) expansion at qi' 1:5 j :5 u,

(8)

2:5j:5 U.

148 MARVIN I. KNOPP

The numbers K1' ..• , Ku are defined by

for 1::5 j::5 u. The numbers A10 ... ,Au are defined as follows. A1 is the smallest

A >0 such that (~ ~) is in r. (Hence S = (~ ~1) generates the stabilizer roo of

q1 = ioo in r.) For 2::5j::5u, put Aj = G ~~) so that Aj(qj) = 00. Define Aj >0 so

that Aj1(~ ~j)Aj generates the stabilizer of qj in r. (Aj is called the width of qj

in r.) The growth condition at the parabolic cusps of an automorphic integral can

now be phrased as: the expansions (8) of F at the qj are all left-finite, that is to say, each of the expansions (8) contains only finitely many terms for which m + Kj < 0, 1::5 j::5 u. Thus an automorphic integral on r is a function F meromorphic in H, satisfying the functional equations (5) and such that the expansions (8) are left-finite. In the spirit of Hecke we call an automorphic integral entire if it is holomorphic in H and the expansions (8) contain no terms with m + Kj < 0, 1::5 j::5 u. (That is, F is analytic at the cusps of R.)

4. We prove the

THEOREM. An entire automorphic integral on an H-group r is in P, for any weight k and corresponding multiplier system v.

The proof is contained implicitly in [7]. We reorganize the reasoning of [7] to give an explicit proof here. Suppose then that F is an entire automorphic integral on r and that {I/Iv} is the parabolic cocycle of period functions with I/Iv E P for V E r. We intend to show that FE P.

Recall that S is the minimal positive translation in r, that is, that roo = (S).

Without loss of generality we may assume that I/Is = 0, so that F(Sz) = (F I S)(z) = F(z). (We adopt the abbreviation F I V for FI~ V, since k and v are fixed throughout.) For, by (7) F I S = F + I/Is = F + 1/11 I S - 1/110 so that (F - 1/11) I S = F I S - 1/11 I S = F + 1/11 I S - 1/11 - 1/11 I S = F - 1/11· Furthermore, F - 1/11 E P precisely when FE P since 1/11 E P.

Simplifying somewhat the definition of [7], we let M be the set of coset representatives of r"" \r determined by: V EM if - Al2::5 Re (Vi) < A/2. (Note that Mn roo = {±I}.) With this choice of representatives, a2+b2+c2+d2::5K(c2+d2),

with K>O, independent of V= (; ~)Er. (This is Lemma 6 of [7].) This simple

On the growth of entire automorphic integrals 149

inequality and a result of Eichler [2, Theorem 1] together lead to the

LEMMA (Lemma 8 of [7]). There exist K, Po, 0"0> 0 such that for all T ERn H and VEM,

\t/lvC T)\ < K(yPo + Y -0"0),

where y = 1m (VT).

In the lemma and henceforth, R is the usual Ford fundamental region for r,

defined by R = {Z EH\\Re z\<,\f2 and icz+d\> 1 for all V= C :)Er-Co}

5. Proof of the Theorem

Since F is an entire automorphic integral and F \ s = F, the expansions (8)

assume the form

(9)

Since there are only finitely many cusps qi and t/li E P, it follows from (9) that for zERnH,

y = 1m z, for KR , P, 0">0. (10)

Put fez) = yk/2\F(z)\, Y = 1m z >0 and let V= (: !)E r. Then f \ S = f and, by (5),

yk/2 f(Vz) = \cz + d\k \(cz + d)kF(z) + (cz + d)kt/lV(Z)\

$ yk/2\F(z)\ + yk/2\t/lv(Z)\ = fez) + yk/2\t/lv(Z)\. (11)

Suppose now z E H. Then, by definition of M and R, there exist V E M, T ERn H and a rational integer m such that z = smVT. Let t = 1m T and y = 1m z. Then (11) implies

\F(z)\ = y-k/2f(z) = y-k!2f(smVT)

= y-k/2f(VT) $ y-k/2{f( T) + tk/2 \t/lV( T)\}

= y-k/2tk/2{\F(T)\ + \ t/lvCT) \}.

150 MARVIN 1. KNOPP

Since T ERn H, we may apply the inequality (10) and the Lemma to obtain

(12)

Now, if VEM and V1I, then c10 and thus \cT+dI2~1 for TER. It follows that y=lm(SmVT)=lm(VT)=t\cT+dl-2::s;t. On the other hand, T=V- 1S- mz, so that t=lm(V-1S - mz)=yl-cz+cm+al- 2. If c=O, then t=y, while c10

implies

I-cz +cm + a12= c2y2+(cm + a -cxf~c2y2>dy2,

where co> 0 depends only on r. Thus,

y::s;t<y+C(j2y - 1. (13)

Now (12) and (13) together show that FE P; the proof of the Theorem is complete.

6. Automorphic forms

When F is an entire automorphic form the inequality showing that FE P can be given in a quite explicit form. In order to derive this we simply reconsider the proof given in §5, observing the improvements which follow from the fact that in (5) I/Iv = 0 for all V E r.

Since tfJi = 0, 2::s; j:s u, in the expansions (9), (10) can be replaced by

IF(z)1 <KR (1 + y- k), y = 1m z, z ERn H.

Then (12) is replaced by

IF(z)1 <KRy-kI2tkI2(1 + t-k),

Applying (13), we obtain

IF(z)1 <K(1 + y- k), ZEH.

y =Im z, ZEH.

It is worth comparing (16) with the standard Hecke inequality

F(z) < Ky-kI2, zEH,

(14)

(15)

(16)

that one obtains when F is a cusp form. The inequality (16) implies in the usual way that

n~oo, (17)

when F is an entire form. In this case the estimate (17) holds as well for the coefficients ~(j), 2::s;j::s;u, in (9).

On the growth of entire automorphic integrals 151

7. Concluding remarks

A strong converse of the theorem is immediate: if FE P, then F is an entire automorphic integral for any H -group and any weight, with corresponding multiplier system. Further, it is not difficult to show that if

F(z) = L am exp {21Ti(m + K)Z/A.} m+K~O

is holomorphic in H, then FE P if and only if am = O(n'Y), n ~ 00, for some "I> O. Thus the theorem may be phrased: For an entire automorphic integral F on an H-group, ~(I)=O(n'Y), n~oo, with some "1>0, where the coefficients am(l) are defined by (9). (The same estimate, holds for the coefficients am (j), 2 $ j $ U in (9).)

We may infer from the remarks above that the space of entire automorphic integrals on an H -group is neither more nor less than the space P. This suggests that P may be too large a space for the cocycle of periods. Indeed, when k E Z, automorphic integrals with rational period functions form a subclass worthy of special attention ([2, 5, 6, 8, 10] and [1, Chapter 5]). It is not yet clear what the corresponding natural subclass is when the weight k is not an integer. (1)

REFERENCES

[1] R. A. CAVAUERE, Automorphic integrals and their period functions, Doctoral Dissertation, Temple University, Philadelphia, Pa., 1984.

[2] M. EICHLER, Grenzkreisgruppen und kettenbruchartige Algorithmen, Acta Arith. 11 (1965), 169-180.

[3] L. J. GOLDS1EIN and M. RAzAR, The theory of Heeke integrals, Nagoya Math. J. 63 (1976), 93-121.

[4] M. I. KNopp, Notes on automorphic forms: An entire automorphic form of positive dimension is zero, J. Res. Nat. Bur. Standards Sect. B 71B (1967), 167-169.

[5] M. I. KNopp, Rational period functions of the modular group, Duke Math. J. 45 (1978),47-62. [6] M. I. KNopp, Rational period functions of the modular group II, Glasgow Math. J. 22 (1981),

185-197. [7]M. I. KNOPP, Some new results on the Eichler cohomology of automorphic forms, Bull. American

Math. Soc. 80 (1974), 607-632. [8] H. Meier and Gerhard Rosenberger, Hecke-Integrale mit rationalen periodischen Funktionen und

Dirichlet-Reihen mit Funktionalgleichung, Res. der Math. 7 (1984), 209-233.

(1) Holger Meier in his doctoral dissertation, "Hecke-Integrale zu den diskontinuierlichen Hecke­Gruppen G(A) mit O<A ,;;2" (Dortmund, 1984), discusses a subclass of period functions of nonin­tegral weight k for the Hecke groups G(A), with A = 2 or A = 2 cos 1T/q, (q E Z, q 2: 3). While this subclass is the natural one for the treatment of the Mellin transforms of automorphic integrals­Dirichlet series with functional equations - it seems likely that a much larger subclass will prove to be the natural analogue for k <Ie Z of the rational period functions which occur in the case k E Z.

152 MARVIN I. KNOPP

[9] D. P. NIEBUR, Automorphic integrals of arbitrary positive dimension and Poincare series, Doctoral Dissertion, University of Wisconsin, Madison, Wise., 1968.

[10] L. A. PARSON and K. ROSEN, Automorphic integrals and rational period functions for the Heeke groups, lllinois 1. Math., 28 (1984), 383-396.

Temple University Philadelphia, Pa.

Eingegangen am 11. November 1984