| Buy books. Otherwise there will be no new books. | |||||
|---|---|---|---|---|---|
| Year | Title | Pub | Erratum | ||
| 1980 | Etale Cohomology, Princeton University Press. | NA | PUP | notes | |
| 1982 | Hodge Cycles, Motives, and Shimura Varieties (with Deligne, Ogus, Shih), LNM 900. | DMOS | SV | notes | |
| 1986 | Arithmetic Duality Theorems, Academic Press. | ADT1 | AP | notes | |
| 1990 | Automorphic Forms, Shimura Varieties, and L-functions, (editor with L. Clozel) | Vol. 1 | AP | NA | |
| Proc. of a Conf. held at the Univ. of Michigan, Ann Arbor, July 6--16, 1988. | Vol. 2 | AP | NA | ||
| 2006 | Arithmetic Duality Theorems, second edition, Booksurge LLC. | ADT2 | Kea | notes | |
| 2006 | Elliptic Curves, Booksurge, LLC. | EC | Kea | Erratum | |
| 2017 | Algebraic Groups: the theory of group schemes of finite type over a field. Cambridge U.P. | AG17 | CUP | Erratum | |
| 2020 | Elliptic Curves, second edition, World Scientific Publishing. | WSP | Erratum | Cover | |
| 2022 | Algebraic Groups: the theory of group schemes ... Corrected reprint. Cambridge U.P. | CUP | Erratum | ||
| 2022 | Fields and Galois Theory, Kea Books. | FT0 | Kea | Erratum | |
| 2025 | Tannakian Categories (penultimate version) | tcdraft |
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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/Kindle Direct Publishing.
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.
Reviews
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
Comments on Print on Demand publishing
In 2006, I rewrote my course notes and made them available as a paperback (the first edition of this work). For this second edition, I have rewritten and updated the notes once again, while retaining most of the numbering from the first edition.
Beyond its intrinsic interest, the study of elliptic curves makes an excellent introduction to some of the deeper aspects of current research in number theory. In the book, I have attempted to place the theory of elliptic curves in this wider context.
In reviewing the theory of elliptic curves, I have been struck by how much of it originated with calculations: those of Ramanujan, which suggested Hecke operators and the (generalized) Ramanujan conjecture; those of Sato, which suggested the Sato-Tate conjecture in its general forms; those of Selmer, which suggested the Cassels-Tate duality theorem; those of Birch and Stephens, which suggested the Gross-Zagier formula; and, of course, those of Birch and Swinnerton-Dyer, which suggested their conjecture and its generalizations.
The first two chapters are concerned with preliminaries on polynomials and field extensions, and Chapter 3 proves the fundamental theorems in the Galois theory of fields. Chapter 4 explains, with copious examples, how to compute Galois groups, and Chapter 5 describes the many applications of Galois theory.
In Chapter 6, a weak form of the Axiom of Choice is used to show that all fields admit algebraic closures, and that any two are isomorphic. The last three chapters extend Galois theory to infinite field extensions, to étale algebras over fields, and to nonalgebraic extensions.
The approach to Galois theory in Chapter 3 is that of Emil Artin, and in Chapter 8 it is that of Alexander Grothendieck.
This book originated as the notes for a first-year graduate course taught at the University of Michigan, but they have since been revised and expanded numerous times. The only prerequisites are an undergraduate course in abstract algebra and some group theory. There are ninety-six exercises, most with solutions.
Deligne removed a major lacuna in the theory of nonneutral tannakian categories, gave an internal characterization of a tannakian category in characteristic zero, and removed some unnecessary hypotheses in the theory or polarizations.
This is a updated account of the theory of tannakian categories, written in the spirit of the 1982 article by Deligne and Milne.
