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A Gateway to Number Theory: Applying the Power of Algebraic Curves

Keith Kendig
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions
[Reviewed by
Holley Friedlander
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A Gateway to Number Theory: Applying the Power of Algebraic Curves by Keith Kendig highlights a beautiful interface of number theory and geometry. Specifically, the book focuses on techniques for solving homogenous Diophantine equations of degree less than or equal to three and the fruitful method of translating a “number theory problem into one of finding rational points on a rational curve.” At the heart of the exposition is a rich set of results and open questions arising in the theory of elliptic curves. As a unique feature, the book also calls attention to the ways in which computation has played a role in advancing the theory. Kendig’s book is an earnest endeavor to make this vast area, which spans ancient through cutting-edge modern mathematics, accessible to a wide audience.
The first chapters of the book introduce how techniques for solving homogeneous Diophantine equations of degree one and two arise from the geometry of the associated algebraic curves. The primary focus is on degree two equations and the associated conics. Chapter two has a particularly nice description of the disc model of the real projective plane, complete with many examples to help the reader build intuition. The next several chapters focus on degree three equations and elliptic curves. Chapters three and four introduce the theory of rational points on elliptic curves and discuss major results including Mordell’s theorem and Mazur’s torsion theorem. Chapter four also includes a quick primer for getting started with the L-functions and modular forms database (LMFDB). Chapter five explains the Birch and Swinnerton-Dyer conjecture and current progress on the problem. Chapters 6 and 7 move to the complex setting and discuss elliptic curves, hyperelliptic curves, and Falting’s theorem, and the final chapter summaries the main results from the book.  
The preface of the book is forthright that although the early chapters require only a background in high school math, the later chapters are more easily digested if one has had some exposure to linear algebra, group theory, topology, and complex analysis. In general, the level of the exposition is a bit inconsistent throughout the text, even within chapters, but for those willing to “black box” some terms or seek out one of the many references provided in the text, this may be only a minor annoyance. 
The book is marketed as suitable for an undergraduate capstone course, and with the right supplementation, it could work as such. In that vein, this would be a good book for an independent study, where one can pick and choose what topics to explore further depending on the student's background. Each chapter includes many worked examples and figures to illustrate the key ideas and theorems, and at the end of each section, there are a few well-chosen exercises. About half of the exercises from the text are computational in nature, and the appendix has suggested code to get readers started on these exercises in their computing program of choice. 
Kendig promises the text to be an “informal and leisurely” introduction to the interplay between number theory and geometry, and it is both. Even with background in the subject, I found the book to be a surprisingly quick read. For those who prefer to not get bogged down in technical definitions or the details of more complicated proofs and would instead like to focus on the big picture, this is a nice book. Those looking for a more rigorous treatment of the subject may enjoy A Gateway to Number Theory as a companion to Silverman and Tate’s Rational Points on Elliptic Curves or Silverman’s The Arithmetic of Elliptic Curves.
Holley Friedlander is an Assistant Professor of Mathematics at Dickinson College.