Number Theory and Geometry: An Introduction to Arithmetic Geometry

Most of the number theory courses I’ve taught follow a pretty standard progression—divisibility, prime numbers, congruences—all building up to a proof of the quadratic reciprocity law (with a few additional independent topics thrown in at the end for good measure). While there is a lot of beautiful material to discuss, it often feels as though we are studying number theory as a historical artifact. Even if students are actively engaged in “discovering” proofs of major theorems, we end up where Gauss left off over 200 years ago. In Number Theory and Geometry: An Introduction to Arithmetic Geometry, standard topics are covered with the unifying goal of finding rational (or, at times, integral) points on curves. These kinds of problems can be traced back to the Greek mathematician Diophantus in his series of books called Arithmetica, and they have continued to inspire an active area of research today. While other texts such as Burton’s Elementary Number Theory do a good job explaining the historical origins of topics, few incorporate the connections to modern research to the extent found in Number Theory and Geometry. In addition, many of the chapters include a lengthy “Applications” section which highlights real-world uses of the material and connections to other mathematical topics. The result is a text which portrays number theory as a field not only with a long and rich history but also with continued relevance in the present day. 

 

The text is divided into three sections which are motivated by the problems of finding points on linear, quadratic, and cubic curves. A one-semester course would typically cover material only from the first two sections, with enough additional material for a second semester-long course in introductory arithmetic geometry. The connection between course content and the overarching theme of finding points on curves is given at the start of most chapters, in key examples within the chapter, and in exercises at the end of each chapter. Though the motivation is different, the standard topics one would expect in an elementary number theory course do appear, with the exception of arithmetic functions. One difference to highlight is that the author includes a chapter on groups, rings, and fields (and an optional chapter on finite fields) immediately after the chapter introducing congruences. On the one hand, this has the potential of streamlining later proofs (such as that establishing the Euler phi function is multiplicative) and providing better continuity for students who have already taken or will take a modern algebra course. However, this also means students will have to grapple with a higher level of abstraction in some of the later chapters than they might in a course based on Burton’s Elementary Number Theory or Marshall, Odell, & Starbird’s Number Theory Through Inquiry. Thus I think this text would be best suited for a course primarily geared towards math majors.

 

Another noticeable difference from other texts I’ve used is the large number of detailed examples and tables included in the sections which serve both to motivate and to illustrate theorem statements. I could see these providing good jumping-off points for in-class discussions, perhaps being given as assigned reading to prepare students for class. Overall, these examples serve to better motivate topics than in other texts I’ve used. For instance, Burton’s Elementary Number Theory and Montgomery, Niven, & Zuckerman’s An Introduction to the Theory of Numbers introduce the section on primitive roots with the definition of order. In Number Theory and Geometry, the chapter on primitive roots begins with an example about the length of the period of a rational number—a topic discussed by Gauss as an application of primitive roots. Before the definition of order, we find a table of powers of congruence classes modulo 7, and in the discussion that follows we see the connection to Fermat’s Little Theorem (covered in the previous chapter) and the observation that some classes reach 1 sooner than others. Even those not interested in adopting the book as their primary course text may find these examples provide useful inspiration when crafting lectures or in-class activities.

 

Overall, I think this book would be a great foundation for a course which is more inspiring—and perhaps more challenging—than your standard course on elementary number theory.

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Abbey Bourdon is an Assistant Professor at Wake Forest University. Her website is here.