This book, authored over a decade ago by UCLA’s Haruzo Hida, should probably be studied in tandem with the same author’s Modular Forms and Galois Cohomology, with this pair of sources then providing sufficient background for the reader to become able to study Andrew Wiles’ work on Fermat’s Last Theorem. Indeed, the three broad areas that Wiles brought into play to prove the Shimura-Taniyama-Weil Conjecture (“all rational elliptic curves are modular, which implies” FLT by work Frey, Serre, and Ribet) are the theory of modular forms, the theory of elliptic curves, and the theory of Galois representations; so the titles of Hida’s books are in themselves a dead giveaway of what’s going on inside them.
From a broader point of view, the topics considered here are autonomously important in the scheme of modern number theory and arithmetic geometry, and therefore this material is certainly of considerable pedagogical value in itself. Hida’s presentation lends itself very well to a pretty quick path of ascent: the opening chapter, “An algebra-geometric tool box,” is a wonderfully compact vade mecum of what was once available only scattered through some half-dozen books including Hartshorne’s Algebraic Geometry, Grothendieck and Dieudonné’s Eléments de Géométrie Algébrique, and (for the intrepid) Grothendieck’s titanic Séminaire de Géométrie Algébrique, all with a focus expressly on algebraic geometry. Even though there are now several treatments available with an explicitly arithmetical leaning, it’s very useful indeed to have the relevant material laid out in this way.
Similarly, Hida’s second chapter (equally long — it can’t be otherwise), “Elliptic curves,” presents a wonderful compendium of the required results, building on the algebraic geometry presented in the first chapter. In sequence we encounter in this chapter Riemann-Roch, Weierstrass theory, working over C and then over the p-adic numbers, and L-functions of elliptic curves: the plot thickens, but we are clearly headed for Wiles’ arithmetic geometry.
Modular forms come next, with the p-adic case treated early on. These are geometric modular forms, and Hida says this:
“We study the module structure of the space of geometric modular forms over a variety of groups and rings … We … prove the horizontal control theorem which … has been used in [Modular Forms and Galois Cohomology] to prove a result of Wiles: identification of a (minimal) universal Galois deformation ring with an appropriate Hecke algebra …” Again, we’re headed in the direction of FLT and its deep surrounding material.
The trend continues in the next chapter, “Jacobians and Galois representations,” which is an exemple of the interplay with algebraic geometry that is present in so much of contemporary number theory. The book then closes with an explicit discussion of the critical theme of modularity, and Hida introduces this climactic chapter by stating Shimura-Taniyama-Weil very precisely and then proceeding to a discussion of Wiles’ original treatment. Because of the timing of the book’s original appearance, Hida is restricted to noting merely that “it has been reported that Breuil, Conrad, Diamond, and Taylor succeeded in proving the conjecture in full generality.” What Hida focuses on (Wiles + Taylor-Wiles) is indeed the stuff FLT is made of and therefore of both historical and (huge) pedagogical value.
Having been tested at UCLA and Hokkaido, Geometric Modular Forms and Elliptic Curves is suited for both the (advanced and specialized) classroom and (well-prepared and highly motivated) reader bent of serious self-study. There is really a non-negotiable set of prerequisites hiding in the shadows: a good deal of number theory is needed, as well as the usual material covered in the first couple of years in graduate school. With cohomology entering in, familiarity with this wonderful machine (be it from the point of view of homological algebra or from that of topology) is strongly indicated. Beyond this, the book’s prose is clear, there are examples and exercises available, and, as always, the serious student should have a go at them: he will reap wonderful benefits.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.