Diophantine Geometry: An Introduction
Marc Hindry and Joseph H. Silverman
Springer-Verlag – Graduate Texts in Mathematics 201
ISBN: 13: 978-0387989815
– 1st ed.
– © 2000
– 561 pages
Diophantine geometry is the study of integral and rational points to
systems of polynomial equations using ideas and techniques from algebraic
number theory and algebraic geometry. The ultimate goal is to
describe the solutions in terms of geometric invariants of the underlying
algebraic variety. This book contains complete proofs of four of the
fundamental finiteness theorems in Diophantine geometry.
-
The Mordell-Weil theorem —
The group of rational points on an abelian variety is finitely generated.
-
Roth's theorem —
An algebraic number has only finitely many approximations of order
2+ε.
-
Siegel's theorem —
An affine curve of genus at least one has only finitely many integral
points.
-
Falting's theorem (Mordell conjecture) —
A curve of genus at least two has finitely many rational points.
Also included are a lengthy overview (with sketched or omitted proofs)
of algebraic geometry, a detailed development of the theory of height
functions, a discussion of further results and open problems, numerous
exercises, and a comprehensive index.
Contents
- Part A: The geometry of curves and abelian varieties
- Part B: Height functions
- Part C: Rational points on abelian varieties
- Part D: Diophantine approximation and integral points on curves
- Part E: Rational points on curves of genus at least 2
- Part F: Further results and open problems
Errata List for Diophantine Geometry
No book is ever free from error or incapable of being improved. We
would be delighted to receive comments, good or bad, and corrections
from our readers. Please send mail to
jhs@math.brown.edu
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