Prime Geodesic Theorems

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8103552

Sa r n a k , P e t e r C l iv e

PRIME GEODESIC THEOREMS

Ph.D.

Stanford University

University Microfilms International

300 N. Zeeb Road, Ann Arbor, M I 48106

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1980

PRIME GEODESIC THEOREMS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

By Peter Sarnak August 1980

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I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

(Princ$>al Adviser)

I certify that I have read this thesis and that in. my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

Approved for the University Committee on Graduate Studies:

Dean of (Graduate Studies and Research

-ii-

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Acknowledgments

I would like to thank my adviser, Professor P. Cohen, for all he taught me over the past few years, and for his encouragement and advice. I would also like to thank Professor R. S. Phillips for his encourage­ ment and the many discussions on and relating to the subject of this thesis.

To my office mate, A. Woo, thanks for all the mathematics

learned and done together. doing the typing.

I would like to thank E. Arrington for

Finally, thanks to Helen for her continual support.

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Table of Contents

0. Introduction ...........................................

1

1. Trace Formulas and Other Preliminaries ...................

7

2. Entropy.................................................. 18 3. Prime Geodesic Theorems for ConstantCurvature ............

37

4. Applications to Number Theory............................. 65 5. Conclusion - Further Questions ..........................

90

6. Appendix................................................ 91 References..............................................

-iv-

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. 96

1

INTRODUCTION A lot of work has been done in recent years concerning closed geodesics on Riemannian manifolds.

In the case that the manifold has

a "large” fundamental group such as a surface of genus

g £ 2,

the

existence of infinitely many such geodesics is easily demonstrated by exhibiting one in each free homotopy class.

The work referred to above

is concerned with showing the existence of infinitely many prime geo­ desics on any compact, closed manifold. It is our aim in this thesis to study the asymptotic distribution of the lengths of periodic geodesics, and consequences thereof.

The

results we obtain have interesting applications to number theory as well as to the behavior of eigenvalues of the Laplace Beltrami operator on Riemannian manifolds. There are three different methods of getting hold of the closed geodesics.

The first is to use the fact that these geodesics are

critical points of the energy integral on a path space.

Secondly, by

viewing these geodesics as periodic orbits of the geodesic flow, one may use methods of topological dynamics.

Finally, in the case of constant

curvature, the lengths of the closed geodesics can be identified through exact trace formulas such as the Selberg Trace Formula.

We will be con­

cerned mainly with the latter two methods, as they are more useful as far as asymptotics of the lengths is concerned. The relevant trace formulas are introduced in Chapter 1.

These

include the Poisson Summation Formula, Selberg Trace Formula, and what we call the Duistemaat-Guillemin summation formula. viewed as a computation of the trace of the operator

These may all be exp(i/At) in two ways.

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2

We also introduce in this chapter the notions of the geodesic flow and the different entropies associated with it. Let r(y)

c Pq

be the set of closed geodesics on

be its length. For

x > 0 define

We are interested in the behavior of

M.

For y e cP^

ir(x) = #{y e cPQ: t(y) $ x}.

tt(x )

as x -+■ °°.

We know it(x)-»•»,

as "x-*-00 for "almost" any manifold (Klingenberg [18]). that

tt(x )

=«>

for some

x

let

(e.g., the two sphere).

It may happen

However, in the

case of a Riemannian manifold with all sectional curvatures negatives, we have the following remarkable result of Margulis [22].

Theorem: tt(x ) ~ ----------Here tt(x )

h

..hx C0 x

for constants

is the topological entropy.

c

and

h.

An asymptotic expansion for

will be called a Prime Geodesic Theorem. In view of this prime geodesic theorem, it is of interest to study

the dependence of the entropy on the geometry of the manifold.

In

Chapter 2, monotonicity and continuity theorems aTe proved for the topological entropy.

If

X

is the "bottom" of the spectrum of the

Laplacian as it acts on the universal covering of the Riemannian mani­ fold whose entropy we are considering, then the following is true

X

h2 £ ~2~ ,

where h = topological entropy of the geodesic flow.

In the case of constant curvature, these are equal, but they need not be so in general.

We also prove the following estimates for the

"natural measure" entropy

h^:

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3

(n-1) i /M / = F 0 0 dA(x)

Cn-1) (£ /M -k’(x)dA(x))% .

In particular for a surface of genus

^

/T(xT dA(x) $

g £ 2

f ( ^{g-l) )

(See Theorem 2.3 for definitions of

k+,k“.)

k = curvature .

In constant curvature all

of the above are equal. Together with h;

h i h^

the above gives us another lower bound for

in terms of the geometry. When the curvature is constant, say equal to minus 1, and we are

in dimension two, we have the prime geodesic theorem.

h= 1 and we may seek higher order terms in In the case of a compact surface (closed),

Selberg and later Hejhal andHuber, determined such higher order asymptotics.

The situation, in the noncompact but finite volume sur­

face, is complicated by the presence of continuous spectrum.

By methods

different to the above authors we prove the prime geodesic theorem for finite volume surfaces.

Our method consists of first obtaining some

estimates on the spectrum of the Laplace Beltrami operator via partial differential equations, and then applying a chosen one parameter family of test functions.

This method avoids the use of the Selberg

Zeta function; however, that method can also be made to work, as is indicated in Selberg [27 ]. If

0 = Aq < Aj $ ... £

the discrete eigenvalues of Aon the surface, and if m = 0,...,k,

< 1/4

are

it = / k m m

-"l/4,

then the prime geodesic theorem reads

\

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4

tt(x

)

where

= Li(ex) + Li(e(1/2 + tj)) + ... + Li(e(1/2 + tk)X) + 0(x2 e3/4 X )

fu Li(u) = J

1

t dt.

We continue Chapter 3 by discussing the asymptotics of the lengths of prime geodesics where we impose conditions on the primes to be counted.

Indeed, this part is analogous to prime ideals of algebraic

number theory and their splitting in extension fields.

Ke develop the

following analogue of the Dirichlet-Chabotarev density theorem. S

be a closed compact surface of constant curvature -1.

a finite regular cover of transformations of the surfaces jugacy

over

degree S.

G,depending

on

n. Let

G

W

be

be the group of cover

Let us call a prime geodesic on any of

simplya prime. Toeach

class in

prime induces.

W

S

Let

Let

prime on

S weassociate a con-

the cover transformation that this

For a fixed class

C

of G we define

the number of primes which induce

C

andwhose length is less than

We obtain the asymptotics of

tt^Cx),

(up to order

in terms of the irreducible representations of

G.

x"

to be

e

x.

^4 X as usual)

If one view the

above in terms of Selberg Zeta functions, one finds the analogue of the Artin conjectures is true. We conclude Chapter 3 by making some eigenvalue estimates which are needed later. that is,

G(y)

z

Specifically, let

G(y)

be one of the "Hecke groups",

is generated by 1 --2

z -+■z + y

« rr

y = 2 cos (-) , q

q £ 3,

q e 2 .

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5

G(y)

is a discrete subgroup of

half plane

H.

For the surface

discrete eigenvalue for

q=3

PSL(2,]R) and acts on the upper U S = we prove that the first H »Jy

X^

is not less than one quarter.

is well known.

This theorem

The prime geodesic theorems of this chapter

may be developed in hyperbolic n-space by similar methods. The next chapter deals with applications to number theory.

The

connection comes when one tries to find the lengths of the prime geodesics for the surfaces subgroup of

PSL(2,Z),

where of level

N.

F(N)

is the principal congruence

In these cases we have a nice

interpretation in terms of number theoretic quantities. Let

D = {n

e Z, n > 0,

For

d e D,

let e ^

equation,

2

x - dy

2

2

or 1 mod 4, n i m }.

be the fundamental solution of the Pellian Let

-A .

n 5 0

h(d)

be the number of inequivalent

classes of primitive quadratic forms of discriminant

d.

The lengths

of the periodic geodesics in the case of the modular group are the numbers

2 log

with multiplicity h(d),

The numbersh(d)

and

log

d e D.

have been a mystery ever

since

Gauss introduced them in Section 304 of his Disquestiones Arithmetica. Gauss, seeking to understand the behavior of these class numbers formed averages of

h

over the sets

{l,2,...,x} HD,

that these have no regular behavior as a function of out that

00 =

\ h(d) log e, de x d This was later proved by Siegel:

2

but noticed x.

He did point

does have an asymptotic expansion.

3/

0>00 ® TgfcsJ x

h(d),

0(x log x^ •

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6

Also through the work of Siegel we understand the behavior of h(d) log

(although not effectively).

separate the quantities the averages of

h(d)

h(d)

and

It seems difficult to

log ed. In Chapter 4 we compute

over the exhausting family of sets

{d: ed £x}.

The results may be summarized as follows:

(i)

I

h (d )

= Li(x2) + L(x3/2 ( l o g x)2)

{dre^Sx}

(ii)

-- ±— — I h(d)=i|Ml2Ll+ |{d:e,£x}| {d:e,£x}

o(x2/5 + e ),

Ve > 0

(i) follows from our considerations about closed geodesics, while (ii) follows from (i) and some arguments on asymptotics of the number of solutions to a diophantine inequality. The above shows that on the average the class number is about the size of the unit.

The question of the finer behavior of

h(d),

raised

by Gauss, seems as difficult today as it has been for many years. Similar considerations for to asymptotic averages of

h(d)

T(p)

where

p

is a prime, give rise

over certain subsets of

precise statements may be found in Theorem 4.34.

The

The asymptotics in

these cases are closely related to the conjecture that the groups

D.

X^ £ 1/4

T(p).

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for

7

CHAPTER 1.

TRACE FORMULAES AND OTHER PRELIMINARIES

In this chapter we will state and give outlines of proofs of various summation formulas that are basic to our later work.

We

begin with the simplest of these.

1.1

Poisson Summation Formula If

SQR)

denotes the Schwartz class on F,

f(n) =

I

neZ

I

x,

f e SQR), then

f(n) .

neZ

In the language of distributions if 6x = point mass at

and

S =

T