Elliptic Curves

Many years ago, when I was an undergraduate (I believe), I first came across Husemöller’s name as half of the entity “Milnor-Husemöller,” i.e., as belonging to one of the co-authors of Symmetric Bilinear Forms, a wonderful book — eminently readable (as books Milnor has something to do with usually are). Somewhat later, in graduate school, I encountered Husemöller’s name again, now as the author of the important book, Fibre Bundles. And now, being considerably longer in the tooth, I have the pleasure of reviewing another book by Husemöller, titled Elliptic Curves. This is quite a departure from algebraic topology, to put a label on how Husemöller’s aforementioned works should be classified. (To strengthen this case, however, just consider the fact that as recently as 2008 Husemöller appeared as co-author of Basic Bundle Theory and K-Cohomology Invariants (Springer Lecture Notes in Physics)). Indeed, elliptic curves do not spring to mind immediately.

But here it is, and the second edition, in fact, and a Springer GTM entry. And, even though, consonant with the musings above, Husemöller starts out his “Acknowledgements to the First Edition” with the modest phrase, “Being an amateur in the field of elliptic curves…,” it’s a wonderful piece of work. But one would expect no less from an author whose expository skills are so well-known and appreciated (as illustrated by the above offerings).

Moreover, continuing with the “Acknowledgements,” we learn from Husemöller that “John Tate’s treatment of an advanced subject, the arithmetic of elliptic curves, in an undergraduate context has been an inspiration for me during the last 25 years while at Haverford” (this writing being dated to the middle 1980s). He goes on to say that “[t]he general outline of the [present] project, together with many of the details of the exposition, owe so much to Tate’s generous help.” Well, this is splendid, of course, since every one who has even a passing interest in elliptic curves knows about Tate’s fantastic work and it looks like it is something of a prelude to what Husemöller is doing. This is very exciting.

And it gets even better. Evidently this book had a long incubation period during which Husemöller was exposed to lecture courses by Serre, and discussed the subject with not only Serre but also with Deligne, Katz, Lichtenbaum, Mazur, and, somewhat later, with Harder, Ogg, and Gross.

Beyond this, specifically as regards the second edition, Husemöller notes that his own interest in the subject of elliptic curves was revived in 1998 due to the fact that “Stefan Theisen, during a period of his work on Calabi-Yau manifolds in conjunction with string theory, brought up many questions in the summer of 1998 which led to a renewed interest in the subject of elliptic curves on my part.” Husemöller then notes, also, that Otto Forster awakened his interest in the role played by elliptic curves in cryptography. Finally, the second edition of Elliptic Curves was then apparently vetted by Richard Taylor, Frans Oort and Don Zagier.

Well, it doesn’t get much better than that. It leaves one with very high expectations; and they are met:

Elliptic Curves is a beefy book of twenty chapters, taking the reader from rather elementary material to some unquestionably avant garde stuff. So, if the reader does things right and spends enough time on the foundational material he will be prepared in pretty short order to enjoy and absorb, e.g., Chapters 4 and 5, on families of elliptic curves and torsion points, and on reduction mod a prime (and torsion points), and then, to be sure, Mordell’s Theorem (on finite generation of the subgroup of rational points).

But were not even half way there: after Mordell, it’s Galois cohomology and classification over arbitrary fields, elliptic and hypergeometric functions, theta functions (an exceptionally evocative treatment: it’s very explicit — and Husemöller even includes coverage of Tate’s p-adic theory in this connection), modular forms.

But hold on: the latter parts of Elliptic Curves covers working over finite fields, local fields vs. global fields, l-adic representations, L-functions of elliptic curves (and, to be sure, Husemöller includes complex multiplication and Eichler-Shimura theory). Happily the book’s 18^th chapter is then devoted to “Remarks on the Modular Elliptic Curves Conjecture and Fermat’s Last Theorem” and presents a very nice compact discussion of the Frey-Serre-Ribet-Wiles-Taylor adventure of the 1990s. Here’s Husemöller’s last line, on p. 343: “…and the modular curve conjecture is established for all elliptic curves over the rational numbers.”

Then we come to the last two chapters: “Higher dimensional analogs of elliptic curves: Calabi-Yau varieties,” and “ Families of elliptic curves.” This is really something. The entity “Calabi-Yau” (to return to an earlier literary conceit of mine) brings to mind such things as string theory, Kähler manifolds, and so on. So what’s going on? Well, we learn a lot in Chapter 19: Husemöller starts with some differential geometry, pretty much up to the statement but not the proof of de Rham’s theorem (which, showing his excellent taste, Husemöller characterizes as “basically a theorem in sheaf theory,” Grothendieck’s proof being by far the slickest). He then hits Kähler manifolds, states Chow’s Theorem (to the effect that complex projective closed submanifolds are algebraic varieties), does some Hodge theory, some Riemannian geometry, and at last gets to characteristic classes and curvature. Well with Chern classes on the scene, we can define Calabi-Yau manifolds: Kähler manifolds with vanishing first Chern class. (Well, again, there’s more: it’s equivalent to say that we have Ricci flat Kähler metric, or that we have a Kähler manifold with a trivial canonical line bundle, for example. See p. 366). And what about Kähler varieties? Well, we have (p. 355) that “[a] complex submanifold of a Kähler manifold is a Kähler manifold so that all smooth projective algebraic varieties are Kähler…” (this being an evident converse to Chow’s theorem).

Chapter 20 subsequently has as its purpose “to return to the concept of families of elliptic curves in the context of [Grothendieck’s powerful but still somewhat austere] scheme theory and to point out some of the many areas of mathematics, and now also even of physics, in which families of elliptic curves play a role,” and it culminates in studying elliptic fibrations. To wit: “…passing to three dimensional varieties X, we can, by extension of the study of elliptic fibrations on surfaces, consider fibrations by K3 surfaces and abelian surfaces. This we can do only for Calabi-Yau varieties…”

Finally, Stefan Theisen adds an appendix on “Calabi-Yau manifolds and string theory,” Otto Forster adds one on “elliptic curves in algorithmic number theory and cryptography, Husemöller himself gives us an appendix on “elliptic curves and topological modular forms,” while the fourth and last appendix, by Ruth Lawrence, is titled, “Guide to the Exercises” (this being a textbook after all). At this point we’re nearly at 500 pages: not for the dabbler without Sitzfleisch.

There’s little left to be said. Elliptic Curves is an exciting book, full of what Feynman used to call “the good stuff,” and, in point of fact, Feynman himself makes an appearance on p. 404 — or at least his diagrams do so. Is it not amazing in itself that in this mathematical day and age we find in a book on what is after all ostensibly arithmetic algebraic geometry not only the Diophantine theme par excellence of Fermat’s Last Theorem but also the hard core physics of Feynman diagrams? How wonderful!

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.