Elliptic Tales: Curves, Counting, and Number Theory

The author's main challenge is to explain in less than 250 pages a very
technical yet interesting conjecture, in a comprehensible but also appealing
way. They introduce the technical details just before to need them. This print
in their pages a quick rhythm.

Part I presents the very basic questions concerning this topic such as the
projective plane or some definitions and facts in Algebra. Authors do not look
for the treatment by the usual sequence definitions and results with proofs;
they focuses on the intuitions behind the mathematical concepts and results
without any loss of rigour. The numerous examples (accompanied by some
pictures) makes this lecture suitable for undergraduate students. However,
although a graduate students can safely omit this part, it is so well written
that it is nonetheless a very entertaining and illuminating reading.

The Part II is devoted to the elliptic curves, the main topic of this book.
Besides the definition and the group law in elliptic curves, this part prepare
to the reader to understand the meaning of rational points and the torsion
points until reach the Mazur theorem.

The Part III focuses on the generating functions, Dirichlet series and finally
they introduces the Zeta functions. Stimulating examples and deep applications
is given to illustrate these concepts. This leads to define more difficult
concepts such as the Hasse-Weil Zeta function and the L-function on a
elliptic curve and prepare to the reader for deals with the
Birch-Swinnerton-Dyer conjecture. At the end of this chapter, Tunnell's theorem
gives the beautiful example, assuming the BSD conjecture.

This book has many nice aspects. Ash and Gross give a truly stimulating
introduction to elliptic curves and the BSD conjecture for undergraduate
students. The main achievement is to make a relative easy exposition of these
so technical topics. Besides, this is more impressive taking into account
that authors do not avoid some technical details, and they are treated as an
insertion in many cases without breaking the main discourse of the book. The
reader will not avoid making its own examples and the proposed exercises.

In conclusion, the authors' ambitious project is done in many aspects: an
undergraduate student could read this book with a bit effort, but this exciting
topic so well introduce in this book becomes such effort into a pleasure.

Jonathan Sánchez-Hernández
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