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
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