There’s a fantastic video online titled “How to write mathematics badly,” the speaker being none other than Jean-Pierre Serre, certainly among the very best mathematical expositors. The video is very funny, even as it strikes at the heart of some common problems with mathematical writing in this day and age, so it’s unquestionably pedagogically useful. But Serre is obviously also enjoying himself no end, poking fun at, for instance, at the rampant tendency to present findings in as abstract (and bizarre) a language as possible.
Relative to the book under review, then, the point to be taken is that the reader is in store for a multiple treat: with Serre wielding the pen, the mathematics is exquisite and the presentation is wonderful. In fact, the book is something of a paradigm regarding the message Serre tries to convey in “How to write mathematics badly.” In this book, although the mathematics brought to bear on the thesis is potentially quite austere, there is a constant push in the direction of Serre’s specific goal, viz. the final observations and conclusions concerning NX(p) are made very clearly and even concretely, and the surrounding motivation is outstanding. The development of the background mathematics and methodology is crystal clear. Even when Serre purposely suppresses proofs; the surrounding landscape is drawn with the right amount of detail. Of course, one does expect this from Serre, whom I often regard as being incapable of phrasing things in any way other than clearly and concisely.
We start with a definition: if X is, say, an algebraic variety, then NX(p) is the number of Fp-points of X, i.e. the number of closed points of X at which the according residue field has p elements. In less arithmetic-geometric terms (i.e. in more old-fashioned number theoretic language), NX(p) is the number of solutions mod.p of the finitely many polynomials (over Z) defining the variety X. (A propos, Serre right off the bat presents the arithmetic geometric characterization of NX(p) in terms of schemes rather than varieties: we want the full machinery of modern algebraic geometry and its Nachlass from Grothendieck). The explicit goal of the book is to describe NX(p) as a function of p, and the book begins with Serre posing the leading questions: “Can it be computed by closed formulae, by cohomology, and/or by efficient computer programs?”
Next, and this may be the most pedagogically exciting feature of the book, Serre also takes these questions as “a good opportunity for reviewing several basic techniques in algebraic geometry, group representations, number theory, cohomology (both l-adic and standard) and modular forms.” It doesn’t get any better that this! Serre treats us to a discussion of half a dozen major mathematical movements, so to speak, set against the background of a very beautiful problem, replete with a number of historical connections.
For example, Section 1.5 already presents “[t]he zeta point of view” and Chapter 3 deals with Chebotarev density. Thereafter Serre presents a good deal of cohomology (of both flavors he indicated above, but with l-adic cohomology certainly in the foreground) and representation theory, which certainly strikes a personal chord with me given that I once studied his classic Linear Representations of Finite Groups with great joy — caveat: the representations that matter (cf. Chapter 6) in the book under review often deal with profinite groups, e.g., the absolute Galois group of an algebraic closure of the rationals.
He then closes the book with discussions of Sato-Tate, the prime number theorem (for schemes of finite type over Z; this does include the classical case, of course), and, again, Chebotarev density. These results are stunningly beautiful: for example, for an irreducible scheme of finite type and of dimension d, the number of points t whose residue field is finite and for which |t| < x, is asymptotic to xd/dlog(x) as x grows without bound.
Yes, this is another terrific book by Serre: it provides a splendid introduction to both a beautiful arithmetic (-geometric) theme and hugely important mathematical methods pertaining to the given theme. It should tantalize the reader and move him to go into these themes in greater depth, using Serre’s exposition as a high-level road map.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.