![]() ![]() It cannot be denied, however, that these are reasonably austere texts, the former book’s terseness notwithstanding (do all the exercises!), and after dealing with them algebraic geometry itself is still ahead. In my own case that meant working through Atiyah-MacDonald, but other classics abound, such as the classic two-volume source by Zariski and Samuel. But it is not the case that one can approach this beautiful subject without first preparing the way, and this means in particular a thorough study of commutative algebra. Thus, now more than ever, the market for algebraic geometry is strong: every aspiring mathematician (except maybe the most exclusive of logicians) needs to learn a sizable chunk of it. Hand in glove with this is the fact that books on algebraic geometry are plentiful, the most prominent still probably being Hartshorne’s Algebraic Geometry. Naturally, this parochial view of algebraic geometry as a tool for number theory is at the same time disingenuous: the subject is alive and well and indeed is flourishing. But to say that it is the final chapter is of course ridiculous: the plot continues to thicken every day. So, in a true sense, the work done by Wiles should be regarded as the climactic culminating chapter of a long story, with the excitement building every step of the way. Accordingly what is now called arithmetic geometry quickly reached the prominence it so properly enjoys.įrom a broader (or deeper) standpoint, the handwriting had already been on the wall for many years with Weil’s visionary work, Grothendieck’s development of the architecture that would support the most dazzling and spectacular constructs, the work of Deligne on the Weil conjectures, and Faltings’s treatment of the Mordell Conjecture. The remarkably fecund interplay between modular forms, Galois representations, and elliptic curves ascended to center stage as an exemplar of a new distribution of emphases both in algebraic number theory and algebraic geometry. Also, the algebraic geometry was primarily geometry (Mumford’s geometric invariant theory was a major theme at UCLA, for example), while the number theory was primarily arithmetic (what a bizarre way to put it!) in any case, there was, for instance, no mention of sheaf theory in my number theory courses.īefore long a profound infusion of algebraic-geometric methods occurred, becoming universal after the stunning proof of Fermat’s Last Theorem as a consequence of settling the Shimura-Taniyama-Weil Conjecture. The algebraic geometers had their seminars, and the number theorists had theirs, and the overlap was rather sporadic. Yes, although what is now called arithmetic geometry was already beginning to knock on the door, Wiles’ work was still over a decade in the future, and the floodgates hadn’t really been fully opened yet. At that time, at least where I was (UCLA, UCSD), the focus fell on such things as representation theory and trace formulas, Hecke algebras, oldforms and newforms, and connections with elliptic curves. My own schooling was heavily oriented toward modular forms: in the late 1970s and early 1980s it couldn’t be otherwise. One of the great pleasures of being a number theorist these days is the unavoidability of algebraic geometry. ![]()
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