|1982||Hodge Cycles, Motives, and Shimura Varieties (with Deligne, Ogus, Shih)||DMOS||notes|
|1986||Arithmetic Duality Theorems||ADT1||notes|
|2006||Arithmetic Duality Theorems, second edition||ADT2||notes|
|1990||Automorphic Forms, Shimura Varieties, and L-functions, (editor with L. Clozel)||Vol. 1||none||Proc. of a Conf. held at the Univ. of Michigan, Ann Arbor, July 6--16, 1988.||Vol. 2||none|
|2016||Algebraic Groups: the theory of group schemes of finite type over a field.||none||none|
In the 1970s, derived categories were still quite new, and known to only a few
algebraic geometers, and so I avoided using them. In some places this worked
out quite well, for example, contrary to statements in the literature they are
not really needed for the Lefschetz trace formula with coefficients in
I also regret treating Lefschetz pencils only in the case of fiber dimension 1. Apart from using derived categories and including Lefschetz pencils with arbitrary fiber dimension, I plan to keep the book much as before, but with the statements of the main theorems updated to take account of later work. Whether the new version will ever be completed, only time will tell.
Following is the blurb for Elliptic Curves that was on Amazon, and would still be, but for the incompetence of the people at BookSurge/CreateSpace/Amazon.
This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. It assumes only a knowledge of the basic algebra, complex analysis, and topology usually taught in advanced undergraduate or first-year graduate courses.
Indeed, the book is affordable (in fact, the most affordable of all references on the subject), but also a high quality work and a complete introduction to the rich theory of the arithmetic of elliptic curves, with numerous examples and exercises for the reader, many interesting remarks and an updated bibliography.
Mathematical Reviews, Álvaro Lozano-Robledo
J. S. Milne's lecture notes on elliptic curves are already well-known
The book under review is a rewritten version of just these famous lecture notes from 1996, which appear here as a compact and inexpensive paperback that is now available worldwide.
Zentralblatt MATH, Werner Kleinert
Algebraic groups play much the same role for algebraists that Lie groups play for analysts. This book is the first* comprehensive introduction to the theory of algebraic groups over fields (including the structure theory of semisimple groups) written in the language of modern algebraic geometry. When Borel, Chevalley, and others introduced algebraic geometry into the theory of algebraic groups, they used the algebraic geometry of the day, whose terminology conflicts with that of modern (post 1960) algebraic geometry. As Tits wrote in 1960, "the scheme viewpoint ... is not only more general but also, in many respects, more satisfactory." Indeed, allowing nilpotents gives a much richer and more natural theory.
The first nine chapters of the book study general algebraic groups. They culminate in a proof of the Barsotti-Chevalley theorem stating that every algebraic group is an extension of an abelian variety by an affine algebraic group. The remainder of the book treats only affine algebraic groups. After a review of the Tannakian philosophy, there are short accounts of Lie algebras and finite group schemes. Solvable algebraic groups are studied in detail in Chapters 13-16. The next seven chapters treat the Borel-Chevalley structure theory of reductive algebraic groups over arbitrary fields. Two final chapters survey additional topics. The exposition incorporates simplifications to the theory by Springer, Steinberg, and others.
Although the theory of algebraic groups can be considered a branch of algebraic geometry, most of those using it are not algebraic geometers. In the present work, prerequisites have been kept to a minimum. The only requirement is a first course in algebraic geometry.
*The only attempt to write such a book that I know of was by Demazure and Gabriel. As they wrote in the foreword of "Groupes Algébriques (1971)" (my translation):
We know that A. Grothendieck has introduced into algebraic geometry two particularly useful tools: the functorial calculus and varieties with nilpotent elements. These tools allow us to better understand the phenomena arising from inseparability, by rehabilitating differential calculus in nonzero characteristic, and they simplify considerably the general theory of algebraic groups. We originally intended therefore to develop in the framework of schemes the now classical theory of semisimple algebraic groups over algebraically closed fields (Borel-Chevalley). It was therefore simply a question of updating the 1956-58 seminar of Chevalley. But we soon realized that there did not exist a suitable reference book for the general theory of algebraic groups, and that it was impossible to refer a nonspecialist reader to the Eléments de Géométrie Algébrique (EGA) by Grothendieck. It was in this way that our original project was considerably modified and developed, and that we present to the "mathematical community" a first volume devoted to the general theory of algebraic groups, where the case of semisimple groups is not discussed.Unfortunately, after the 700 page first volume, no second volume appeared.