Preface

This book grew out of a set of notes from a graduate course in which I tried

to introduce the students, in a single course, to both algebraic number theory

and the theory of curves. Most books in the literature that touch upon both

subjects discuss the function field case from a very algebraic point of view. The

modern language of arithmetic geometry is very geometric, and I hope that it

will be useful to have in the literature a book such as this with a more geometric

introductory presentation of the function field case,

Let us define arithmetic geometry in this preface to be the study of the solu-

tions in

kn

of a system of polynomial equations in n variables with coefficients

in a ring k (such as k = Z, k = Q, or k = Z/pZ). While the central problem

of arithmetic geometry is thus easily described, mastering the powerful tools de-

veloped in the last thirty years to study solutions of polynomials is extremely

challenging for

beginners1.

A student with a basic knowledge of algebra and Ga-

lois theory will first have to take a course in algebraic number theory, a course

in commutative algebra, and a course in algebraic geometry (including scheme

theory) to be able to understand the language in which theorems and proofs are

stated in modern arithmetic geometry. Moreover, a student who has had the

opportunity to take those three courses wilL be faced with the additional hurdle

of understanding their interconnections.

An Invitation to Arithmetic Geometry tries to present in a unified manner,

from the beginning, some of the basic tools and concepts in number theory, com-

mutative algebra, and algebraic geometry, and to bring out the deep analogies

between these topics. This book introduces the reader to arithmetic geometry

by focusing primarily on the dimension one case (that is, curves in algebraic

x An example of a polynomial equation in two variables is the Fermat equation xn +yn = 1.

The celebrated Fermat's Last Theorem states that the only solutions in rational numbers to

the equation xn + yn — 1 are the "obvious" ones if n 2. Ribet's proof, in 1986, that Fermat's

Last Theorem holds if a weak form of the conjecture of Shimura-Tanyiama-Weil holds is an

example of modern tools in arithmetic geometry producing new and deep results about easily

stated problems in number theory. Wiles announced on June 23, 1993, that he can prove

this weak form of the Shimura-Tanyiama-Weil conjecture and, therefore, that Fermat's Last

Theorem holds [Rib], [Fer].

xiii