The subject matter of this book is wonderful, and so is the book itself. Indeed, the author wisely plugs bigtime — both in her title and throughout her book — into the most wonderful thing about the subject — that it’s an “alternative” to real analysis. Thus all the formulas and theorems in real analysis have analogous formulas and theorems in p-adic analysis. Briefly: in real analysis we complete the normed space of rationals, taking the norm to be the usual absolute value. Why not do the same “completion thing” using some other norm or norms? There are as many of these “new” normed spaces as there are primes, and for each prime we denote our completion by Qp.
It doesn’t take long to define what these “new” p-adic norms are, and Katok does this on page 20 (after wisely including the basics of metric spaces and their completions). It then becomes more and more apparent that there are two salient facts that often make p-adic analysis “weird”: (1) The norms have discrete values and (2) they’re non-Archimedean (and therefore satisfy the Strong Triangle Inequality).
There are many places where the author does the kind of thing I like — namely, use language, sometimes colloquial, to illuminate her points. For example, she often, throughout the book (pp. 76, 77, and 118, for example) dubs various consequences of the strong triangle inequality “the strongest wins” (although she doesn’t do this on p. 11, where she first introduces the strong triangle inequality; I think it would be great if she did).
Among my favorite parts of the book are p. 43 and p. 47, where she answers two questions that had immediately occurred to me when I first read her definition of p-adic norms: (1) Are there any other norms on the rationals that we can play with (and compare to real and p-adic)? And (2) What if p isn’t prime? (However, I’m having trouble seeing why the usual absolute value norm is supposed to correspond to p = ∞.)
It’s clear, and commendable, that the author does intend to give the “gyst” of most of her proofs (though not all) and she also gives appropriate and motivating (and curiosity-satisfying) counterexamples — for instance, on p. 118 — A differentiable function with never-vanishing derivative but not “injective at 0” — meaning, as she puts it, not injective in any neighborhood of 0 — so f has no inverse. This, she points out, is different from real analysis — and again commendably, she almost always takes care to let us know the differences between the results of p-adic vs. the real analysis counterparts.
However, I felt that her very-first mention, and definition, of the p-adic norm (p. 20), was unsatisfying. She gives the technical definition, then immediately embarks on remarks, propositions, etc., without first giving the reader a sense of what the initial definition means in plain English. This to me feels like a missed opportunity. If it were my book, I would add, immediately, to that technical definition: Every rational number x is the quotient of two integers, each of which has a prime decomposition. If we write x ‘in lowest terms’, we of course can have a power of p in either only the numerator or only the denominator, not both. Well, ord(x) tells us “how much p is in the numerator‘ — which includes the idea that if p turns out to appear in the denominator, then we ‘make ord(x) negative’. And then the “actual” p-adic norm of x (as opposed to the just-defined ord(x)) would be p to negative that power — unless, of course, x = 0, in which case our norm has to be 0, since that’s what happens with all norms. Thus we can say, rather intuitively, that the p-adic norm of x gives us the absolute value of the reciprocal of “the p-part’ of x.”
< p > Perhaps that’s cumbersome but it’s a good “optional” description. It isn’t technically necessary
to do that, and students can figure it out for themselves, but they probably wouldn’t while listening to a lecture. but to me it adds to the beauty (and I wonder whether, in her classes, the author does indeed do it that way, or some other intuitive way.) Similarly, I would give at least a few quick examples of sequences that converge to 0 in the p-adic norm.
But it’s only on pp. 20-21 that I notice this phenomenon. In general, the author is — happily — not afraid to state the obvious when that’s indicated. She also often does a beautiful job of stating and clarifying the non -obvious..
I’ll end with some further wonderful differences between p-adic and real. I counted fifteen “basic” ones; I’ll present my favorite five:
- All open balls are also closed.
- There are only a countable number of open balls in Qp.
- A series converges if and only if the sequence of its terms converges to 0. (Ah, how simple real analysis would be if it could boast such a theorem! On the other hand, the theory of p-adic series is not trivial!)
- A convergent series always converges unconditionally.
However, lest we conclude that life in p-adic lanes is easy street: The power series corresponding to exp(x) does not converge everywhere. In fact, it doesn’t even converge for x = 1, so there is no p-adic analogue of e. Perhaps we should count our blessings that the “usual” norm is not p-adic, that p-adic is only optional, for those mathematicians like the author, and like me, who love to delve into the unusual.
Marion Cohen has a new book of poetry about the experience of math, Crossing the Equal Sign. She teaches part time at Arcadia University. Check out her other writings and math limericks on her site marioncohen.com, and email her at: firstname.lastname@example.org