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.