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 leaders through
the mathematics they need to understand this captivating problem.
The key in 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.
This book explains the Birch–Swinnerton-Dyer Conjecture
in the mathematical field of elliptic curves.
We get 14 chapters of background before the conjecture is stated in chapter 15;
by that point we have learned a wide range of interesting mathematics,
and are in a position, if not to fully understand,
at least to appreciate the Conjecture and its importance.
The route to this point covers a lot of ground.
Each new piece of mathematics introduced is (relatively!) straightforward,
but by the end, there is just so much machinery in play,
that it all becomes a little overwhelming.
That feeling is good for understanding just how deep this Conjecture is.
I was reminded of things I learned long ago,
and learned lots of new interesting pieces of mathematics:
an algebraic definition of the projective plane,
how points on a curve can have a group structure,
group generators, analytic continuation, series expansions, and much much more.
What is great about this book is the way each new piece
is slotted into the picture with a route map of where each chapter is going,
explanations of how the pieces fit,
and explanations of the importance and meaning of the concepts.
There are exercises along the way, of a form that deepens understanding,
there for the serious reader.
I was more of a visitor, looking at the interesting details,
frankly skimming a few of them, but not putting in the work needed to live there.
But I had a good time as a tourist.
