Order Number 9S07755

Rational period functions for the modular group and related discrete groups

Gethner, Ellen, Ph.D.

The Ohio State University, 1992

UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 R ational P eriod F unctions for the M o dular G roup a n d R elated D iscrete G roups

dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Ellen Gethner, A.B., M.A.

The Ohio State University

1992

Dissertation Committee: Approved by

L. Alayne Parson

Francis W. Carroll ndvifidvisor Robert Gold Department of Mathematics A cknowledgements

I am deeply grateful to my advisor Alayne Parson for her generosity with her time and her patience, and for many enticing mathematical discussions. I am indebted to Frank Carroll and Bob Gold for serving on my dissertation committee, and par­ ticularly to Frank Carroll for his moral and mathematical support throughout my tenure at Ohio State University. I would like to thank Marvin Knopp who made a conjecture, the proof of which evolved into the material in Chapter 2, and who sug­ gested the problem which ultimately became the body of Chapter 3. Special thanks go to Stan Wagon who graciously supplied the Mathematica code which generated the graphics in Figure 4.

This project could not have been completed without the patience, good cheer, and moral support of my family and close friends. I especially thank my family for their unwavering belief in me. I am grateful to Bobby Craighead, Griff Elder, Alan

Foreman, Meg Hartenstein, Steffanie Sabbaj, Lynn Singleton, and Barry Spieler, each of whom helped in their own unique way. Finally, my deepest gratitude goes to Joan

Hutchinson, Alayne Parson, and Dottie Sherling who were there to enjoy the best of times, and who helped me hang on through the worst of times. V it a

July 10, 1959 ...... Born in Chicago, Illinois.

1981 ...... A.B., Smith College, Northampton, Massachusetts.

1983 ...... M.A., Department of Mathematics, University of Washington Seattle, Washington.

1986-1992 ...... Graduate Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio.

P ublications

1. Connected graphs with complementary edge-orbits, with Joan P. Hutchinson, Ars Combinatoria 12(1981) 135-146.

2. Rational period functions with irrational poles are not Hecke eigenfunctions, to appear in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Con­ temporary Mathematics Series, AMS.

F ie l d s o f S t u d y

Major Field: Mathematics

Studies in Modular Forms and Complex Analysis T a b l e o f C o n t e n t s

ACKNOWLEDGEMENTS ...... ii

VITA ...... iii

LIST OF FIGURES ...... vi

CHAPTER PAGE

I Introduction ...... 1

1.1 Some Preliminaries ...... 1 1.2 Modular Functions and Forms ...... 2 1.3 Existence of Entire Modular Form s ...... 10 1.4 T he Hecke O p e ra to rs ...... 10 1.5 Modular Integrals and Rational Period Functions ...... 13 1.6 History of Rational Period Functions ...... 16 1.7 The Effect of T(n) on Some Examples of R P F s ...... 21

II The Main Result ...... 25

2.1 About the Poles of Rational Period Functions ...... 25 2.2 More About the Induced Hecke O perators ...... 40 2.3 The Main Theorem ...... 43

III An Analogous Result ...... 50

3.1 The Hecke Groups G(A„) 50 3.2 Automorphic Integrals and Rational Period Functions ...... 51 3.3 The Poles of ^ ( q x ) ...... 56 3.4 Relationship Among T\(n), and V>a ...... 65 IV The Zeros of Rational Period Functions ...... 78

4.1 An Explicit Description ...... 78 4.2 A C o n je c tu re ...... 84

APPENDIX

A Graphs of Zeros of Rational Period Functions ...... 86

BIBLIOGRAPHY ...... 94

V L is t o f F ig u r e s

FIGURE PAGE

1 The parallelogram 71 determined by and u> 2 ...... 4

2 A covering of C by all translates of % by elements of Gn ...... 5

3 The region T ...... 7

4 A covering of H by ‘translates’ of T by all elements of T (l) ...... 7

5 Potential poles of q\ related to Z{ ...... 62

6 Potential poles of q\ related to zmi ...... 64

7 The zeros of 9 9 ,5...... 86

8 The zeros of 9 1 9 ,5...... 87

9 The zeros of

10 The zeros of

11 The zeros of q9l$ + t(l - jij-) ...... 89

12 The zeros of 921,5 + *0 —......

13 The zeros of 9 9 ,8 ...... 90

14 The zeros of 9 1 7 ,8 ...... 90

15 The zeros of 9 3 ,1 2 ...... 91

16 The zeros of 9 5 ,1 2 ...... 91

vi 17 The zeros of 97,13 • ■ • •

18 The zeros of 919,13 • • • •

19 The zeros of 93,5 + 93,17 •

20 The zeros of 99,5 + 99,8 •

vii CHAPTER I

Introduction

1.1 Some Preliminaries

Felix Klein viewed geometry as the study of those properties of a space which re­ main invariant under the action of a group. Therefore, different geometries may be identified by their respective spaces, groups, and group actions [Lei, p. 11]. Perhaps the most familiar geometry is Euclidean geometry, where the space is the Euclidean plane, and the group is the group of isometries of the Euclidean plane. Abstractly this notion of geometry, or alternatively, of invariance, provides a means by which we may view a mathematical object (a space) from a variety of viewpoints. That is, we have a ‘before picture’ and an ‘after picture’ of a space, where the ‘after picture’ is arrived at by applying a group element to the ‘before picture.’ Though these pictures may look fundamentally different, they are, in reality, the same, and therein lies a basis for comparison.

The topic of this thesis is rational period functions for the modular group and related discrete groups. We will return often to Klein’s simple yet powerful vision of geometry, in the context of vector spaces of meromorphic functions which are invariant under an action of the modular group, in order to motivate and clarify some of the

1 2 fundamental ideas that occur naturally in this subject.

To understand fully the answers to the questions which make up a large part of this thesis, we must first motivate the questions themselves. Therefore, in this chapter, we begin with a brief overview of the theory of modular functions and forms, and then more to the matter at hand, a history of the development of the theory of rational period functions.

In Chapter 2 we address the main purpose of this thesis, which is to answer the following question [Kn3]: ‘Are there rational period functions (defined on the modular group) with irrational poles which are eigenfunctions of the induced Hecke operators

Answering this question will enable us, in Chapter 3, to answer an analogous question about rational period functions defined on the two Hecke groups G(y/2) and

C?(\/3). Lastly, in Chapter 4, we will shift emphasis by beginning an investigation into the location of zeros of rational period functions.

1.2 Modular Functions and Forms

We wish to motivate the definitions of modular functions and forms from first princi­ ples. Therefore, before giving these definitions, we begin by considering the notion of periodicity in the context of the so-called elliptic functions. More precisely, suppose wi,u/2 € C are non-zero such that ^ is not real. If / is meromorphic in C and satisfies f ( z + u>i) = f(z) and f(z + w2) = f(z) for all z € C, then / is an elliptic function. The condition that jj£ ^ R implies that u>i and w2, considered as vectors in the complex plane, are independent. That is, / has two ‘different’ periods, and therefore may legitimately be thought of as a doubly periodic function. An example of such a function is the famous Weierstrass g function, which is defined as follows: with E C as before, let fl = {mwj + nw? : (m,n) € Z x Z}, where fl is called a period lattice. Then

*w = ? + S (u) */(0,0) The group Gq = {Tw(z) = z+u>: u> 6 fl} is generated by Tu,i (z) — z+u>i and T^(z) — z + u>2 . All questions of convergence aside, a straightforward computation shows that g(z) is invariant under the action of Gq, where by action we mean composition of the function g with the function T,L, E Gq. That is, g^^z)) = g(z) for every Tw E Gq and for all z E C. The convergence of g(z) is settled in the usual way by showing that g{z) converges uniformly on compact subsets of C (excluding the obvious places where g(z) has singularities, namely at points in fl). See, for example, [Scl, p. 156] and [Ap, p. 10].

In the language of Felix Klein, we have ‘identified’ the following geometry: the vector space of all elliptic functions with period lattice fl. Such a space is invariant under the group Gq, where the group action is given by composition of functions.

One nice property of the above geometry is that we can understand completely the behavior of any function with period lattice fl, for example g(z), by restricting the domain to the parallelogram, TZ, determined by the two vectors u>i and u)?. That is, 1Z = {z : z = t\u}i + <2w2 : 0 <

TZ, the closure of TZ, by elements of Gq cover the complex plane, with overlap only at the boundaries of translates of TZ. See Figure 2. In fact, by virtue of invariance under the action of G q, any elliptic function with period lattice fl behaves the same way 4

10

-2 -1

- 5

-10

Figure 1: The parallelogram TZ determined by u?i and u;2

on each translate of TZ. In some sense, TZ is a ‘smallest’ such region and therefore is fundamental in determining the overall behavior of q (z ). For further details, see also

[Fo, Section 61]. In order to define modular function as an analogue of elliptic function, we shift to a slightly different point of view. Consider those functions meromorphic in

Ti, the upper half-plane (rather than all of C), whose associated group of invariance is the modular group. Next we define the modular group.

Definition 1.2.1 The modular group, T fl), is defined to be the set of linear fractional transformations M z = where M = ^ ^ ^ with a,b,c,d€ Z and ad — be = 1.

The modular group is generated by S = ( J j J, the translation, and T = 5

10

-2

-1 0

Figure 2: A covering of C by all translates of TZ by elements of G q

- 1 ^ , the inversion. We refer to S and T as the standard generators ofT(l). 0 The use of matrix notation is convenient because matrix multiplication corresponds to composition of linear fractional transformations. Also, since —M and M are identical linear fractional transformations, we have T(l) = SL(2, Z )/ ± I, and consequently,

(ST)3 - T 2 = I (whereas in 51(2, Z), (ST)6 = T4 = I).

We now give

Definition 1.2.2 A function f meromorphic in Ti is said to be a modular function if (a) f(Mz) = f(z) for all M € r(l), and

(b) there exists a positive real number yo such that for all Im(z) > yo, f has an 6 expansion of the form J2T=-m OnC2,Tm*-

The first condition describes the invariance of modular functions under the action of T(l). The second condition describes the behavior of / at the point z = too. That is, if we let q = e2*'*, the expansion in (b) may be rewritten as f(q) = anqn.

Since z e2*'* maps H onto the punctured unit disc {z:0<|z|

It is straightforward to check that the family of modular functions is a vector space over C. Therefore, with the group given by T(l), and the group action given by composition of modular functions with linear fractional transformations in T(l), the space of modular functions is a geometry in the spirit of Felix Klein.

To complete the analogy of modular functions with that of the elliptic functions, we return to the idea of giving a suitable ‘small’ connected subset of Tt on which to study the overall behavior of modular functions. To this end, let F = {z : z + z < 1 and |z| > 1}, which we show in Figure 3. Applying all linear fractional transformations in T(l) to .F, the closure of T, produces a cover of H with overlapping only at the boundaries of copies of T. See Figure 4. Further, by virtue of Definition

1.2.2 (a), a modular function behaves the same way on each copy of T. 7

-i o i

Figure 3: The region T

-l

Figure 4: A covering of Ti, by ‘translates’ of T by all elements of F(l) Remark. In Figure 2, the copies of Tt under translates by elements of Gn are all congruent with respect to the usual Euclidean metric. In Figure 4, with a different metric, the same can be said of J- and all of its copies. That is, with the right metric, the ‘translates’ of T by elements of T(l) are all congruent. This, in fact, is Klein’s upper half-plane model of hyperbolic geometry. See, for example, [Ma, Chapter 1],

[Be, Section 9.4], or [Lei, p. 29].

Perhaps the best known modular function is J(z), Klein’s famous J-invariant, which we will define shortly. See [Ap, p. 15]. At present, we merely offer J(z) as collateral for the existence of modular functions, which, in turn, provides a spring­ board from which we will motivate the definition of a modular form. If / is a modular function, then for every M € T(l), we have f(Mz) = f(z). Formally, we may take the derivative of / as follows: /'(A fz)d^*) = f'(z). Then ( cz + d)~2f'(Mz) = f'(z).

We see that / ' is not quite a modular function because it is not quite invariant under the group action given by composition of functions. In particular, the extra factor

(cz + d)~2 prevents / ' from satisfying condition (a) in Definition 1.2.2. We turn this possible obstruction to our advantage by generalizing the notion of composition of functions in the following definition.

Definition 1.2.3 The slash operator is given by

(F\rM)(z) = (cz + d)~rF(Mz), (1.2)

where M = | *. ] and r 6 R. d) Therefore, if / is a modular function, then we have seen that (/'|2Af)(z) = /'(z) for every M € T(l). This idea of not quite composition of functions gives new energy to Klein’s notion of the geometry of a space of functions. In particular we will use the slash operator to describe a group action for T(l), other than composition of functions, to give the definition of entire modular form. Before doing so, it is useful to note that {(F\TMx)\TM2){z) =

Definition 1.2,4 A function f analytic in H is said to be an entire modular form of weight 2k, for k G Z if

(&) (fUkM)(z) = f(z) for all M G T(l), and

(b) f has a Fourier expansion given by anC2,r*"*, valid for all z G 7i.

If ao=0, then fis said to be a cusp form of weight 2k.

One can show that the existence of nontrivial entire modular forms requires that k > 2. See for example [Ap, p. 116]. Condition (a) in Definition 1.2.4 describes the invariance of modular forms under the action of T(l), and condition (b) requires that / be analytic at ioo. It is not hard to see that the family of entire modular forms of a given weight forms a vector space over C. Finally, we note that quite often the Fourier coefficients of entire modular forms turn out to be number theoretic functions, and thus these coefficients provide a link between complex function theory and multiplicative number theory. We will say more about this beautiful connection presently. 10

1.3 Existence of Entire Modular Forms

As evidence for the existence of entire modular forms of weight 2k, we offer the

Eisenstein series C? 2fc(z), defined by

. ‘ (i-3) m,n.-oo \ (m 1+ nzr* / (m ,n)^(0,0) where, for k > 1, the series Gjki*) converges uniformly on compact subsets of H, and therefore represents an analytic function on H. In fact, for k > 1, C?2fc(z) is an entire modular form of weight 2k, and moreover, every entire modular form is a polynomial in G 4(z) and G&(z). For details, see [Ap, p.12]. When k = 1, G?(z) does not satisfy part (a) of Definition 1.2.4 because (? 2(z) converges only conditionally in H. This

‘failure’ will be one of the motivating forces behind the definition of Modular Integral with associated Rational Period Function in Section 1.5 of this chapter.

Finally, we note that if /i and / 2 are modular forms of the same weight, then

jjj- and £ are both modular functions. In fact, the modular function J(z) given in section 1.2 of this chapter, is the quotient of 60G4(2:)3 and G 4(z)3 — 27Ge(z)2, which, incidentally, are both modular forms of weight 12, and are both polynomials in G4(«) and Ge(-z).

1.4 The Hecke Operators

We mentioned earlier that the Fourier coefficients of entire modular forms often turn out to be number theoretic functions. More precisely, a function g is said to be 11 multiplicative if, for m and n positive integers, g satisfies

Hecke was able to find all entire modular forms with multiplicative Fourier coefficients by creating a family of linear operators {72fc(n)} which are defined as follows.

Definition 1.4.1

r»(B)/ = n“ - £ (i.5) ad = n d> 0 b(mod d)

e / u ( ; JV (i.5) ad = n d > 0 b(mod d)

With the right tools from algebra, it is not hard to see that Twin) maps the space of entire modular forms of weight 2k to itself, and in fact is independent of the choice of representatives of b(mod d). See, for example [Gu, p. 58] or [Ap, pp. 122-124].

Furthermore, Hecke showed that these operators commute, that is, T 2fc(n)T2it(m) =

T2 k(m)T2ie(n), and using this, that the only entire modular forms with multiplicative

Fourier coefficients must be eigenfunctions of all of the operators 72*(n), n > 1.

In addition, Hecke took /, an entire modular form of weight 2k, with Fourier expansion f(z) = ao + ^T^Li one2rtnx and created the Dirichlet series

Because of certain growth conditions on the sequence {<*„},

Next, Hecke showed that ip has an Euler product, which means that v?(s) has an infinite product expansion of the form

n (i -p"-1-1*)-1, (i.7) p prim e if and only if the Fourier coefficients are multiplicative. Moreover, the Euler product converges in the same half-plane with

Theorem 1.4.2 If f is an entire modular form of weight 2k, then the following are equivalent:

(a) f is an eigenfunction ofTik{n) for all n > 1,

(b) the Fourier coefficients of f are multiplicative,

(c) the Dirichlet series for f has an Euler product expansion of the form y>(s) = lip prime (1 ~ <*pP_' “ p2*"1" 2’)"1.

Condition (c) provides evidence that the behavior of a multiplicative function

(namely, the Fourier coefficients of /) is completely determined by its values at prime numbers. For details of Hecke’s work, see [Hel, p. 668] or [He2, p. 37],

Both Gi(z) and Ge{z) are examples of entire modular forms which are eigenfunc­ tions of all of the Hecke operators T!|(n) and Te(n) respectively, and hence by Theorem

1.4.2 have multiplicative Fourier coefficients. In particular,

G,(z) = ^ (l + 240 f )

( l - 504 f > s(r.)e3" " ') . (1-9) where <7;(n) = J3

Dirichlet series can also be created from certain generalizations of modular forms, in particular from modular integrals, which will be defined in the next section.

1.5 Modular Integrals and Rational Period Functions

The theory of modular forms lends itself naturally to a variety of generalizations, and we will focus on one such for the remainder of this chapter.

In section 1.3 we claimed that the Eisenstein series G^z) was not an entire modular form. Specifically, G2(2 ) fails to satisfy the functional equation in Definition 1.2.4 (a).

To understand the nature of this failure, since it suffices to examine and G 2I2T, where S — ^ ^ and T = ^ ^ ^ ^ are the standard generators of T(l). In particular, = Gi(z), but (G ^T^z) =

G2 {z) — See for example [Hu].

In [Knl], Marvin Knopp used the example provided by G^iz) to motivate the following generalization of modular form:

Definition 1.5.1 Suppose f is meromorpkic in ?{ and satisfies

(fhkS)(z) = f(z) (1.10) and

(f\2 kT)(z) = f(z) + q(z), (1.11) 14

where k is an integer and q(z ) is a rational function. If, in addition, f is meromor- phic at ioo, then f is a modular integral of weight 2k with associated rational period function (abbreviated as RPF) q(z).

Under such circumstances, we say that q(z) is an RPF of weight 2k. Note that when q{z) = 0, / is a (not necessarily entire) modular form.

Remark. This definition is also a generalization of an Eichler integral, where q(z), a rational function, is replaced by p{z), a polynomial. Eichler integrals are (2k — l)-fold integrals of cusp forms and entire modular forms of positive weight 2A;. For further details, see [Ei] and [Kn4].

For many reasons, it is worthwhile to note that the Hecke operators { ^ (n )} defined in Section 1.4 also act as operators on the space of modular integrals of weight 2k. In other words, if / is a modular integral of weight 2k, then so is T2k(n)f.

Moreover, in [Knl], Knopp defined the induced Hecke operator, T2k(n), an operator on the space of RPFs of weight 2k, as follows.

Definition 1.5.2 If f is a modular integral of weight 2k with associated RPF q(z) then T2k(n)q = (T2k(n)f)\2kT - T2k(n)f.

A _ The induced Hecke operator, T2k(n), inherits the multiplicative property of T2k(n).

Moreover, it is of particular interest to note that if / is an eigenfunction of T2k(n), A then q is an eigenfunction of T2k(n). To see why, suppose that T2k(n)f = sf for some s ^ 0 in C. Then

T2 k(n)q = (T2Jt(«)/)|2*T — T2k(n)f

= (sf)\2kT - sf 15

= s(fUT-f)

= sq.

In other words, if / is an eigenfunction of Twin), then q is an eigenfunction of 72fc(n),

and in that case, / and q share the same eigenvalue. Equivalently, if q fails to be an eigenfunction of T2fc(n), then / is not an eigenfunction of T2*(n).

Following the lead of Hecke in [He3], Goldstein and Razar in [GR] gave a method

with which to associate a Dirichlet series to the Fourier expansion of functions that are more general than the entire modular forms defined in Section 1.3. In [Knl],

Knopp used the methods of Goldstein and Razar to associate a Dirichlet series to the

Fourier expansions of modular integrals, and these results have recently been applied further in [Kn5] and [HK].

All of these developments naturally led to the question of an analogue to Hecke’s

work as described in Section 1.4. In particular, it becomes important to know which

modular integrals are eigenfunctions of the operators {72fc(n)} and to study proper­

ties of Dirichlet series associated with modular integrals. Alternatively, it becomes

important to know when a modular integral / is not an eigenfunction of !T2jfc(n),

which in turn, is equivalent to knowing when q , the RPF associated with /, is not an

A eigenfunction of the induced Hecke operator r 2ifc(n). Since RPFs are actually rational

functions, the problem of determining when q , and hence /, is not an eigenfunction

is more accessible. In fact, the main result in this thesis, a proof of a conjecture of

Knopp in [Kn3], is the following theorem, which we will restate in Chapter 2.

T heorem . If q is a rational period function of weight 2k with at least one real 16 quadratic irrational pole, then q is not an eigenfunction of f-nfn) for any n > 1.

To better understand the generality and substance of this theorem, we next pro­ vide a history of the development of the theory of rational period functions. For a more detailed treatment of this history, see [Kn3].

1.6 History of Rational Period Functions

In [Knl] and [Kn4], Knopp showed that a rational function q is an R PF of weight 2k if and only if the following two functional equations are satisfied:

q\ikT + q = 0 (1.12) and

gb(sr)2 + 9h(sr) + g = 0. (1.13)

Moreover, in [Knl], Knopp added to the collection of known RPFs (those with a pole at zero, and the Eichler polynomials, which have poles at oo) by giving the first example of RPFs with real quadratic irrational poles, namely for k a positive odd integer,

(z*-z- 1)* + ( z* + z - l )k’ *1,14) A and proved that q is never an eigenfunction of J 2jt(n). Note that the poles of q are

which are in Q(V5).

In [Kn2], using functional equations 1.12 and 1.13, Knopp completely classified

RPFs with poles only at 0 as follows. If q is an RPF of weight 2k > 0 with poles only in Q, then 17

f fco(l — ? r ) if fc > 1 q(z ) = < (1.15) I &o(l — ? ) + ^ if A: = 1 for some bo, bi € C.

If k = 1, bo = 0, and &i ^ 0, then q(z) = ^- is the RPF associated with a suitable multiple of G2, the Eisenstein series of weight 2. In contrast to the RPFs given in equation 1.14, Knopp proved that q is always an eigenfunction of 7a(n) by showing that C?2 is always an eigenfunction of T2(n).

In fact, we can show that any RPF with poles only at zero, given by 1.15, is always

* . A an eigenfunction of T2fc(n) as follows. Since is always an eigenfunction of T2(n), it

1 A suffices to show that 6o(l — is always an eigenfunction of T2*.(n), for k > 1.

To this end, we first show that f(z) = —bo is a modular integral of weight 2k with associated RPF bo{l — jsr). In particular,

( f\ 2 kS)(z) = -b o U S

= — bo, and

( f\ 2 kT)(z) = -bo\2kT

= - z ~ 2kb0

— —bo + fco(l 5r)

-/(*) + 18

Next, we show that f(z) = —60 is always an eigenfunction of Tjfc(n). Specifically,

W f = »"-■ E / l - f 0 J) ad„ j ~ n \ / d> 0 b(mod d)

= n ad = n d > 0 b(mod d)

= n 2* - 1 £ d~2k(-bo) ad = n d> 0 b(mod d) = Cnf(z), where

C„=n2k-1 £

Thus, f(z) = —&o is always an eigenfunction of J ^ n ), and hence 6o(l — jsr) is A always an eigenfunction of T ^n).

What ultimately turned out to be of greater consequence and interest in [Kn2] was that Knopp proved that the finite non-zero poles of RPFs are necessarily real quadratic irrational numbers. To help understand why this is so, we give a definition.

Definition 1.6.1 A linear fractional transformation M = ^ ^ ^ T(l) is hy­ perbolic if |TVace(Af)| = |a + £j > 2.

In fact what Knopp proved in [Kn2] was that the finite non-zero poles of RPFs are necessarily fixed points of hyperbolic elements of T(l). Therefore, if zq is such a 19

pole and is fixed by Af = ^ ^ ^ ^ ^ ^ > ^en ^*° = Z° ’m^ ’es that

^ = * (1.17) 720 + o

If 7 = 0, then since det(M) — 1, we have a — 8 — ± 1, which means that M is not hyperbolic, a contradiction. Therefore, we may assume without loss of generality that

7 ^ 0 . In that case, equation 1.17 yields 7 z$ + (6 — a)z0 — 0 = 0, and thus

<* - 6 ± j( 6 - a)* + 40 7 z° = • (L18)

Therefore, ______a — 6 ± \/{ot + £ )2 — 4 2„ ------^ ------’------(1.19) because a8 — 0 7 = 1. Further, |7Vace(Af)| = |a + £| > 2 implies that (o + 8)2 — 4 >

0. Moreover, we claim that (a + 6 )2 — 4 cannot be a square in Z. If so, then

(a + 6 )2 — 4 = r 2 for some integer r. In fact, since |a + > 2 , we must have r > 0 .

Then (a + 5)2 — r2 = 4. But this is impossible, since the difference between two distinct non-zero squares in Z cannot equal 4. Consequently, if N (a + tf)2 — 4, we have zq E Q('/W)\Q. Hence, if zo is a finite non-zero pole of an RPF, then Zq is a real quadratic irrational number. Therefore, since Eichler integrals have RPFs which are polynomials, and the RPFs given by 1.15 have poles at zero, we see that RPFs may only have poles at 0 , 0 0 , and at real quadratic irrational numbers.

This elegant revelation about poles provoked a variety of papers on the subject of

RPFs. For our purposes, it is those papers describing RPFs with quadratic irrational poles which are of primary interest. 20

In [PR], A. Parson and K. Rosen found examples of RPFs with poles in Q(\/3) and Q(\/21), and showed that these examples were never eigenfunctions of 7afc(n).

In [Ha], J. Hawkins began a study of quadratic irrational poles of RPFs by intro­ ducing the notion of irreducible systems of poles which are minimal sets of quadratic irrational numbers, that are arrived at because of the nature of the functional equa­ tions 1.12 and 1.13. For example, if zq ^ 0 is a finite pole of an RPF qy then by functional equation 1.12, Tz0 = ^ must also be a pole of q. Hawkins also showed that for k < 0, an RPF q of weight 2k must be a polynomial, and for k = 0, q must be identically 0. In other words, RPFs with finite non-zero poles must be of strictly positive weight.

In [Ch], Y-J Choie developed two methods by which to construct RPFs of positive weight with poles in Q (y/N) for any positive non-square integer N.

In [CPI] and [CP2] Choie and Parson began a more detailed classification of RPFs of positive weight, and to do so, drew extensively from the theory of indefinite binary quadratic forms. In particular, it follows from [CPI] that

k t W (1 -20) b2 —4 ac=D>0 is always an RPF of weight 2k when k > 0 is an odd integer.

In [As], A. Ash also explained the existence of RPFs from the point of view of cohomology of groups.

In [Pal], Parson gave a formula for certain modular integrals / corresponding to RPFs as classified in [CPI]. She then used the formula to describe the Fourier coefficients of these modular integrals, and in addition gave an explicit formula for 21

T2k(p)f for p prime, which in turn gave a formula for Ttk{p)q, where q is the RPF associated with /.

Finally, in [CZ] and in [Pa2], Y-J Choie and D. Zagier, and A. Parson have inde­ pendently completed the classification of RPFs by providing an explicit construction for all RPFs of a given positive weight.

Even though RPFs with quadratic irrational poles have been classified, it is dif­ ficult, in general, to find a formula for T 2*(n)g. However, it is possible to show that those RPFs with at least one quadratic irrational pole are not eigenfunctions. This is the topic of Chapter 2.

1.7 The Effect of T(n) on Some Examples of RPFs

The purpose of this section is two-fold. First, we will use some of the examples to demonstrate the effect of the induced Hecke operators T(p) for p prime. Then, by examining the effect of the induced Hecke operators on these examples, we will gain some insight into the foundation of the proof of Theorem 1.5. To this end, we draw from the families of RPFs given by 1.20, and for contrast, from 1.15 in Section 1.6.

To begin, we introduce some convenient notation. Let

r —v 1

qk D ~ ahi c (aZ2 + bz + C)k 6s — 4ac=D >0

By [CPI], we know that when a,b,c 6 Z, and k is a positive odd integer, then q k,D is an R PF of weight 2k with quadratic irrational poles. By means of a Basic A program, the following is a sample of the effect of T(p) (see Definition 1.5.2), where p is prime, on some explicit RPFs given by qtt,D’ From now on, we will assume that k is a positive odd integer.

Example 1.7.1 The family of RPFs given by qk,& = + (;a+].i)fc is the original example of RPFs with quadratic irrational poles found by Knopp in [Knl].

We have

1 , 1 , 1 T (2^ = (*2 + 2z - 4)* + (z2 - 2z - 4)fc + {4z2 + 2z - 1)*

1 1 1 + (4z2 - 2 z - l) fc + (5z2 - 1)* + (z2 -4 z - l)k 1 + (*2 - 5 ) fc‘

= T. ------(a z 2 + bz + c)k b3 —4ac=20 = ?fc,20*

The poles of qk ,5 are { — Therefore, since none of the poles of T(2)qk,s are poles of qk, 5, we have that qk ,5 cannot be an eigenfunction of T(2). In fact, in general, we need only find one pole of T(n)q that is not a pole of q in order to prove that q is not an eigenfunction of T(n).

Example 1.7.2 23

= £ 1------(az* + bz + c)k b?-4ar=45 ~ 9k,45-

We see that ~3+3Vs }s a pCle (arising from the first term) of T(3)qk,s which is not a

A pole of qk, 5, and therefore qk ,5 is not an eigenfunction of T(3).

For the remaining two examples, D and p are large enough so that the functions

A qk,D and T{p)qk,D are cumbersome to write in their entirety, and so we simply give the result of the effect of T[p) on qk,D-

Example 1.7.3 T(7)qkfi = ^.392 + 2(7)~kqk,8

In the above example, although all of the poles of qk# are poles of T(7)qk,8, it is still the case that qk ,s is not an eigenfunction of T(7) because >/98 is a pole of T(7)qk,s that is not a pole of g*, 8 . In particular, >/98 is a pole of fzi J^ k which is one of the terms of qk,3 92-

In our final example, we show the effect of an induced Hecke operator on an RPF with poles at zero as well as at real quadratic irrational numbers.

Example 1.7.4

f(5)[,t.21 + to (l - p j)]

= T (i)q k,„ + f(5)[4, (l - ij)]

= 9fc,525 + 2 (5 )- *q*,Jl + Cffto ( l — ^2fc)]' 24 where C is computed by using equation 1.16 in Section 1.6 . In particular,

C = 52*"1 £ d- i k a da 5 b(mod d)

= 52fc“1[l + E 5 - 2*] i = 0 = b2k-'[l + 5(5)-2k]

= 52fc-1 + 1.

We note that gjt .21 + M l — J3r) >s n°t an eigenfunction of T(5) because kiVSS? js a pole of T(5)[qk, 2 1 + 6o(l — ^55-)] which is not a pole of qk# 1 + 6b(l — In particular,

1+/525 js a pQje q£ ^ which is one of the terms of 9^,525. CHAPTER II

The Main Result

2.1 About the Poles of Rational Period Functions

In this chapter we will prove the main result of this thesis, which will also appear in

A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemporary

Mathematics Series, AMS., and is stated as follows.

Theorem 2.3.1 If q is a rational period function with at least one quadratic irrational pole, then q is not an eigenfunction of T 2 k(n) for any n > 1 .

Before proving Theorem 2.3.1, it is useful here to note that RPFs can be de­ fined on the family of Hecke groups which are defined as follows: for n > 3 in Z,

G(A„) = < SXn,T >, where S\„ = ^ J 2c°*n j and T - ^ J q1 ) are the stan­ dard generators, and satisfy ( Sx„T)n = T 2 = I. Note that when n = 3, A„ = 1, and in that case C?(A3) = G(l) which we will use as an alternative notation for T(l), the modular group. Note also, in analogy to the functional equations given by 1.12 and

1.13, that a rational function q is an RPF of weight 2k on G(A„) if and only if the following two functional equations are satisfied [Kn4]:

q\ikT + q = 0 (2 .1)

25 26 and

E « l » ( S».r )i + « = °- 2 . (2 ) 1 = 1 In Chapter 3 we will derive an analogue to Theorem 2.3.1 for RPFs defined on

G(A4) = G(y/2 ) and G(Xe) = G(y/3). Much of the machinery which we will develop for RPFs on T(l), particularly concerning poles, has direct analogues for RPFs on

G{y/2) and G(y/Z) which will help facilitate the proofs of the results given in Chapter

3. Therefore, we spend the remainder of this section deriving results about finite non­ zero poles of RPFs defined on G(An) for n = 3,4, and 6 . From now on, we will write

A in place of An, and will assume that A = 1, y/2, or y/3 unless otherwise specified.

We mentioned in the Section 1.6 , that in [Kn2], Knopp proved that the finite non-zero poles of RPFs on T(l) are necessarily fixed points of hyperbolic elements of r(l). We begin by giving an analogue for RPFs defined on G(y/2) and G{y/3).

First, we give a definition, which, incidentally, is a broader form of Definition

1.6.1.

Definition 2.1.1 Let A = 1, y/2, or y/3, and suppose zq is fixed AT = ^ ^ ^ ^

G(A). Then

(a) M is hyperbolic if \Trace(M)\ = |a + £| > 2 , and

(b) M is parabolic if \Trace(M)\ = |a + 6 | = 2.

Remark 2.1. If M = ^ ° G *S hyperb°lici ^en ^>7 / Otherwise, if, say, 0 = 0, then a8 — @7 = 1 implies that a = 8 = ± 1, in which case \Trace(M)\ = 2, a contradiction. The same argument holds if 7 = 0. 27

Theorem 2.1.2 If z0 is a finite non-zero pole of an RPF q\ defined on (7(A), for

A = 1, y/2 or y/Z, then zo is fixed by a hyperbolic element of (7(A).

The proof that we will give of Theorem 2 .1.2 for A = y/2 and y/Z is almost verbatim that of the proof that Knopp gave in [Kn2] for the case A = 1. Also, in [Sc]

T. Schmidt has an alternative proof of Theorem 2.1.2 using A-continued fractions.

Remark 2.2. It is convenient to note that for A = y/2 or y/Z, the elements of

G(A) fall into two categories, the eren elements ^ ^ ^ ^ and the odd elements

( °c dX ) ’ w^ere £ 2 and ad — bcX2 = 1 and adX2 — be = 1, respectively.

In fact, this description also holds for the elements of T(l) simply by observing that the two categories of elements actually coincide since A = 1. For details, see [Hul] and [Yo].

Before proving Theorem 2.1.2, we give two lemmas.

Lemma 2.1.3 For A = 1, y/2, or y/3, if zq is fixed by a hyperbolic or parabolic element of (7(A), then for any M 6 (?(A), Mzq is fixed by a hyperbolic or parabolic element of

(7(A) respectively.

Proof: Suppose z0 is fixed by N G (7(A). If M € (7(A), then Mzo is fixed by M N M ~ l because M N M ~ 1(M z0) = MNzo = Mzo- Since Trace(MNM~i) — Trace(N) (see

[Le2, p. 5]), if N is hyperbolic or parabolic, then M N M ~l is hyperbolic or parabolic respectively. □

Lemma 2.1.4 Ifzo / oo is fixed by a nontrivial parabolic element ofG(X) forX = y/2 or y/Z, then zq € AQ, where by AQ, we mean ‘rational multiples of X.’ 28

Proof: By Remark 2.2, if z0 is fixed by /V = ^ ° ^ ^ en 8*nce la + ^1 =

2 G Z, N must be an even element. Therefore, /? = A/3' and 7 = A7 f, where /?', 7 ' € Z.

If 7 ' = 0, then a = 6 = dbl, in which case, since zo is finite, we have /?' = 0 and so

N = I, a contradiction. Therefore, Nz0 = z0 implies that

az0 + A/9' _ A7'zo + 6 Z°’ and so A7 '(z0)2 + (^ — <*) zq — A/3' = 0. Therefore,

a — 6 ± yj(a — 6)2 + 4A2/3'7' 2A7 '

a — 6 ± ^ ( a + 6 )2 — 4 = 2 Ay ’ because a£ — \ 20'Y = 1. Since |a + = 2, we have

a — 6 a — 6 x 20 = 2A7' = 2A2Y ’ which is in AQ, as desired.

Proof of Theorem 2.1.2

The proofs for A — y/2 and A = y/Z are essentially the same as the proof that

Knopp gives in [Kn2] for A = 1, and we demonstrate this by presenting the case

A = y/2.

Recall by functional equations 2.1 and 2.2 that if q\ is an RPF on G(y/2) of weight

2k then

(2.3) 29 and

G aM -S^T )3 + qxlikiSjtT)2 + 9Abfc(^r) = -qx. (2.4)

For the remainder of the proof, we write S in place of and | in place of |2*.

Therefore,

9a|T2 = qx\(ST fT + qx\(ST)2T + qx\{ST)T, (2.5) and hence

qx = qx\{ST)2S + qx\(ST)S + qx\S. (2 .6 )

We proceed with the proof by rewriting equation 2.6, and in order to do so, we introduce some convenient notation. In particular, let M\ = S = ^ ^ ^ =

( * )■ M > ~ {S T 'S - ( f J s ) = (* *)’“d ^ = ( s r y s =

( \ / 2 J ) = ( o' ^ )’ ^°tC entr'es * = ^ are aH non­ negative. We may now rewrite equation 2.6 as follows:

qx(z) = (c3z + d3)~2kqx(M3z ) + (c2z + d2)~2kq\{M2z) + (cxz + di)~2kqx{Miz) (2.7)

The remainder of the proof is given in two cases:

Case 1: z0 is a finite non-zero pole of qx such that zq $ \/2Q

Then zo must be a pole of ( CrZ+dr)~2kqx(Mrz ), one of the terms on the right-hand side of equation 2.7, where r 6 {1,2,3}. Since z0 \/2Q , we have CrZo + dr ^ 0, which means that the factor (CrZ + dr)~2k cannot account for a pole even when k < 0 .

Therefore, zo must be a pole of qx(Mrz), or alternatively, Mtzq is a pole of q\{z). By writing Mr as a word in S and T, the standard generators of G(y/2), it is easy to see that Mrz0 is non-zero and finite, and Mtzq \/2Q because zq is non-zero and finite, and zo $ y/2Q. Therefore, we may invoke the above procedure again with zq replaced by Mrzo. By doing so repeatedly, say s times, we generate a (partial) list of poles of q\: {20, A/r, So* Afrj A/r, Zo, ..., (A/rj...Mri)z0}) where r, E {1,2,3}, and the r/s are not necessarily distinct. As a rational function, qx has only finitely many poles so that after invoking the procedure, say t times, an element in the list must be duplicated. That is, for some 0 < j < i, we have Mr,...Mrizo = MTj...Mrizo which means that Zj := MT)...MTizo is a fixed point of N := Afr,...A/r>+1. Also, note that since \Trace(Mi)\ > 2 for i = 1,2,3, and since the entries of Mi are all non-negative, we have \Trace(A )| > 2. In other words, Zj is fixed by a parabolic or hyperbolic element of G(y/2 ). Since, z0 — (Afr>...Mr,)- 1Zj, Lemma 2.1.3 applies, and thus, zq is fixed by a parabolic or hyperbolic element of G(x/2). But if z0 is fixed by a parabolic element, then by Lemma 2.1.4, we have zq € \/2Q, which is a contradiction. Thus, z0 is fixed by a hyperbolic element of G(y/2).

C ase 2 : zo is a finite non-zero pole of qx such that zo G v ^ Q

First, by equation 2.1 both z0 and Tzo = ~ are poles of qx and hence by Lemma

2.1.3, we may assume without loss of generality that z0 > 0. Therefore, the arguments given in Case 1 hold for zo, except in this case, CrZ0 + dT ^ 0 because z0 > 0. Thus, as before, Zj is fixed by TV = ^ ^ ^ ^ which is either an even or an odd element of G(y/2), such that |7Yace(TV)| > 2. Recall that Zj = MTj...MTlzo. Since zq > 0 is finite, and z0 E y/2Q, the same can be said of Zj. Because Zj is finite and is fixed by because a8 — 0 7 = 1. By Remark 2.2 if N is an odd element, then

a '\/2 — S'y/2 ± \/2(q/ + 81)2 — 4 *< ------2 7 ------’ (2 9) or else if N is an even element, then

a — 8 ± \I(a + 8 )2 — 4 t f r ' (210) where a, a', 7 , 7 ', 8,8' £ Z.

If JV is an odd element, then Zj is given by equation 2.9, and so

Since zj 6 \/2Q, we must have that (a' + 6 ')2 — 2 is a square in Z, which is impossible, because the difference between a square and 2 cannot equal a square. On the other hand, if N is an even element, then by equation 2.10, we have

Since Zj £ \/2Q> we must have (a + 8)2 — 4 is a square in Z. Since the difference between two distinct non-zero squares can never equal 4, this means (a + 8)2 = 4, so that (a + £ )2 — 4 = 0. Since a , £ > 0, we must have a = 8 = 1 and in this case implies that Zj = 0, which is a contradiction. Thus, zq y/2Q. In total, then, if zq is a non-zero finite pole of qx, then zq must be fixed by a hyperbolic element of (7(A). □

In some sense, Theorem 2 .1.2 describes what the finite non-zero poles of rational period functions are, namely, the fixed points of hyperbolic elements of (7(A). The following corollary of Theorem 2.1.2 sheds more light on the nature of these poles by describing where they are. But first, Theorem 2.1.2 inspires the following definition.

Definition 2.1.5 If Zo is a finite non-zero pole of an RPF qx defined on (7(A) for

A = 1, y/2, or y/3, then zo is said to be a hyperbolic pole of qx.

C orollary 2 .1 .6 If zq is a hyperbolic pole of an RPF qx on (7(A) for A = l,\/2, or y/3, then

(a) Zq is a root of a quadratic polynomial of the form P(z) — Xaz2 + bz + Xc, where a, b, c G Z such that a, c / 0, gcd(a , b, c) = 1, and b2 — 4A2ac > 0. Consequently,

(b) Zq G Q(A, y/N)\XQ, for some positive integer N, where if N is a square or if

N — X2(N')2 for some positive integer N ', then z q G Q(A)\AQ.

Remark 2.3.

(i) When A — 1, Corollary 2.1.6(b) simply restates Knopp’s theorem regarding hy­ perbolic poles of RPFs defined on T( 1). That is, such poles must belong to Q{y/N)\Q, where, in this case, N is a positive non-square integer, and therefore poles are real quadratic irrational numbers.

(ii) It also follows from [MR, Theorem 2] that the finite poles of RPFs defined on

G(y/2 ) and G(-\/3) must be in Q(A, y/N )\AQ, for some positive integer N. 33

(iii) The condition that finite non-zero poles of RPFs are fixed points of hyperbolic elements of G(A) is necessary, but not sufficient.

Proof of Corollary 2.1.6

(a) Suppose zq is a hyperbolic pole of q\, an RPF on (7(A). By Theorem 2.1.2, zo is fixed by Af = ^ ^ ^ ^ > Remark 2.1, we have /? ,7 ^ 0, so that the standard computation following from Mzo = z0 yields 7 ZQ + (S—a)zo—0 = 0.

If M is an odd element, then by Remark 2.2, we have 7 'Xzg + (6 — q )zq — 0* A = 0, where a,0',j\6, €. Z. Then let

a = gcd( y, 6 - a, 0>)

S — a b - gcd(y, 6 - a,0’) and

c = 0' gcd{ Y, 6 - a , 0’) ’

Therefore, z0 is a root of P(z) — aXz2 + bz + cA where, gcd(a, b, c) — 1. Moreover,

= (gcd(\',6-a,0')Y<'i° + 6)1 ~ 4> > 0 , because \Trace(M)\ = |a + > 2 .

On the other hand, if M is an even element, by Remark 2.2, we have 7 z\ + A(6' — a')z0 — 0 = 0, where a',0, 7 , € Z. Multiplying through by A yields 7 A2Q + A2(£' — a')zo — 0X = 0. Then we simply proceed as in the case when M is an odd element. 34

(b) Let N = b2 — 4A2ac. Then from (a), since z0 is a root of P(z) = aXz2 + bz + cA with c ^ 0 , we have

- T , (2 , 1) which means that zQ 6 Q(A, JV). Further, by Case 2 in the proof of Theorem 2.1.2, we have zq ^ AQ.

In total, if Zq is a hyperbolic pole of qx, then z0 € Q(A,y /N )\AQ, for some positive integer N. D

Corollary 2.1.6 motivates the following definition.

Definition 2.1.7 Suppose 0 ^ zo 6 Q(A, \/]V)\AQ for some positive integer N. If

Zo is the root of a quadratic polynomial of the form P(z) = aXz2 + bz + cA where a,b,c £ Z , ac 0 and gcd(a,b,c) = 1, then P{z) is said to be an associated quadratic for 00-

R em ark s.

(i) Since zq ^ 0 and Zq ^ AQ, a priori, an associated quadratic P(z) = aXz2 + bz + cA must satisfy ac ^ 0. Moreover, since Zo is real (and z0 AQ) we must have b2 —

4A2ac > 0.

(ii) When A = 1, an associated quadratic for zq is (up to multiplication by —1) the minimal polynomial for z q , and hence is uniquely determined.

(iii) When A = y/2or \/3, associated quadratics are not necessarily minimal polyno­ mials. For example, if zq = * is a rational number in lowest terms, then q2Xz2 — p2X is an associated quadratic for z q , but is not the minimal polynomial for z q . More 35 importantly, though, associated quadratics are unique up to multiplication by — 1, as will be verified in the following proposition.

Proposition 2.1.8Suppose 0 ^ zq E Q(A, \/jV)\AQ for some positive integer N.

Then an associated quadratic for z0 (if it exists) is unique up to multiplication by —1.

Proof: Since we wish to prove that an associated quadratic for Zo is unique up to multiplication by — 1, it suffices to show that if an associated quadratic for Zo exists, then there is exactly one associated quadratic for Zq with a positive lead coefficient.

To this end, by way of contradiction, suppose P\(z) = ai Az2 + 6jz + cjA and

P2(z) — a2Xz2 + fcjz + C2A are distinct associated quadratics for Zq with aita2 > 0.

Without loss of generality, assume that a\ is minimal with respect to all associated quadratics for Zo (with positive lead coefficient).

By the Euclidean Algorithm, there are positive integers r and s such that a2 = a\3 + r with 0 < r < aj. Let P(z) = Pi{z) ~ sPi(z). Then

P{z) = (a2 — sai)Az2 + (6j — sfci)z + (c2 — sci)A

= rAz2 + (&2 — sb\)z + (c2 — sci)A.

Since z 0 is a root of P\{z) and Pz(z), zq is a root of P(z). If r > 0, then because zo i aq, the constant term of P(z) cannot equal 0. Therefore, by the method used in Corollary 2.1.6(a), P(z) can be normalized to be an associated quadratic for zo with positive lead coefficient strictly less than ai, which contradicts the minimality of aj. Thus, r = 0 and since Zo ^ AQ and is a root of P(z) = P2(z) — sP1(z), we must have P2 (z) — sPi(z) = 0 , or alternatively, that Pi(z) = sPi(z). In that case we have 36 a 2 = soi, &2 = sbi, and c 2 = acj. Since 0 cd(ai,6i,ci) = #cd(a2, 63, 03) = 1, we have a = l, and hence P\(z) = P2(2 ), which is a contradiction because P\{z) and P 2(z) are distinct.

Thus, if an associated quadratic for z0 exists, it must be unique up to multiplica­ tion by — 1 . □

By Corollary 2.1.6, all hyperbolic poles of RPFs defined on G(A) have associated quadratics. To prove the main results in this chapter and the next, quite often we will study real numbers that are potential poles of RPFs because they are roots of polynomials of the form Q(z) = aXz2 + bz + cA. However, these quadratic poly­ nomials will not necessarily be associated quadratics. That is, it may be the case that gcd(a,b, c) ^ 1. These quadratic polynomials provide a means by which we can associate a potential pole of an RPF with a triple of integers, namely [a,b,c]. For convenience, then, we write Q(z) = [a, 6,c] in place of Q(z) = a \z2 + bz + c\ wherever appropriate. Also, we say that a is the lead coefficient of Q(z) (as opposed to aA), 6 is the second coefficient of Q(z), and c is the constant term (as opposed to cA).

Definition 2.1.9 Suppose z0 is in Q(A, y/N )\AQ, for some positive integer N, and has associated quadratic given by P(z) = [a, 6, c]. Define disc(zo) to be the discrimi­ nant of the polynomial P(z). That is, disc(zo ) — b2 — 4A2ac.

Note that disc(P(z)) = disc(—P(z)), so that by Proposition 2.1.8, disc(zo) is well defined. In order to make a distinction between the definition of disc(zo) and the dis­ criminant of any quadratic polynomial of which zq is a root, we give the following definition.

Definition 2.1.10 Suppose P(z) = rz2 + sz + t is in R[z], Then Dp (,) = s2 — 4rt.

In other words, Dp(x) is the discriminant of the quadratic polynomial P(z).

If in fact P(z) is an associated quadratic for zo, then disc(zo) = Dp(x).

Lemma 2.1.11 Suppose z0 is a root of the polynomial P(z) = rz2+sz+t in R[irj, and let M = ^ ^ ^ ^ be a linear fractional transformation such that det(M) = d / 0 .

Then

(a) M zq is a root of the (at most quadratic) polynomial Q{z) — (P|_ 2M')(z), where M ' = ( ^ ^ ]. and V - 7 « ) (b) Dq(z) = (PDptf.

Proof: First note that

(P\-2M')(Mz0) = {-j{Mz0) + a)2P(M'(Mzo))

. M M * ) + a? P * ! * ) * )

= (-7(Mz„) + a)2i'( ^ )

= (—'f(Mzo) + a)2P(z0)

= 0, because z0 is a root of P{z). 38

Moreover, Q{z) is a polynomial of at most degree 2 because

o(2)= (-^ +Q)> (-(^ y +«(:^y+•)

= r(6z - ft)2 + s{Sz - /3)(- 7 z + <*) + <(— 7 * + a)2

= (rS2 — s'yS + (7 2 )z2 + ( s ( a 6 + ^ 7 ) — 2 (r/36 + ta~j))z + (r/?2 — safi + t o 2).

Finally, the second statement follows from the first since

Dq(z) = (s(a6 + £7 ) - 2[rfiS + ta-y))2 - 4(r/?2 - s/?a + fa2)(r£2 - S7 ^ + <7 2)

= d2(s2 — 4rf)

= d2 Dp(x) , as desired. □

Corollary 2.1.12 Suppose zq is a hyperbolic pole of an RPF qx on (7(A) for A =

1,\Z2, or v fr If an associated quadratic for zo is P(z) = [r,s,f], then for any M 6

(7(A),

(a) an associated quadratic for M zq exists, and is given by Q{z) = (P |_ 2A/- 1)(z), and consequently,

(b) disc(zo) = disc(Mzo).

Proof: (a) In the special case that M € (7(A), we have AT = M -1 (M ' as defined in Lemma 2.1.11). Furthermore, because the entries of M are in Z[A], it is clear that since zq G Q(A, y/N) then so is M zq. Also, since S\zo = zo + A and T zq = we 39 have M zq £ AQ, because Zo £ AQ, and because S\ and T generate G(A). Therefore,

M z0 satisfies the hypotheses of Definition 2.1.7, and hence may have an associated quadratic.

In fact by writing M -1 as a word in S\ and T, it is easy to show that Q{z) =

(P |_ 2A/_ 1)(,2) is a polynomial of the form aA.z 2+ 6z+cA, where a, 6,c € Z,gcd(a,b,c) =

1, and since M zq $ AQ, we have a c ^ 0. Specifically, (P |_ 25a)(2) = [r, 2rA2 + s,rA 2 + s+<] and ( P\-iT)(z) = [f, —s, r], and therefore since the polynomial P{z) when viewed as a triple of integers is relatively prime, then the same can be said of (P^SaH *) and (P |_ 2!r)(2 ), and hence of Q{z) = {P|_ 2Af_1)(2 ). Thus, Q(z) is an associated quadratic for Mz0.

(b) By Lemma 2.1.11 (b), since det(M) = 1, we have Dp (t) = an

For G(A) = T(l), with the same hypotheses as in Corollary 2.1.12, statement (b) follows directly from the fact that we may view P(z) and Q(z) as binary quadratic forms. That is, if we let Qi(:r,y) = ax2 + bxy + cy2 and Qi(x,y) be the binary quadratic form with the same coefficients as (P|_ 2M- 1)(z) (so that P(z) = Q\{z, 1) and (P |_ 2M- 1)(z) = Q2(2 , 1)), then since Qi{x,y) and Qi[x,y) are equivalent in the narrow sense, we have that the discriminant of P(z) is the same as the discriminant of (P |_ 2Af- 1)(z). In other words, disc(Mzo) = disc(zo). (For more information on binary quadratic forms, see [Bu] and [Za].)

Lemma 2.1.13 Let q be a rational period function on G( A) for A = l,\/2 or \/3- 40

If Zo is a hyperbolic pole of q, then given a fixed prime p, there is a hyperbolic pole

zi of q satisfying disc(zi) = disc(zo) and with associated quadratic [r, s, <] such that gcd(r,p) = 1 .In other words, q has a hyperbolic pole with associated quadratic whose

lead coefficient is relatively prime to p.

Proof: Because the proofs for A = 1, y/2, and y/3 are analogous, we present only the

case A = 1. To this end, suppose an associated quadratic for z0 is [a,b,c]. Without loss

of generality, assume gcd(a,p) = p. By functional equation 1.12, Tzo = ^ is a pole

of q(z) and by Corollary 2.1.12 (a) has associated quadratic [c,-b,a]. If gcd(c,p) = 1,

we are done. Otherwise, gcd(c,p) = p. Now, by functional equation 1.13 either

(ST)-1zo = ( ST)2zq or ( ST)~2z0 = STz0 is a pole of q(z) (or possibly both). An

associated quadratic for ( ST)2zq is [a + 6 +c, —2a — 6,a], and an associated quadratic

for STzo is [c, —2c — 6, a + b + c]. In the first case, since gcd(a , p) = gcd(c, p) = p and

gcd(a,b,c) = 1, we must have gcd(a + b + c,p) = 1. In the second case, consider

which has associated quadratic [a -f b -|- c,2c + b, c]. As before, gcd(a + b + c,p) = 1.

By Corollary 2 .1.12, in either case, q(z) has a hyperbolic pole z\ such that disc{z\) =

disc(zo), and whose associated quadratic has lead coefficient relatively prime to p. □

2.2 More About the Induced Hecke Operators

For the remainder of this chapter, we restrict our attention to RPFs defined on T(l),

the modular group, which is the setting for the main result of this chapter.

We begin by taking a more detailed look at the induced Hecke operator 7!tfc(n)

defined on the space of RPFs on T(l). To this end, in Section 1.4 we saw that for / an entire modular form of weight 2k, and n a natural number the Hecke operator

T-ik(n) was defined as follows by

r«M/ = £ /l*(S j ad = n d> 0 b(mod d)

A A and in Section 1.5, that the induced Hecke operator Tn(n) was given by Ta*(n)g =

(T2 k(n)f)\2 kT — Twin)/, where q is the RPF associated with /, a modular integral of weight 2k. From now on, for convenience, we choose the set of representatives of

A b(mod d) given by {0,1,2, ...,d— 1}, and write T(n) and T(n) in place of Tjk(n) and

A Tjjk(n) respectively.

By following [Ap, pp. 120-124], we can write the induced Hecke operator as follows.

First, define T„ = {Mb,d = ^ g ^ ^ |a, b,d e Z,a, b,d > 0,0 < b < d,ad = n}.

Then for each Mb4 in T„, there is a unique Vb,d in T(l) and a unique Mb'4 ' in r n such that Mb,dT = Vb,dMb>,d>- For example, if Mb4 = Mo,„ — ( n ^ V then since

( 1 M r - r f " 0 j , we have Mb'4 ' = A/0,1 and Vb4 = T. ^ 0 n ^ 0 1 A Next, examine one term of the sum representing T(n)q(z) corresponding to a fixed Mb4 . Let Vb,d and Afy,^ he the unique elements of T(l) and r„, respectively, satisfying A/t^T = Vb4 Mb>4 >• Since Vb,d is in T(l), and T(l) is generated by S and

T, we can write = SaiT S0 2 T...TSar, where a,- € Z. Then for the RPF q(z), the induced Hecke operator can be written as follows: 42

= "” - ‘ Z fU M ^ T - T(n)f

= n2k~l £ f \ 2 kVb4MV4,- T { n ) f.

Since /|a*5 = /, we have

fllkVbjMvji

= f\2k{SaiT S a2T...TSarMb,td,)

= {f\2kSa')\2k{TSa'T...TSarMb'4l)

= f\2k(TS0,’T...TSa'M b'4,)

= ( f + q)\ 2 k(Sa'T...TS°'M b,,d')

= ((f\ 2 kSa'+ q \ 2 kSa*)\2 kTS°>T.„TSa')U M b.id.

= ( ( / + q\ 2 kSa3)\2 kTSa>T...TSa')\2kMb.,dl

= ((fhkT + q\2kSa3T)\2kS 0l3T...TSa')\2kMb,,d'

= (((/ + q) + q\2kSa2T)\2kSa3T...TSaT)\2kMb',d'

— ( f + qbtd)\2kMb>4t, where qb4 is the sum of terms of the form gla/tAf, M 6 T(l). In total, then,

f ( n), = n«-> £ (/+ ?6,j)UA/v,j. - T(n)/ Mt, d,€r„ 43

= r(n)/+n”-' £ - T(n)f

=_ n —2fc_1 . qbjUkMvji.

Rem ark 2.4. For the proof of the main result, it is important to note that when

Zq is a pole of q(z ), then nzo is a potential pole of jT(n)qf(z). To see this, recall that when Mb,d = M0,i “ ^ q j ^ we must have Vb,d = K>,i = T and Mv,d> —

1 0 \ ( q n J . In particular, the term qb,d\ikMb’,d‘ of T(n)q corresponding to

Afo.i = ^ q j ^ is n -2fc? (” ) and is arrived at as follows:

qb.dUkMbi'di = f\2kVb,dMb>4’ ~ f\2kMb»,d'

= 'l“ T ( j n )

-< /+ « > !.(; ° )

= «!“ ( 0 n )

Therefore, n- 2feg(^) is one term in the sum which represents T(n)q(z) and hence nzo

A is a potential pole of T(n)q(z).

It is the above description of the induced Hecke operator T(n), and Remark 2.4 which provide the foundation for the proof of the main result. For further details, see also [Knl]. 44

2.3 The Main Theorem

We are now ready to prove Theorem 2.3.1, which we restate using the terminology developed in Section 2.1.

Theorem 2.3.1 Let q{z) be a rational period function with at least one hyperbolic

A pole. Then q(z) is not an eigenfunction of the induced Hecke operator T(n ) for any n > 1 .

A The idea of the proof is to show that the RPF T(n)q(z) has a hyperbolic pole xo which is not a pole of q(z). In particular, for any hyperbolic pole zp of q(z), xq will have the property that disc(xo) > disc(zp). The proof is given in three steps: for n = p®, where p is an odd prime and s is a positive integer; for n = 2 *; and finally for

A any integer n > 1, where we use the multiplicative property of the operator T(n).

In the proof of Theorem 2.3.1 we need two lemmas.

Lemma 2.3.2 Suppose ^ ° ^ is in r p*, where p is prime, and s is a positive integer. Suppose Zq is a hyperbolic pole of an RPF on F(l) with associated quadratic

Q(z) = [a, 6, c], satisfying disc(zo) = D. Then ^ ° ^ ^ p’zq has an associated quadratic such that disc (s'D (p*zo) is at most p**D.

Proof: Let

*-(«*) i p ' z ° ] = (o Z)(f !)w 45

_ / a y V \ (*)• "VO *) First, since o', b',d', p* E Z with a'*/' ^ 0, and since zo is a real quadratic irrational number (see Remark 2.3(i)), we must have that A is a real quadratic irrational number. Thus, X satisfies the hypotheses of Definition 2.1.7, and hence has an associated quadratic, which, in this case, is actually the minimal polynomial.

q £ I >s P2*» we have that A" is a root of P(z) = (a(d')2, d'(—2abr + a'p* 6), a(6 ')2 - ba'b'p* + (a')2p 2*c] and Dp(t) = P* 3 Dq(x). That is, A is a root of a quadratic polynomial with integer coefficients that has discriminant pi3Dq ^x). Note that since A ^ Q, by necessity, neither the lead coefficient nor the constant term of P(z) may be zero. Therefore, an associated quadratic for A is P(z) or else a normalization of P(z), and thus disc(X) is at most p 4 aDq(zy □

Lemma 2.3.3 Suppose z 0 is a hyperbolic pole of an RPF defined on T (l) with associ­ ated quadratic [a, 6, c], such that gcd(a,p) = 1 . Then p3Zo has an associated quadratic which satisfies disc(p 3 Zo) > disc{zQ).

Proof: It is straightforward to see that paz 0 is a root of the quadratic polynomial

[a, 6p*,p 2*c]. Moreover, since gcd(a,p,) = 1 and ac / 0, this quadratic polynomial is an associated quadratic for p 3 zo- Thus, disc(pazQ) = p 2i (62 — 4ac) = p 2adisc(zo), and hence, disc(p 2 aZo) > disc(z0). □ 46

Proof of Theorem 2.3.1:Recall that we intend to show that T(n)q(z) has a pole which cannot be a pole of q(z).

Step 1: n — pB, p an odd prime and s a positive integer

Let Z\ be a hyperbolic pole of q(z) such that disc(zi) is maximal with respect to all hyperbolic poles of q(z). By Lemma 2.1.13 we can find a hyperbolic pole of q(z), z0, with associated quadratic [a, 6, c] satisfying gcd(a,p) = 1, and such that disc(zi) = disc(za). For convenience, let D — disc(zo). In the alternative description of the induced Hecke operator T(p*), we had

t(p‘)q=(p-)2i-' £ «Ml

so that T(p‘)q is the sum of terms of the form q\ 2 kMMb>,d> with M € F(l) and

Mb’d• £ rn. In order to follow the initial reasoning in Knopp's original argument for the RPF [Knl]i let M = ^ ^ ^ ^ and Mb>,d’ = ^ o d1

Then

«luM M w = [~iaz + 7&' + •

Since the factor ( 7 a'z + 7 b' + 6 d!)~2k contributes only rational poles, a hyperbolic pole of T(p*)q must be a pole of q(MMb\d>z) = ? o*'!1®1’ words, in order for x 0 to be a hyperbolic pole of T(pa)q, we must have that = z:2, where z2 is a hyperbolic pole of q. Specifically, by Remark 2.4, we know that p3Zo is a pole of the term of T(p*)q(z) arising from Mb\d• = ^ ) and = T = ^ J q J . It is a, enough to show that no other term of T(pa)q(z) has a pole at p * zq s o that cancellation does not occur. In particular, by the maximality of disc(z0) and by Lemma 2.3.3, we will have shown that T(n)q(z) has a pole, namely p*zo> which is not a pole of q(z). 47

To this end, suppose there exists Af = ^ ^ ^ ^ € r(l)and M yj < = ^ q ^

Tp. with

( “ «)(o f ) &'’*•) = *»•

where z2 is a hyperbolic pole of q(z). We will show that that Mv,d> = ^ o p» j which by Remark 2.4 implies that M = TVbj = TT = /, and therefore that z2 = zq.

Since z2 is a hyperbolic pole of qx , Corollary 2.1.12 (b) applies, and hence

disc{z2) = disc ^ “ £ ) ( 0 d' ) = disc [ ( 0 d’ ) ’

Moreover, by Lemma 2.3.2 disc(z2) = disc ° ^ (p* 2o)| is at most p*‘D.

Since D is maximal, we must have disc [( 0 d' ) ^ a t *n *he computation given in Lemma 2.3.2, we saw that X = ^ ° ^ (paz0) was a root of P(z) = [a(d')2,d '(—2a6' + arp'b),a( 6')2 — ba'lfp * + (a') 2p 2i c], and Dp(x) = pA*D.

Therefore, in order for z2 to be a pole of q(z), we must have P(z) = mQ(z), for some integer m with m > p2*, where Q(z) is an associated quadratic for z2 and necessarily has discriminant less than or equal to D. In fact, we show that m = p2i as follows. First, we argue that the only prime number that can divide all of the coefficients of P{z) is p. For, if there is a prime number r distinct from p such that r divides all of the coefficients of P(z), then in particular r must divide a(d')2, the lead coefficient. But a'd' = ps, implies that d! and a! are both powers of p, and hence r|a. Now if r|d '(—2a6' + a'p’b), the second coefficient of P(z), then since gcd(r,d') = gcd(r,a') = gcd(r,p) = 1, and r|a, we have r\b. A similar argument for the constant term of P(z) shows that r|c. Recall that a, 6, and c are the coefficients for an associated quadratic, so that gcd(a, b, c) = 1. Therefore, we have shown that p is the only prime number that can divide each coefficient of P(z), and hence we have that P(z) = p*Q(z), where p* > p2* for some positive integer t. Thus, each coefficient of P(z) must be divisible by, at the least, p2*, or possibly a higher power of p. But a n since gcd(a,p) = 1 and, by the definition of T(p*), we have 0 < d1 < p , the largest power of p which can divide o(d ') 2 is p2*. That is, since the discriminant of P(z) is p 4, J9, we have p* < p2*, and so P(z) — p 2 ,Q(z). Therefore, p3‘ must divide each coefficient of P{z). In the case of the lead coefficient, because gcd(a, p) = 1, we have p 2*|(d ') 2 which gives d! = p‘ and hence a' = 1. Then the second coefficient of P(z), which is also divisible by p2*, becomes — 2a6'pa + p 2*6, so that p 2, |(—2a 6'pa + p 2j 6), which implies pJ \W. Finally, since 0 < 6' < p*, we have b' — 0.

The above computations show that the only term of T(p*)g(z) for which p'zo is a pole arises from My# = ^ q p* ^ and M — /, as desired.

S tep 2: n = 2s

By the arguments presented in Step 1, we have a' = 1 and d! = 2*. It remains to show that bf = 0 . It is still true that 2 2aJ(-2a6'2* + 22a6) so that 2*-1 divides b'.

This leaves only two possibilities, namely that V = 0 or else 6' = 2,_1. In the first instance, we are done; in the second, examine the constant term of P{z) which is a22*-2 — 622i + 22*c. As before, 22a must divide this coefficient, but this is impossible since gcd(a, 2) = 1. Thus, bf — 0, as desired.

A Note that the proof of Steps 1 and 2 produces a hyperbolic pole Xo of T(p*)q(z) with the property that for any hyperbolic pole zp of q(z)} disc(x0) > disc(zp). 49

Step 3: n > 1.

Let n = pt**pt-i*'- 1...pi'1, where pi,p 2,...,P( are distinct primes, and si,s 2 l...fst are ft positive integers. By the multiplicative property of operator T(n), we have

T(n)q(z) = f(pt‘')f(pt- 1 ‘*-')...f(p 1 ‘')q(z^

Let hi(z) be the RPF given by

for i = 1,2,,..., t. By the proof of Steps 1 and 2, for each i, hi(z) has a hyperbolic pole

Xi with the property that for any hyperbolic pole zPi of /i,_i ( 2 ), disc(i,) > disc(zPi). * _ In particular, ht(z) = T(n)q(z ) has a pole which is not a pole of q(z). Thus, q(z) is not an eigenfunction of T (n) for any n > 1. This completes the proof of the theorem. □

Corollary 2.3.4 Suppose q(z) is a rational period function with at least one hyper-

* bolic pole. Then the RPF T(n)q has a hyperbolic pole X 0 with the property that for every hyperbolic pole zq of q, disc(X 0 ) > disc(zo).

Proof: This follows directly from Case 3 in the proof of Theorem 2.3.1. □

In the next chapter we will prove an analogous theorem for RPFs defined on

G(y/2 ) and G(a/3). CHAPTER III

An Analogous Result

3.1 The Hecke Groups G(A„)

In Chapter 2 we saw the modular group T(l) as one of the infinite family of Hecke groups G(A„) = < S\„,T >, where Sxn = ^ J 2“ 5" ^ and T = ^ In particular, T(l) =

For example, one intriguing topic is the notion of generalizing on the idea of a simple (infinite) continued fraction expansion. That is, it is well known that a real number a has a periodic continued fraction expansion if and only if a is a real quadratic irrational number [Za, pp. 57-95]. The connection with real quadratic irrational numbers, continued fraction expansions, and the modular group is that any real quadratic irrational number is the fixed point of a hyperbolic M element of

T(l), and the entries of M together with the Euclidean Algorithm yield the periodic continued fraction expansion [Bu, pp. 35-48].

A generalization of continued fraction expansion, in the sense that T(l) is replaced by G(A„), is given by D. Rosen in [Rol]. The classification of the periodic continued fraction expansions in the more general context is a much more difficult problem, and has applications in diophantine approximation. Using results of Rosen, J. Lehner in

50 [Le3] and [Le4], has addressed the generalized notion of diophantine approximation.

For other results using Rosen’s continued fraction expansions, see also [Ro2], [RS] and [SS].

Another well-researched topic is the study of Fourier coefficients of automorphic forms, and for our purposes, those forms defined on G{An). In analogy to results obtained for the Fourier coefficients of modular forms, there is a variety of papers on estimates of Fourier coefficients of automorphic forms defined on (7(An). For a sampling of these results, see, for instance, [Ak], [Pa3], and [Ra].

Our main interest, at present, is that RPFs are defined on G(An), and this is the focus of the remainder of this chapter.

3.2 Automorphic Integrals and Rational Period Functions

In Section 2.1, we saw that rational period functions could be defined on the Hecke groups G(A„) =< S \nJT > for n > 3 6 Z. To make this more precise, we give a definition.

Definition 3.2.1 Suppose f is meromorphic in H for n > 3 in Z, and satisfies

(f\ 2 kSxn)(z) = /(* ) and

(f\ikT)(z) = f{z) + q(z), where k is an integer and q{z) is a rational function. If, in addition, f is meromorphic at zoo, then f is an automorphic integral of weight 2 k with associated rational period function (abbreviated as RPF) q(z). 52

Under such circumstances, we say that q(z) is an RPF of weight 2k on G(A„). If

<7 = 0 , then / is an automorphic form of weight 2 k. In the special case that n = 3, since G(A3) = T(l), / is a modular integral, and if, in addition, q = 0, then / is a modular form.

To give substance to the notion of RPFs defined on G(An), we note that Parson and Rosen in [PR] give an infinite family of (non-trivial) RPFs for each group G(A„) as follows:

?»(*) = _ hz _ j)* + (Z2 + b z _ !)* ’ (3>1) where A: > 1 is an odd integer, and b — An+1/ An ±1 ------j . “ ® 2 A„+VV+4 In this chapter, we obtain an analogue to Theorem 2.3.1 for RPFs defined on

G(A<) = G{y/2) and G(A6) = G (\/3).

We must restrict ourselves to the two groups G(>/2) and G(-^t) essentially because of the existence of Hecke operators on the space of automorphic integrals of a given weight, which, in turn, induce operators on the corresponding space of RPFs. To better understand why this is so, we give a definition.

Definition 3.2.2 Suppose G\ and G2 are subgroups of a group G such that for some g,h e G, [Gi : g(Gi < 0° [ @ 2 ■ h{G\ < 00 (ie., g{G\ and h{G\ of finite index in Gj and G 2 respectively). Then

Gi is said to be commensurable with Gi-

The Hecke groups, G( An), are subgroups of SL{2, R). Leutbecher, in [Le], showed that the only Hecke groups which are pairwise commensurable are T(l), G(y/2), 53 and G(\/3). In [BK], J. Bogo and W. Kuyk used the pairwise commensurability of r(l), G(y/2 ), and G(\/3) to show the existence of, and subsequently define Hecke operators on the space of automorphic forms on G(\/2) and G(y/$). Implicit in their construction of Hecke operators was the use of map defined by Hecke, which maps the space of automorphic forms on G(A), of weight 2 k for A = y/2 or y/H, to the space of modular forms of the same weight.

In [PR], A. Parson and K. Rosen applied results of [BK] to the space of automor­ phic integrals and the corresponding space of associated rational period functions. By doing so, they created new modular integrals and rational period functions defined on the modular group, from automorphic integrals and RPFs defined on G(\J2) and

G(>/5).

Moreover, it is straightforward to see that the Hecke operators defined in [BK] also act as operators on the space of automorphic integrals of weight 2 k, and Parson and Rosen showed that these Hecke operators induce operators on the corresponding space of RPFs. Therefore, in view of Theorem 2.1.2, that the finite non-zero poles of RPFs on G(yJ2 ) and G(\/3) are fixed points of hyperbolic elements of G(y/2) and

G(\/3), and in light of the fact that induced Hecke operators exist on these spaces, we are irresistibly drawn to an analogue of Theorem 2.3.1.

To this end, the results mentioned above which are relevant to the results of this chapter are summarized in the following theorem and the next two definitions, which can be found, collectively, in [PR] and [BK]. 54

Theorem 3.2.3 For A = a/2 or A = a/3,

(a) if fx is an automorphic integral of weight 2 k on G( A), then if>\{fx) is a modular integral of weight 2 k, where

MM = MM + A"" E h (^ ) , and

(b) if qx is the RPF associated with fx, then ifi(qx) is an RPF on r(l), where

^ ( 9 a ) = (ff’xifx^UkT - V » a ( / a )

= qx(y/ 2 z) + ( a / 2 )~2kqx j

if X = a / 2 , and

+ (v/3)—“, A + (v^)-“ 9»

if A = a /3.

Definition 3.2.4 For A = a/2 or a/3 and fx an automorphic integral of weight 2k on G(A), the Hecke operators Tx{n) are are defined as follows.

(a) If X2 fn, then

Tx(n)h = E 0 6(morf 0 b(mod d) (b) i!n = \ \

Tx(\2)h = )+ E M » f A0‘ (), ad = A2 V / *=i V / d>0 b(mod d) and

(c) if n = (A2)r for some integer r > 1,

t*((aj)'+,)/a = iM A^KA’n A - (AJ)‘rj((A J)’)/A

-(A ^^’nttA2)'-1)/,.

As was the case for the spaces of modular forms and modular integrals, the Hecke operators defined on the space of automorphic integrals on G(\) are multiplicative

(see [PH] and [BK]), and are independent of the choice of representatives b(mod d).

Therefore, we again choose the set of representatives of b(mod d) given by {0, 1, , d—

1}. Moreover, in analogy to Definition 1.5.2, the Hecke operators in Definition 3.2.4 induce operators on the corresponding space of RPFs, which are given as follows.

Definition 3.2.5 For A = \/2 or y/3, if f\ is an automorphic integral of weight 2 k on G(A) with associated RPF qx, then Tx{n)qx = (Tx(n)fx)\ 2 kT — Tx(n)fx.

It is worth noting that the induced Hecke operators inherit the multiplicative property of Tx{n), as well as the recursion formula in Definition 3.2.4 (c). 56

We now have enough information to state the main theorem of this chapter, which is a direct analogue of Theorem 2.3.1.

Theorem 3.4.4 If q\ is a rational period function with at least one hyperbolic pole,

A < then q\ is not an eigenfunction ofTx(n) for any n > 1 .

The proof of Theorem 3.4.4 will be a proof by contradiction, in which we will use

Theorem 2.3.1 and the fact that ‘essentially’ *j>(T\(n)q\) = T(n)tj>(q\). However, in order to apply Theorem 2.3.1 in such a proof, we must guarantee that if q\ has a A | hyperbolic pole, then so does ij>(q\). This is entirely the purpose of the next section.

3.3 The Poles of

For the remainder of this chapter, assume that A = l,\/2, or \/3, and that all auto­ morphic integrals and RPFs are of weight 2k, A; is a positive integer, unless otherwise specified.

The goal of this section is to prove the following proposition.

Proposition 3.3.1 For A = y/2 or y/3, suppose q\ is an RPF defined on G(A) with

A A a hyperbolic pole. If iftx is the map defined in Theorem 3.2.3 (b), then il>s(q\), an RPF defined on r(l), has a hyperbolic pole.

Before proving Proposition 3.3.1 we give one lemma.

Lemma 3.3.2 Suppose q\ is an RPF on G(y/2) with a hyperbolic pole Zq such that disc(zo) = l(mod 2). Then there is a hyperbolic pole z\ of q\ satisfying disc(z^) = 57 d*sc(zo) with associated quadratic [r,s,<] such that 2|r. In other words, q\ has a hyperbolic pole with associated quadratic whose lead coefficient is divisible by 2.

Lemma 3.3.2 is quite useful in proving Proposition 3.3.1 for the case A = >J2.

Unfortunately, the analogous lemma for A = y/3 is false, as evidenced by the following example. For k a positive odd integer, let

,(2) = - * - ,/3)‘ + (VS»> + * - v 5 )‘ (3'2)

First, we verify that q is an RPF of weight 2k on G(\/3) by checking functional equations 2.1 and 2.2 from Section 2.1.

For convenience, we use the notation introduced in Section 2.1 after Proposition

2.1.8 to write q(z) = [1, —1,1]-* + [1,1, — l]-*. In that case, we must show

£ [i,-1, - i r ‘M sr)' + £[1,1, -1]-*MST)' (3.3) 1 = 1 1 = 1

= -([l,-l,l]-* + [l,l,-l)-‘). (3.4)

To this end, note that since k is odd, we have — [a, 6, c]“* = [—a, — b, — c]“*. Hence, it is straightforward to see that q satisfies 2.1, and therefore, we check only 2.2. To this end, £[i, -i, -irV sry + £[i,i, -irV sT)' i=l i=l = [1, -5, l]~k + [-1,-1, l]~fc + [-3,7,-l)~k + [-3,11, -3]-fc + [-1,7, -3]-*

+[3, -7 ,1]"* + [3, -11, 3}~k + [1, -7,3]-* + [-1,1,1]"* + [-1,5, -l]~k

= -([1,-1, !]-* + [!, 1,-1]"*), 58 as desired.

Next, note th at the poles of q are and ~1* ^ 3 which have associated quadrat­ ics [1 , —1, —1] and [1,1, —1], respectively, neither of which have lead coefficient divis­ ible by 3, and both of which have discriminant equal to 13 (= l(mod 3)).

Therefore, we must resort to another technique, an algorithm, in the proof of

Proposition 3,3.1 for the case A = >/3*

Proof of Lemma 3.3.2 Suppose an associated quadratic for z0 is [a, 6,c]. Without loss of generality, assume 2 J(a. By functional equation 2.1, Tzq = ^ is a pole of q\ which, by Corollary 2.1.12 (a) has associated quadratic [c, — 6, a]. If 2|c, we are done.

Otherwise, gcd(c, 2) = 1. Now by functional equation 2.2 and the fact that z0 V^2Q

(Corollary 2.1.6 (b)), at least one of (S^T^zo, (S^T^Zo, or (S'^T’Jzo is a P°le of q\, and by Corollary 2.1.12 (a), these potential poles have associated quadratics

[2a + b + c, —4a — 6, a], [a + 5 + 2c, —4a — 36 — 4c, 2a + 6 + c], and [c, —6 —4c, a + 6 + 2c], respectively. By hypothesis, disc(zo) = 62 — 8 ae is odd, so that 6 is odd. Thus, since a and c are odd, the first two associated quadratics have even lead coefficients, and therefore if either (5v^T)3^o or {S^j,T)2zo is a pole, we are done. Otherwise, (5V^T)2 0 is a pole whose associated quadratic has even constant term, and in that case we need only use which has associated quadratic [a + b + 2c, b + 4c, c] with even lead coefficient. □

The proofs of Proposition 3.3.1 for A = \/2 and A = >/3 are analogous only up to a point. Therefore, we present the proof of Proposition 3.3.1 for A = y/2 for as long 59 as the analogy holds, which will, in fact, completely take care of the case A = \/2. To finish the proof, we then address what remains of the case A = a/3*

P ro o f of P ro p o sitio n 3.3.1 We wish to show that if 9a, an RPF on G(A) has a hyperbolic pole, then V»a(9a), an RPF on Tfl), has a hyperbolic pole. To this end, recall from Theorem 3.2.3 that if A = a/2,

92(2 ) :=

= q>(V2 «) + 2-‘«» +r ‘»» + (1 - * )-“ «a , (3.5) and if A = a/3,

q 3 (z) := tj>x(«>(«))

= M + r*« (^ ) + 3-‘,a ( ^ ) + r*« ( ^ )

+ (* + 1)-% + (1 - *)-“4a . (3.6)

For i = 2,3 we wish to show that

transformation of determinant A, we look to the poles of q\ in order to search for

potential poles of 9,-.

For example, if is a hyperbolic pole of q\, for A = a/2, then a /2 2 2 is potentially

a pole of 92 because the second term in equation 3.5 is 2 ~kq\(-j-). On the other

hand, \f\\z3 may be a removable singularity of 92 if any of 2 z2, v^^ ~ 1, and are

poles of 9a. Note here that (1 — z)~2 kq \ ( ^ ) is the fourth term in equation 3.5. By

Corollary 2.1.6 (b), since z 2 ^ a/ 2 Q , we have 1 — a / 2 2 2 ^ 0 and so (1 — z)~2k cannot

provide a zero-denominator when z = a/222. 60

Similarly, if 23 is a hyperbolic pole of q\ for A = \/3, then by examining the second term in equation 3.6, we see that \/323 is potentially a pole of q\, but we have no guarantee that y/3zs is not a removable singularity of g3.

Therefore, we proceed as follows. First, for the sake of clarity, we temporarily restrict the discussion to the case A = y/2 . We wish to find a hyperbolic pole z2 of q\ such that 2 z2, v^ ^ ~ 1 and are not poles of q\, and in doing so, will guarantee that y/2z2 is a pole of q2. This pole will necessarily be hyperbolic because since z2 is non-zero and finite, then y/2z2 is also non-zero and finite. By Theorem 2.1.2, then, we will have that y/2 z2 is a hyperbolic pole of q2.

To this end, let z2 be a hyperbolic pole of q\ with associated quadratic [ 03, b2, c2] such that D 2 = b2 — 8 a 2C2 is maximal with respect to all hyperbolic poles of q\.

For convenience, let Nt = ^ 1 ) ’ = ( ^ ^ ) ’ and = ( - ^2 1 ) ’ S° that N iz2 — 2z2, N2 z2 = v ^ ^ ~ 1 , and N$z2 — t_2^ a • Next, we apply Lemma 2.1.11

(a) to N\ z2, N 2 z2, and N$z2 to find that they satisfy the (not necessarily associated) quadratic polynomials Pi ( 2 ) — a2 ^/2z2 + 2b2z + 4c2-\/2, P2 {z) = 2a2 y/2z2 + (4o 2 +

262)2 + (o 2 "f 62 + 2c2 )y/2, and P 3(2 ) = (a2 4- 62 -f- 2c2 )\f2zi -f- (262 -I- 8 c2)2 + 4 c2^ J , respectively. First, note that we must show that Pi( 2 ), P2 (z) and Ps{z), or else normalizations of .Pi( 2 ), P2 (z) and P^{z), are of the necessary form with respect to

Definition 2.1.7 so that N\z2l N 2 z2, and JV322 may legitimately be hyperbolic poles of q\. To this end, we need only show that the lead coefficients and constant terms of P i( 2 ), P2 (z) and P^{z) are all non-zero. This is clearly true of Pi( 2 ) because a2 c2 ^ 0 since [a 2, 62, c2] is an associated quadratic for z2. Therefore, it remains to 61

show that 02 + 62 + 2 c2 ^ 0. But, if 02 + 62 + 2c2 = 0, then N2 z2 — is a root of P2 (.z) — 2 a2V^222 + (4a2 + 262)^* That is, 2 a2 y/2 (N2 z2)^ + (4a2 + 2b2 )N2 z2 = 0, which means that either N2 z2 = 0 or else 2a2 y/2N2 z2 + (4a2 + 262) = 0. This is a contradiction because in either case, we have N 2 z2 and hence 22 is a rational multiple of \/2. Thus, N\Z2, N 2 z2, and N 3 z2 all have associated quadratics, and hence, we return to the usual notation for the Pi{z), namely P\(z) = [a2, 2 &2, 4 c2],

P2 (z) — [2a 2, 4 o 2 -t- 262,02 + 62 + 2C2}, and /^ (z ) — [02 + 62 + 2 c2, 2&2 + 8 c2, 4 c2].

Since for j = 1,2,3 the determinant of Nj is 2, by Lemma 2.1.11 (b), we have

Dpj(z) = 4£>p(*), or, alternatively, Dp^) = AD2 because P(z) is an associated quadratic for z2. If it turns out that for j = 1,2,3, Pj(z) is an associated quadratic for N jz2, then NjZ2 will not be a pole of q\ by the maximality of D2, and then we are done. That is, if Pj{z), when viewed as a triple of integers, does not have 2 as a common factor, then Pj(z) is an associated quadratic for NjZ2, since the remaining condition is to check that the coefficients are relatively prime as a triple.

A similar discussion regarding potential hyperbolic poles of q 3 provides the fol­ lowing information. First, choose 23 with associated quadratic [ 03, 63, c3] so that

Z?3 := disc(z3) is maximal. Then in order to guarantee that y/323 is a (neces­ sarily hyperbolic) pole of q3, we must ensure that 323, and

are not poles of q\ for A = \/3* We apply Lemma 2.1.11 (a) to each of the above numbers, and discover that they satisfy, respectively, the quadratic poly­ nomials Qi(z) = [a3, 363, 9c3], Q2( z) = [3a3,6a3 + 363, a3 + 63 + 3c3], Q3 {z) =

[3a3, —603 + 363,03 — 63 + 3c3], $ 4(2 ) = [03 — 63 + 3c3,363 — 18c3,9c3], and Qs{z) = 62

If A = y/Z Pole Corresponding Pole Corresponding of q\l Quadratic Polynomial of q \? Quadratic Polynomial PM = Q M = 2 Zj [a2, 2 &2, 4 c2] 3 z 3 [a3, 363, 9c3] P M = Q 3 (z) = V ^ -l V '^pl [2 a2,4a2 + 2 b?, a2 + h? + 2c2] [3a3, 6 a3 + 363, a3 + 63 + 3c3]

ft(» ) = QM = V3^+l [a2 + ^2 + 2c2,262 + 8 c2, 4 c 2] [3a3, —6a3 -1- 363, a3 — + 3c3] i q f e QM = 3 z.i 63 y /3 * .i+ l [fl3 — + 3c3, 3&3 18c3,9c3] Q M = 3*3 [<*3 + ^3 + 3c3, 363 -f- 18c3, 9 c 3] , 1,-Vki Figure 5: Potential poles of q\ related to z,-

[ 0 3 + ^ 3 4- 3c3, 36s + 18c3, 9c3]. Note also that by Lemma 2.1.11 (b) that D q^z) = 9D3 for j = 1,2,3,4,5; if it then turns out that each Qj(z) is an associated quadratic, we are done by the maximality of £>3. That is, if each Qj(z), when viewed as a triple of integers, does not have 3 as a common factor, then Qj{z) is an associated quadratic.

We condense the information obtained thus far in Figure 5.

The remainder of the proof is given in two cases, the second of which includes an algorithm. Recall that D2 and £>3 are maximal and fixed for the rest of the proof.

Case 1: £>, = 0(mod A2)

Since Di — 62 — 4A2ajCi, we have 6,- = 0(mo«f A2). By Lemma 2.1.13, we may assume without loss of generality that gcd(a{, A2) = 1. Then every potential pole of q\ in

Figure 5 is eliminated because each corresponding quadratic polynomial has either lead coefficient or constant term relatively prime to A2. In other words, all polynomials 63 in Figure 5 are associated quadratics. Therefore, Az; is a hyperbolic pole of

C ase 2: Di ^ 0(mod A2)

Since Di = bf — 4A2a,Cj, we have 6, ^ 0 (mod A2).

The remainder of Case 2 is given in two steps, the first of which gives a procedure A for producing a hyperbolic pole of il>x(q\), although under somewhat restrictive cir­ cumstances (due, in part, to the failure of Lemma 3.3.2 for A = \/3)- The purpose of the second step is to show that even in the worst case, we may always return to the first step.

S te p 1:

If there exists a pole zmi with disc(zmi) = Di, and with associated quadratic [rj,s;,t;] such that A2|r*j, and if among all such poles, we choose zmi so that \zmi\ is maximal, then all potential poles of q\, shown in Figure 6 are eliminated. That is, 2zma and

3zm 3 are eliminated by virtue of the maximality of |2m<|i and all of the remaining potential poles of q\ are eliminated because each has corresponding quadratic poly­ nomial with either lead coefficient or constant term relatively prime to A2.

Remark. By Lemma 3.3.2, when A = y/2, we can always find a hyperbolic pole of qx satisfying the hypotheses given in Step 1. Therefore, when A = a/2» if q\ has a hyperbolic pole, so does and we are done. On the other hand, when A = a/3, as promised, the situation is more complicated, and we deal with it in the next step.

Step 2: A = \/3) D$ ^ 0(mod 3) 64 % -< II II Pole Corresponding Pole Correspondi ng of qx? Quadratic Polynomial of qx? Quadratic Polynomial 2? [r2,2 s2,4<2] 3^n»3 [r3,3s3, 9/3] V3*ms-1 [2r2,4 r2 4- 2s2, r2 -I- s2 4- 2t2] [3rs, 6 r 3 -I- 3 s3, rs 4- S3 4- 3 ^3] V2 2Zma [r2 + s2 + 2 f2, 2 s2 + 8 t 2,4t2] [3r3, —6 r 3 4 - 3 s3, r 3 — S3 4- 3 (3] 1 — 3frm +1 — S3 + 3* 3 ,3s3 — 18*3, 9*3] 1—VSlm, [r3 + S3 + 3<3, 3 s3 + 1 8 <3 , 9(3]

Figure 6 : Potential poles of qx related to zmi

Recall that Z3 is a hyperbolic pole of qx with associated quadratic [<13,63, C3] such that D3 — disc(z3) is maximal. Note that since £>3 ^ 0(mod 3), we must have gcd(b 3 , 3 ) = 1.

If 3 ja3, then go to Step 1. Otherwise, gcd(a3, 3) = 1. In that case, 3^3 is eliminated as a possible pole of q 3 because the lead coefficient of Q\ (see Figure 5) is not divisible by 3.

It remains either to eliminate as potential poles of q3, or else use to our advan­ tage, the final four potential poles of qx in the second column of Figure 5. To this end, suppose X 4 = ^;*3+] is a pole of qx- By the maximality of D3, and because gcd(3, 6 3 ) = 1 , we must have that [°3~(>|'t'3ca , 63 — 6C3,3c3] is an associated quadratic for X 4 . In that case, is a pole of qx with associated quadratic whose lead coefficient is divisible by 3. Go to Step 1. Similarly, if X$ = 1_3^ J is a pole of qx, then ^ is a pole of qx with associated quadratic whose lead coefficient divisible by 3. Go to Step

1.

Now, without loss of generality, assume that neither X 4 nor X s are poles of qx- 65

We will eliminate both X 2 = v^ ~ 1 and X 3 = as potential poles of qx as follows. Observe that the constant terms of Q2 and Q3 (the corresponding quadratics for X 2 and Jf3) are (a 3 + 63 + 3c3) and (a 3 — 63 + 3c3), respectively. Therefore, we may eliminate one of X 2 or X 3 depending on whether or not a 3 = bz(mod 3). By a solicitous choice of z3, we may eliminate X 2 and Xa simultaneously.

In particular, if a 3 = ^{mod 3), then among all such poles of q\ with maximal discriminant and o 3 = b$(mod 3), choose z 3 to be the largest (furthest to the right on the real axis). Then X 2 is eliminated because the constant term of Q2 is not divisible by 3, and X 3 is eliminated because = z 3 + ^- > z3.

Similarly, if o 3 ^ ^(mod 3), then among all such poles of maximal discriminant, choose z 3 to be the smallest. Then X 3 is eliminated because the constant term of Q3 is not divisible by 3, and X 2 is eliminated because = z3 — < z3. Therefore, y/3za is a hyperbolic pole of qa. This concludes Step 2 and Case 2.

In all cases, we have shown that if qx is an RPF on G(X) with a hyperbolic pole, then i>\{qx)i an RPF on T(l), has a hyperbolic pole. □

>> A A 3.4 Relationship Among T(n), T\(n), and

A A A In this section we find a formula for the relationship among ipx, T(n ), and Tx(n),

A A where T(n) and T\(n) are the induced Hecke operators on the space of RPFs on T(l) and G(A) respectively. See Theorem 3.2.3 (b), Definition 1.5.2, and Definition 3.2.5.

In the next lemma, we begin by giving a formula for the relationship among

T(n), T\(n), and V’a, the usual Hecke operators, and the map from the space of 66 modular integrals on G(A) to the space of modular integrals on T(l), respectively.

See Definitions 1.4.1, 3.2.4 and Theorem 3.2.3 (a). Then we will show that the corresponding formulas hold for the induced map and operators t/>A, T(n), and T!x(n).

Lemma 3.4.1 For A = \/2 or \/3, if f\ is a modular integral of weight 2k on G(A), then

(a) MTx(n)fx) = T(n)Uh) i/* 2

and

(6) = T (n )M h ) + (*2 - = *2-

Proof: The proofs for A = \/2 and y/3 are analogous, and so we present only the case A = y/2.

(a) First, for convenience we write / and rj) in place of f ^ and respectively.

Next, by Definition 3.2.4 (a), since 2 /n, we have

T ^ { n )f = n ^ f U ( “ W 2 ^ (3.7) ad = n 0 < b < d and by Theorem 3.2.3 (a),

V-(/) = / k ( j 1 ) /l“(o /2 v ) ’ (3.8) so that

n 2k -l £ /U (o hf) 12/e (o ad = n

{ 0 < b < d 67

+ n 2*—1 E /!» ( o "f) 2Jt ad — n (f:) 0 < b < d

+ n 2k—1 e /i-(; bf) ( o V i ) ad = n 0 < b < d

£ /la(o i % ) + £ ad = n ad = n 0 < b < d 0 < 6 < d

+ E /I- (o V i ) • (3.9) ad = n V 6 y 0 < b < d On the other hand, by Definition 1.4.1

T(n)W ) = „“-' £ /|M( J ^ ) b ( “ *) ad = n 0 < b< d

+»-* ad = E n /l«(f?)k(o^) 0

+«“-• e /k(; j) (3.10) ad = n 0 < 6 < d 68

— n3/t-l E ^ ( o ^ ) + E ad = n ad= n 0 < b < d 0

* z (3.11) ad = n 0 < b< d In order to see that ^ (T ^ n ))/ = T(n)ifi(f), it suffices to show that the summa­ tion of the first and last terms of equation 3.9 equals the summation of the first and last terms of equation 3.11, because the second terms in both equations are identical.

Specifically, it suffices to show that

E

E (3.13) ad = n 0 < 6 < d

For convenience in computations to follow, we write ^ in place of a.

First, observe that we need only examine the upper right-hand entries of the matrices in 3.12 and 3.13, since all other corresponding entries are identical. By doing so, we may rewrite 3.12 and 3.13 as follows. Recall that 2 /n so that d and ^ are odd. Let El = {0,2,...2d-2}, 01 = {*, *+2,..., *+2d-2}, 02 = {l,3,...2d-l}.

In other words, El is the list of consecutive even integers from 0 to 2d — 2, 01 is the 69

list of consecutive odd integers from ^ to 5 + 2d — 1, and 0 2 is the list of consecutive odd integers from 1 to 2d — 1. In that case, 3.12 may be written as

£ 'I”( 0 *0 + E 'l’‘(o dU) ad. — n ad — n be El be01f)02

E (3.14) ad — n

b e Ol\(01 f| 0 2 ) and 3.13 may be written as

E /b*(o E ^‘(0^) ad — n ad = n &e£i S

e /i-(; ^)- (3.15) ad = n

b e 0 2 \ ( 0 1 fl 0 2 ) Therefore, it suffices to show that the last terms in 3.14 and 3.15 are the same. That is, we must show that

E /l»(o ^)- (3.16) ad = n b e 0 l\( 0 1 f) 0 2 ) is the same as e /u(; d%) (3.17) ad = n

b e 0 2 \( 0 l n 0 2 )

To this end, note that for d ^ n, 01\(01f|02) = {2d+l,2d+3,..., ^ + 2d— 2 }, and

O2\(01 fl02) = {1,3,...,^ — 2}, and we see that the difference between every pair 70 of corresponding elements in the above two lists is 2d. When d = n, 0 1 \(0 1 f] 02) and 02\(O1 f| 02) are empty. Therefore, we may rewrite 3.17 as

£ / i - ( o <318> a d = n N ' b € 0 1 \ 0 1 f ] 0 2 which means that 3.16 and 3.17 are identical because

( o

= /l“ ( o jU) since / is periodic with period y/2. This proves (a).

(b) We will compute ^l>(T^(2)f) and T(2)iJ?(f), “ id then compare the results. To this end, by Definition 3.2.4 (b), we have

r^(2)(/) -/U (i °)+/u(j f )

+ /la ( o 1 ) + /lu ( ' f ^2 ) ’ (319) and by Definition 1.4.1

T(2)0(/) = V(/)la ( J " ) + W )l“ ( 0 2 ) + W )l“( 0 1 ) ' (3 20)

Then combining 3.8 and 3.20 yields

T (2 )W ) = /l«(f °)+/l»(j 2^)+/l-(l j J j ) +/u('f 'f)+/l“(j 2^2 ) +/,2‘ ( 0 2J2 )

+ /k ( J ) + /U ( q ^ 2 ) + /l» ( 0 VS ) ' <3-21>

On the other hand, combining 3.8 and 3.19 yields

*PV*(2)/> = r^(2)/|« ( 'f J)+JV5(2)/I«(J ^ ) + 7 Vj( 2 ) /I » ( J )

= / b ( ^ 2 ) + / |! i ( f ) +/l» (2f l ) +/l"(o Vj)

+/l“ (o 2Vs)+/ll*(o 2V5 ) + /,M ( 0 ^ ) +/l" ( ' f ^ )

+/l2‘(o 2Vs)+/l“ (o 2Vs)+/l"(o ^ ) +/l” ('^ 22^)

= r (2)W ) + / k ( f f ) + / b ( o ^)+/l«(f 2f )

= T(2)V-(/) + 2-‘/ + + 2-*/ ^ V H i+ M j

= r(2)0(/) + 2-‘/ + (V2)-kf(V2z + V5) + 2~kf(-~ + V2).

Since / is periodic with period \/2, we conclude that

V-(rvi(2)/) = r(2)^(/) + ( ^ ) - sS h /), as desired. □

The corresponding formulas hold for q, the RPF associated with / , as is shown in

the next corollary.

Corollary 3.4.2 For A = y/2 or y/S, if q\ is an RPF on G(X), then

(а) fc(ii(n)9A) = f (»)&(«) i/V/n

and

(б) ij>x{f\{n)q\) = T{n)$x{q\) + (A2 - l)X~2kri>\{qx) ifn = X2.

Proof: (a) First, for convenience we write q and ij> in place of q\ and respectively.

Recall by Theorem 3.2.3 that if / is an automorphic integral of weight 2k on (7(A) with corresponding RPF q , then $(q) is an RPF on T(l) with a corresponding modular

A______integral ^(/), and that ij>(q) = (i>x(f))\2 kT — Moreover, by Definition 1.5.2, we have T(n)ti>(q) = (r(n)0(/))|2fcT — T(n)0(/). Finally, note that for any positive integer n, T \(n )f is an automorphic integral on G(X) of weight 2k with associated

RPF T\(n)q. In that case,

(a) since A2 fn, Lemma 3.4.1 (a) applies, and so

rj>(Un)q) = (*l>(Tx(n)f))\2kT - 1>(Tx(n)f)

= (r(n)0(/))|2fcr-r(n )V ’(/)

= f(n)j>(q), as desired. 73

(b) Since n = A2, Lemma 3.4.1 (b) applies, and hence

«n (A 2)?) = (V’(ri(A>)/))|a r - iH W 2)/)

- +(*2 - i)a-”w))i*t - m \ ' ) w ) + (a2 - i)A-2V(/)>

= l(T(A2)^ (/))|„ r - r(A2)0(/)] + (A2 - i)A-2‘[(i6(/))|2lr - *(/)]

= f( A2)^(,) + (A2-l)A -2*^(,).

□

We need one final lemma in order to prove Theorem 3.4.4.

Lemma 3.4.3 Suppose q\ is an RPF of weight 2k on G( A) for A = y/2 or y/%,and

A A _ suppose q\ has a hyperbolic pole. If s > 1 is an integer, then the RPF ip(Tx((X )‘)q\)

(defined on T(1)J has a hyperbolic pole z, with the following property: if zo is any hyperbolic pole of tf>(q\), then disc(zt) > disc(zo).

Proof:We proceed by induction on r. Specifically, we will show for all integers r > 1

A A that il>(Tx( ( \2)r)qx) has a hyperbolic pole zr such that for any hyperbolic pole z'T_j of ‘4’(T\((X2)r~1)q\), we have disc(zr) > disc{z'T_i). To finish the proof, we apply this

A result to the case r = 1, or equivalently, to the hyperbolic pole of ij>(q\).

Let r = 1. Then

^(fj,((A2n 9x) = ^(Ta (A2)9a) (3.22)

= f (A2)^(9x) + (A2 - 1)A-2‘<%X) by Corollary 3.4.2 (b). Moreover, by Corollary 2.3.4, T ( \ 2)$(q\) has a hyperbolic pole z\ such that disc(zi) > disc(zo) for any hyperbolic pole z0 of V>(flA), and therefore, by equation 3.22, so does i/>(Tx(A2)gA).

Now suppose the induction hypothesis holds for all positive integers j such that

1 < j < r - It remains to show that ^(!fA((A2)r+1)gA) has a hyperbolic pole zr+1 such that dzsc(zr+i) > d isc(z/) for any hyperbolic pole zT' of ^(Tx{(X2)r)q\). To accomplish this, we use the recursive definition of TA((A2)r+1) as given by Definition

3.2.5 (c) in Section 3.2. Specifically,

& ii((A J)'+'),j) = ^ (fA(AJ)#\((A2r ) ?A - (A2)‘r A((A2)'),A

- (Aa) " - , tA((A1)’- , )?i). (3.23)

A Since ifj is linear, and by applying Corollary 3.4.2 (b), we can rewrite equation 3.23 as

*(ft((Aan*A) = f(\2m f x((\2Y)qx)+ (a2 - i)A -2fcv-(rA((A2r ) 9A)

-(A 2) ^ ( T A((A2r ) 9A) - (A2)2*-1^(T,A((A2)r- 1gA)

= T (A2)V-((fA((A2r ) 9A)

+[{X2 - 1)A-2* - (A2)fc]V»(fA((A2r ) 9A)

—(A2)2*-1V>(7A((A2)r-1)gA).

In total,

ii(n((A,)'+,),J) =

+ C,^(fA((A2)')?A) + C2^(Tx((X2)T~l)qx), (3.24) 75 where Cx = (A2 - l)A"2fe - (A2)* and C2 = -(A 2)2*"1.

A A By the induction hypothesis, il>(T\((X3)r)q\) has a hyperbolic pole, zr such that for any hyperbolic pole z'T_x of rj>(T\((X2)T~l)q\)t we have disc(zr) > disc(z'_,). More­ over, by Corollary 2.3.4, f(X 2)ij>(f\((X2)T)q\) has a hyperbolic pole zr+1 such that

A A. for any hyperbolic pole zr' of xJ)(T\((X )r)q\), we have disc(zr+1) > disc(zr'). There­ fore, by equation 3.24, the same can be said of ij>(f\((X2)T+1)qx). This completes the induction on r.

To finish the proof, simply note that for 1 < j < s, we know that \}>(Tx((X2y)q\) has a hyperbolic pole Zj such that disc(zj) > disc(z'j-i) for any hyperbolic pole

A A . A z'j-j of y )q\). In particular, if z0 is any hyperbolic pole of il>[q\), then disc(Zg) > disc(zQ), as desired. □

We are now ready to prove Theorem 3.4.4, which we restate.

Theorem 3.4.4 For A = y/2 or y/3, if q\ is an RPF on G(A) with at least one hyperbolic pole, then q\ is not an eigenfunction of the induced Hecke operator T\(n) for any n > 1.

Proof: We give a proof by contradiction, which is accomplished in two steps: for any integer n > 1 such that A2 /n, and for n = (A2)*n', where s > 1, n' > 1 and A2 fn'.

Step 1: n > 1 is an integer with A2 fn.

By way of contradiction, suppose f\(n )q \ = Cq\ for some C ^ 0 in C. Then by

Corollary 3.4.2 (a),

Mfx{n)qx) = f (ntffoO, (3.25) 76 and by assumption,

ij>(fx(n)qx) = $(Cqx) = C^(qx) (3.26) so that

T(n)j>(qx) = Cto,q>)- (3-27)

By Proposition 3.3.1, ij>{qx) has a hyperbolic pole, and hence Theorem 2.3.1 applies, and therefore equation 3.27 gives a contradiction. Specifically, by Theorem 2.3.1,

A A r(?(qX) is not an eigenfunction of T(n).

Step 2: n = n'(A2)*, where s and n1 are positive integers, and A2 /n #.

By way of contradiction, suppose J\(n#(A2)')^ = Cqx for some C ^ 0 in C, so that

A A A if>(Tx(n'(X )a)qX) = Ci>(qx). Since the induced Hecke operator is multiplicative and by Corollary 3.4.2 (a), we have

In total, with our original assumption, we have

( f A(( X2Y)qx) = W (qx). (3.28)

By Lemma 3.4.3, tj>(Tx(X2Y)qx) has a hyperbolic pole, z, such that disc(zt ) > disc(z0)

A A A A for any hyperbolic pole zo of il>{qx). Moreover, by Corollary 2.3.4, T(n')tJ>(Tx( ( \ y)q x) has a hyperbolic pole Zn such that disc(Zn) > disc(zt ). In other words, since

A A A disc(Zn) > disc(zo), T(n')V»(Tx(A2)*)^) has a pole, Z„, which cannot be a pole

A of ip(qx), and this contradicts equation 3.28. 77

Therefore, qx is not an eigenfunction of 7>(n) for any integer n > 1. CHAPTER IV

The Zeros of Rational Period Functions

4.1 An Explicit Description

In [PR], A. Parson and K. Rosen asked the following open-ended question: ‘What can be said about the zeros of a rational period function?’ The purpose of this chapter is to begin an investigation into the location of zeros of RPFs defined on G(An) for n > 3, and A„ = 2cos(^). To this end, we will give an explicit description of the zeros of the following family of RPFs on G(An) given in [PR] by

*»(*) = - b z - 1)* + (*2 + bz- 1)*’ where k > 1 is an odd integer, and b = Xn+V >in +_ ------? In addition, we will ° 2 A„+VV+4 give an explicit description of the zeros of the family of RPFs defined on G(\/3) given by

= (* » - ^ z - 1)* + <4 ' 2 > where, again, k > 1 is an odd integer. Note that when n = 3 the family of RPFs given in equation 4.1 is the original family of RPFs with quadratic irrational poles found by Knopp in [Knl] as described in equation 1.14 in Section 1.6. Moreover, the RPFs given in equation 4.2 are distinct from the RPFs in equation 4.1 for n = 6 because

78 79 the poles of qG are whereas the poles of q are ±v/3±'/3g Finally, recall that the RPFs in equation 4.2 were given as example 3.2 in Section 3.3.

To help locate the zeros of the RPFs given above, we begin with a proposition.

Proposition 4.1.1 Let f(z) = i)'* for any b ® in W f(x + iy) = 0 them either x = 0 or else x 2 + y 2 = 1. In other words, the zeros of f lie on the unit circle or on the imaginary axis.

Proof: Suppose /(x + iy) = 0. Then

= 0 (4.3) ((a? + iy)2 - b(x + iy) - l)fe ((x + iy)2 + b(x + iy) - l) fc implies that

1 1 (4.4) ((x + iy)2 - b(x + iy) - !)*■' ((x + iy)2 + b(x + iy) - 1)* which means ' (x + iy)2 - b(x + iy) - 1 = - 1, (4.5) (x + iy)2 + b(x + iy) - 1 and therefore (x -1- iy)2 - b(x + iy) - 1 = 1 , (4.6) (x + iy)2 + b(x + iy) - 1 or alternatively,

|(x + iy)2 -b(x + iy)~ 1 = (x + iy)2 + b(x + iy) - 1 (4.7)

We simplify equation 4.7 and get

|(x - y + bx - 1) + i(by + 2xy)\ = |(x -y - bx - 1) + i{-by + 2xy)| . (4.8) 80

Observe that most of the terms on the left-hand-side of equation 4.8 are the same as those on the right-hand side, so that when computing the norms, cancellation occurs, and we are left with

26x(x2 — y2 — 1) + 46xy2 = —26x(x2 — y2 — 1) — 46xy2, (4.9) which gives

6x(x2 + y2 — 1) = 0. (4-10)

Therefore, x = 0 or x2 + y2 = 1 since 6^0. □

Once we know the general location of the zeros of functions of the form of / as given in Proposition 4.1.1, we can explicitly compute the zeros of those functions as is shown in the next corollary.

Corollary 4.1.2 With k > 1 odd and the same hypotheses on f as in Proposition

4.1.1, the zeros of f are given as follows.

(a) The purely imaginary zeros of f are of the form yai, where

-H a n ( ? ) ±

V‘ 2------for all integers s such that 1 < s < k — 1 and \tan | > | .

(h) The zeros of modulus 1 of f are of the form zt = ± y j 1 — yj + iyt , where

b fir s\ y. = 2tan [t ) for all integers s such that 0 < s < k — 1 and \tan | < | . 81

Proof: (a) By equation 4.5 with k odd, we have

/ iiv? - Hiv) - 1 V \-(iy )2 - b(iy) + 1)

_ f —y2 - b(iy) - l \* V y2 - K*y) + 1 / = 1, (4.11)

or alternatively, that is a fcth root of unity. In that case, to find y , we must solve —y2 — b(iy) — 1 (2 irs\ . . ( 2 irs\ = coa( ir ) +'*'”(— ) <412> for integers s such that 0 < a < k — 1. That is, we must determine which fcth roots of unity provide a real solution for y in equation 4.12. With this in mind, we rewrite equation 4.12 as

5,2 (l + CO, ( ^ ) ) + ybsin ( * f ) + (l + cm (H H ))

= •' ( y ‘“ n ( n r ) +s,l>( 1_ 003 ( i t )) + ain ( if) ) ■ (413) Therefore,

y2 ^1 + cos + ybsin (-jp ) + ^1 + cos (pp)) = 0 (414) and

„=3in ( ? I £ ) + y i (, _ CM (?p)) + 3m (Hp) = o. (4.15)

When s = 0, equation 4.14 has no real solutions, and equation 4.15 provides no information about y. However, when s ^ 0, we can rewrite 4.14 as

( stn (t*) \ » + M;— H b s » + 1 = 0 (4.i6) 82 since 1 + cos (^f*) ^ 0 for k odd. Similarly, we can rewrite 4.15 as

y2 + b \ . |y+1 = o (4.i7) I «"(t ) I since sin (^f4) / 0 for k odd. Note that equation 4.16 is the same as equation 4.17 because ”"(50 _ 1 -<”*( t ) , / « \ , , ... l+cos(2f) sin (2f«) Vtj when s / 0. Therefore, we must determine for which s

y2 + btan y + 1 — 0 (4.19) gives a real solution for y. Specifically,

-Wan (?) ± • /**„,»(?)- 4 > = ------\ ------— ----- (4.20) and thus y is real for those s satisfying |fan | > | , as claimed.

(b) Since the proof is similar in spirit to that of a), we merely give a sketch. To this end, assume that z = x + iy is a zero of / such that x2 + y2 = 1. Then following the proof of Proposition 4.1.1 up through, equation 4.5, we may conclude that

( {x + iy)2-b(x + iy)-l \ k V-(* + ■»)* - M* + •'») +1/ { } and hence

(e + iy)2 - b{x + iy) — 1 f2 irs\ , . . / 2irs\ - j i + iyj* - m >+ iy ) +1=103 ( t ) +“m ( t) • (4'22) Next, we simplify equations 4.21 and 4.22, and then equate the real and imaginary parts.

x2- y 2 -l+ b x = (-x2 + y2 + 1 + bx)cos - y(~2x + b)sin (4.23) 83 and

2xy + yb = ( - 2 xy + yb)cos + (2jI2 + bx)sin • (4.24)

We can simplify equation 4.23 further by using a:2 + ya = 1. Hence,

- 2y2 + bx = (2y2 + bx)cos ( p p ) + (2 xy - by)sin ( p p ) • (4.25)

We combine 4.24 and 4.25 by multiplying 4.25 through by x , and multiplying 4.24 through by y , and then adding the resulting equations. By repeated use of the fact that x2 4- y2 = 1, we get

b = 6cos + 2ysin ( ~ jp ) (4.26) so that b-bcos(^)

( 4 -2 7 ) or b /ir s \ (4-28) because = tan (x)- case> since x is real and x = ± y /l — y2, we must have 1 — (|tan (x ))2 ^ or alternatively, that |tan (y ) | > | , as desired. □

Note that in b), x = ±1 are zeros (of modulus 1) of /, which arise when s = 0, and in fact correspond to 1, the trivial feth root of unity. In the case when 6=1, it is not hard to show that |fan(^r)| = 2 cannot occur. Hence, there is no overlap in the solutions given by a) and b). Therefore, each fcth root of unity provides two zeros, and thus in total, we have found 2k zeros, which, in fact, can easily shown to 84 be distinct. These account for all of the zeros of / because solving equation 4.5 is equivalent to finding the roots of a polynomial of degree 2k. On the other hand, if for some b we can have |fan(^)| = 5 , then ±i occur as solutions in both a) and b).

Since we don’t know about the multiplicity, we have listed only 2k —2 zeros. It seems quite likely that j

Note also that, except when s = 0, the zeros of modulus 1 obtained in b) come in groups of four, namely {±\/l ~ y»2 ± iy*}. Similarly, the purely imaginary zeros can be grouped in pairs {j/,*, = {±j/ai}, because tan( y ) = —ta n (^ k~ ^ ). This accounts for some of the beautiful symmetry in the graphs of the zeros of functions like /. See, for example, Figure 7 and Figure 8 in Appendix A.

Since the families of RPFs given in 4.1 and 4.2 all satisfy the hypotheses of Propo­ sition 4.1.1, Corollary 4.1.2 essentially provides an explicit description of their zeros.

In fact, the results of this section also apply to the families of RPFs in 4.1 and 4.2 with k a negative odd integer. Under such circumstances, these RPFs are polynomials, but in the computations in Corollary 4.1.1 we replace k with —k.

4.2 A Conjecture

In general, not much else is known about the zeros of RPFs, as most RPFs do not satisfy the hypotheses of Proposition 4.1.1. However, after extensive experimentation with the aid of Mathematica (see Appendix A), and with the results of this section, it appears that the graphs of the zeros of RPFs with at least one hyperbolic pole are highly symmetric. The following open-ended conjecture is suggested. Conjecture. Suppose q is an RPF with at least one hyperbolic pole. If z = x + iy is a zero of qt then either x — 0 or the ordered pair (x, y) is a solution of A x2 + B y2 +

Cx + Dy -f E xy + F = 0, where A, B , C, DtE,F € R. In other words, the zeros of q are purely imaginary, or else lie on the graph of some conic section. Appendix A

Graphs of Zeros of Rational Period Functions

In keeping with the notation of Chapter 4, we write qk,D in place °f H 3 «>°>^ (a^+L+c)^

1 .5

0 .5

- 1 - 0 .5 0 .5 1

- 0 .5

- 1

- 1 .5

Figure 7: The zeros of < 79,5

86 87

Figure 8: The zeros of g19i5 88

0.5-

-1.5 -1 -J.5 0 .a 1.5

-0.5--

■ - 1 + *

Figure 9: The zeros of qg,s + 40(1 — ^rr)

Figure lQi The zeros of qn.s + 40(1 — ^it) Figure 11: The zeros of9 9 ,5 + i( 1 — ^ij-)

i...

0.5--

-7 i l

-0 .5 "

• -i--.

Figure 12: The zeros of 921,5 + *(1 - ^ 7 ) 90

0 J S

-1-

-1.5-

Figure 13: The zeros of

-0.5

-1-

-1.5-

Figure 14: The zeros of 8 91

“2 -1 i 2

-O.S ■

-X -

Figure 15: The zeros of q 3iu

1.5

1

0.5

• • -0.5 ■

-1 ■

-1.5r

Figure 16: The zeros of 95,12 92

a. 5

0". 5

Figure 17: The zeros of 97,13

1 .5 +

- 1 .5 +

Figure 18: The zeros of 919,13 93

0.4+

0 .2+

- 3 -2 -1

- 0 . 2+

-0.4+

Figure 19: The zeros of q 3,5 + 93,17

0 . 6 - 0.4-

0 * 2 -

-2 - 1 - 0 . 2- -0.4-

-0 . 6-1

Figure 20: T he zeros of 99,5 4- 99,8 B ibliography

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