*Elliptic Tales* describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics—the Birch and Swinnerton-Dyer Conjecture. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.

The key to the conjecture lies in elliptic curves, which may appear simple, but arise from some very deep—and often very mystifying—mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and, in the process, venture to the very frontiers of modern mathematics.

"The authors present their discussion in an informal, sometimes playful manner and with detail that will appeal to an audience with a basic understanding of calculus. This book will captivate math enthusiasts as well as readers curious about an intriguing and still unanswered question."—Margaret Dominy, *Library Journal*

"Minimal prerequisites and its clear writing make this book (which even has a few exercises) a great choice for a seminar for mathematics majors, who at some point should have such an excursion to one of the frontiers of mathematics."—*Mathematics Magazine*

"The authors of *Elliptic Tales* do a superb job in demonstrating the approach that mathematicians take when they confront unsolved problems involving elliptic curves."—Sungkon Chang, *Times Higher Education*

"One cannot help being impressed, in reading the book and pursuing a few of the references, by the magnitude of the enterprise it chronicles."—James Case, *SIAM News*

"Ash and Gross thoroughly explain the statement and significance of the linchpin Birch and Swinnerton-Dyer conjection. . . . [A]sh and Gross deliver ample and current intellectual and technical substance."—*Choice*

"I would envision this book as an excellent text for an undergraduate 'capstone' course in mathematics; the book lends itself to independent reading, but topics may be explored in much greater depth and rigor in the classroom. Additionally, the book indeed brings together ideas from calculus, complex variables and algebra, showing how a single mathematical research question may require an integrated understanding of the various branches of mathematics. Thus, it encourages students to reinforce their understanding of these various fields, while simultaneously introducing them to an open question in mathematics and a vibrant field of study."—Lisa A. Berger, *Mathematical Reviews Clippings*

"The book is very pleasantly written, and in my opinion, the authors have done an admirable job in giving an idea to non-experts what the Birch-Swinnerton Dyer conjecture is about."—Jan-Hendrik Evertse, *Zentralblatt MATH*

"The book's most important contributions . . . are the sense of discovery, invention, and insight into the habits of mind used by mathematicians on this journey. I would recommend this book to anyone who wants to be challenged mathematically or who wants to experience mathematics as creative and exciting."—Jacqueline Coomes, *Mathematics Teacher*

"[T]his book is a wonderful introduction to what is arguably one of the most important mathematical problems of our time and for that reason alone it deserves to be widely read. Another reason to recommend this book is the opportunity to share in the readily apparent joy the authors have for their subject and the beauty they see in it, not least because . . . joy and beauty are the most important reasons for doing mathematics, irrespective of its dollar value."—Rob Ashmore, *Mathematics Today*

"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."—Jonathan Sanchez-Hernandez, *Mathematical Society*

"A carefully thought out presentation."—Danny Yee, Danny Reviews

"Assuming only what every mathematically inclined freshman should know, this book leads the reader to an understanding of one of the most important conjectures in current number theory—whose proof is one of the Clay Mathematics Institute's million-dollar prize problems. The book is carefully and clearly written, and can be recommended without hesitation."—Peter Swinnerton-Dyer, University of Cambridge

"The Birch and Swinnerton-Dyer Conjecture is one of the great insights in number theory from the twentieth century, and Ash and Gross write with care and a clear love of the subject. *Elliptic Tales* will have wide appeal."—Peter Sarnak, Princeton University