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The theory of elliptic curves involves a pleasing blend of algebra,
geometry, analysis, and number theory. This volume stresses this
interplay as it develops the basic theory, thereby providing an
opportunity for advanced undergraduates to appreciate the unity of
modern mathematics. At the same time, every effort has been made to
use only methods and results commonly included in the undergraduate
curriculum. This accessibility, the informal writing style, and a
wealth of exercises make Rational Points on Elliptic Curves an
ideal introduction for students at all levels who are interested in
learning about Diophantine equations and arithmetic geometry. Most
concretely, an elliptic curve is the set of zeroes of a cubic
polynomial in two variables. If the polynomial has rational
coefficients, then one can ask for a description of those zeroes
whose coordinates are either integers or rational numbers. It is
this number theoretic question that is the main subject of Rational
Points on Elliptic Curves. Topics covered include the geometry and
group structure of elliptic curves, the Nagell–Lutz theorem
describing points of finite order, the Mordell–Weil theorem on
the finite generation of the group of rational points, the
Thue–Siegel theorem on the finiteness of the set of integer
points, theorems on counting points with coordinates in finite
fields, Lenstra's elliptic curve factorization algorithm, and a
discussion of complex multiplication and the Galois representations
associated to torsion points. Additional topics new to the second
edition include an introduction to elliptic curve cryptography and
a brief discussion of the stunning proof of Fermat's Last Theorem
by Wiles et al. via the use of elliptic curves.
Additional Contributors Include H. A. Bethe And R. F. Bacher.
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