arXiv subjects
Publications and source records attributed to James S. Milne.
In the 1950s and 1960s Tate proved some duality theorems in the Galois cohomology of finite modules and abelian varieties. As for most of Tate's work this has had a profound influence on mathematics with many applications and further developments. In this article, I discuss Tate's theorems and some of these developments.
Much of the work on Shimura varieties over the last thirty years has been devoted to constructing the theory that would follow from a good notion of motives, one incorporating the Hodge, Tate, and standard conjectures. These conjectures are believed to be beyond reach, and may not even be correct as stated. I argue in this article that there exists a theory of motives, accessible to proof, weaker than Grothendieck's, but with many of the same consequences.
Let G be the pro-algebraic group attached to the tannakian category of polarizable rational Hodge structures. We show that the quotient of G by its derived group is the Serre group, the derived group of G is the simply connected covering of the adjoint group of G, and that the adjoint group G is a product of specific simple algebraic groups. As the Mumford--Tate groups are exactly the algebraic quotients of G, this also describes them.
In two earlier articles, we proved that, if the Hodge conjecture is true for ALL CM abelian varieties over the complex numbers, then both the Tate conjecture and the standard conjectures are true for abelian varieties over finite fields. Here we rework the proofs so that they apply to a single abelian variety. As a consequence, we prove (unconditionally) that the Tate and standard conjectures are true for many abelian varieties over finite fields, including abelian varieties for which the algebra of Tate classes is not generated by divisor classes.
Grothendieck's standard conjecture of Lefschetz type has two main forms: the weak form $C$ and the strong form $B$. The weak form is known for varieties over finite fields as a consequence of the proof of the Weil conjectures. This suggests that the strong form of the conjecture in the same setting may be the most accessible of the standard conjectures. Here, as an advertisement for the conjecture, we explain some of its remarkable consequences.
We prove, following Deligne and André, that the Hodge classes on abelian varieties of CM-type can be expressed in terms of divisor classes and split Weil classes, and we describe some consequences. In particular, we show that Grothendieck's standard conjecture of Lefschetz type implies the Hodge conjecture for abelian varieties (Abdulali, André, ...).
A fundamental theorem of Barsotti and Chevalley states that every smooth algebraic group over a perfect field is an extension of an abelian variety by a smooth affine algebraic group. In 1956 Rosenlicht gave a short proof of the theorem. In this expository article, we explain his proof in the language of modern algebraic geometry.
The p-cohomology of an algebraic variety in characteristic p lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc. and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for p-cohomology in characteristic p.
Already in the 1960s Grothendieck understood that one could obtain an almost entirely satisfactory theory of motives over a finite field when one assumes the full Tate conjecture. In this note we prove a similar result for motivic complexes. In particular Beilinson's Q-algebra of "correspondences at the generic point" is then defined for all connected varieties. We compute this for all smooth projective varieties (hence also for varieties birational to such a variety).
We show that conjugation by an automorphism of the complex numbers (as an abstract field) may change the topological fundamental group of a locally symmetric variety over C. As a consequence, we obtain a large class of algebraic varieties defined over number fields with the property that different embeddings of the number field into C give complex varieties with nonisomorphic fundamental groups.
These are my notes for a talk at the The Tate Conjecture workshop at the American Institute of Mathematics in Palo Alto, CA, July 23--July 27, 2007, somewhat revised and expanded. The intent of the talk was to review what is known and to suggest directions for research. v2: Revised expanded (24 pages).
We show how the Z_p(r)-cohomologies of a smooth projective algebraic variety can be obtained via its de Rham-Witt complex.
For each field k, we define an abelian category of rationally decomposed mixed motives with integer coefficients. When k is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near integers to certain Ext groups. This is the submitted version, with minor corrections and additions from the first version.
This is the text of an article that I wrote and disseminated in September 1981, except that I've updated the references, corrected a few misprints, and added a table of contents, some footnotes, and an addendum. The original article gave a simplified exposition of Deligne's extension of the Main Theorem of Complex Multiplication to all automorphisms of the complex numbers. The addendum discusses some additional topics in the theory of complex multiplication -- the origins of the theory, Hilbert's Twelfth Problem, why algebraic Hecke characters are motivic, and the periods of abelian varieties of CM-type. (43 pages)