INTRODUCTION

The main objective of this thesis is to prove the following two theorems.

Theorem 1. If m is sufficiently large, then there exists a lattice with trivial automorphism

group (i.e. {±1}) in every genus of positive definite unimodular integral lattices of dimen-

sion m. More precisely, (i) the assertion holds if m 43 for odd unimodular lattices, (ii)

the assertion holds if m 144 for even unimodular lattices.

Theorem 1 is obtained from the following stronger theorem.

Theorem 2. Let w be the mass of the given genus of positive definite unimodular lattices

of rank m and let a/ be the mass of all the classes in the given genus with nontrivial auto-

morphisms. Then the ratio of the mass

UJ'/LJ

is bounded above by

30(\/27r)m/r(y)

for odd

unimodular lattices of dimension m 43 and by 2

m + 1

(V^)

m

/r(y) for even unimodular

lattices of dimension m 144. In particular, this ratio

UJ'/LJ

approaches 0 very rapidly as

m increases.

Here is a brief historical background concerning this problem. The existence of lattices

with trivial automorphism group is known. O'Meara [9, 1975] gave an algorithm to con-

struct such a lattice starting from any given lattice. In this process the discriminants of the

lattices increase in each step. Biermann [1, 1981] proved the existence of a lattice with triv-

ial automorphism group in every genus of positive definite integral lattices of any dimension

with sufficiently large discriminant. In his proof the fact that the discriminant is very large

is crucial. We are, instead, interested in lattices with small discriminant. It seems that the

existence of any unimodular lattice with trivial automorphism group has not been known.

This was, however, anticipated in [6]. For a treatment over localizations of polynomial rings,

see [13].

On the other hand, Watson [17, 1979] has shown the existence of an indecomposable

lattice in every genus of positive definite integral lattices if the dimension m 14. Clearly

a lattice having trivial automorphism group is indecomposable, but not vice-versa. But his

idea of estimating the mass of decomposable lattices in the given genus is, however, very

Received by the editors January 31, 1989.

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