Curriculum Vitae: James S. Milne.

Born 1942, New Zealand.
Citizen of New Zealand/Aotearoa.
United States permanent resident.

Education

Otago University, Dunedin, NZ (January 1960 -- December 1963)
Harvard University, Cambridge, MA, USA (September 1964 -- August 1967).

Degrees

B.Sc. (Hons), 1964, Otago University, Dunedin, New Zealand.
A.M., 1966, Harvard University, Cambridge, MA, USA.
Ph.D., 1967, Harvard University, Cambridge, MA, USA, (thesis adviser, J. Tate).

Positions

University College, London, UK.
    Lecturer 1967-69.
University of Michigan, Ann Arbor, MI, USA.
    Assistant professor 1969-72
    Associate professor 1972-77
    Professor 1977-99
    Professor emeritus 2000-present

Visiting Positions

Kings College, London, UK.
   Academic year 1971--72.
Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, France.
    Winter 1975.
    Fall 1978.
Institute for Advanced Study, Princeton, USA.
    Academic year 1976-77.
    Winter 1982.
    Fall 1988.
Université de Rennes, France.
    Winter 1978.
Mathematical Sciences Research Institute, Berkeley, USA.
    Academic year 1986-87.
CMS, Zhejiang University, Hangzhou, China,
    Spring 2005.

Fellow of the AMS (Inaugural Class of Fellows) Declined.

Some Research Results

• Proved the conjecture of Birch and Swinnerton-Dyer for constant abelian varieties over function fields. In particular, this gave the first examples of abelian varieties over global fields whose Tate-Shafarevich group is known to be finite [InvM, 1968, 1968].
• Proved the finiteness of the Selmer group of an abelian variety over a global field [BLMS 1970].
• Proved the conjecture of Artin and Tate for surfaces satisfying the Tate conjecture. In particular, this gave the first proof that the Brauer group of an elliptic K3 surface over a finite field is finite [Annals 1975].
• Introduced the correct $p$-analogue of the étale cohomology of the sheaves $\mu_{\ell^n}^{\otimes r}$, and proved a duality theorem for it (conjectured by Artin in the case of surfaces). [AnnENS 1976, AJM 1986].
• Proved the conjecture of Langlands and Rapoport for Shimura varieties defined by totally definite quaternion algebras [PSPM 1979; PM 1979].
• Proved the conjecture of Langlands (Corvallis, p.234) describing the action of complex conjugation on a Shimura variety [Annals 1981, with Shih].
• Introduced the notion of a canonical integral model of a Shimura variety, and proved that it is unique when it exists [Montreal 1982].
• Proved the conjecture of Langlands (Corvallis, p.223) describing the conjugates of a Shimura variety [case of abelian type, LNM 1982, with Shih; general case, Shafarevich volume, 1983].
• Proved the existence of canonical models of Shimura varieties (Shimura's conjecture) [Shafarevich volume 1983, MMJ 1999].
• Proved the existence of canonical models of automorphic vector bundles [InvM 1988].
• Extended the Artin-Tate conjecture on the special values of zeta functions from surfaces to all algebraic varieties and motives, and proved it for varieties satisfying the Tate conjecture [AJM 1986; JAMS 2004; AJM 2015; JIMJ 2015; arXiv:1311.3166; ...; some with Ramachandran].
• Proved that Shimura varieties of abelian type with rational weight can be realized as moduli varieties for abelian motives (Deligne's dream) [Seattle 1994].
• Proved that the various cohomology classes on abelian varieties predicted to be algebraic by Grothendieck's standard conjectures are not only algebraic, but even Lefschetz; in particular, the Lefschetz standard conjecture holds for Lefschetz classes on abelian varieties [Duke 1999].
• Proved that the Tate conjecture and Grothendieck's standard conjectures hold for abelian varieties over finite fields if the Hodge conjecture holds for complex CM abelian varieties [CompM 1999, Annals 2002]. Deduced new cases of the two conjectures from known cases of the Hodge conjecture [arXiv:2112.12815].
• Introduced the notion of a good theory of rational Tate classes extending Deligne's theory of absolute Hodge classes to nonzero characteristic, and proved that it is unique if it exists [MMJ 2009].
• Found a new simple characterization of algebraic Lie algebras [arXiv:2012.05708].

Some Published Books

• Etale Cohomology, Princeton University Press, 1980; paperback reprint, 2017. I wrote this so that, as Grothendieck put it, people wouldn't have to read the thousands of pages of the original seminars.
• Arithmetic Duality Theorems, Academic Press, 1986; second edition 2006. A collection of duality theorems, some published and some not previously published, some known and some not previously known, in algebraic number theory and arithmetic geometry.
• Elliptic Curves, Kea Books, 2006; second edition, World Scientific Publishers, 2020; paperback reprint 2023. A narrative account of the theory of elliptic curves from the basics to the proof of Fermat's last theorem.
• Algebraic Groups, Cambridge University Press, 2017; corrected reprint 2022. Most of the theory of algebraic groups is written in a language that pre-dates schemes, and which has become increasingly incomprehensible to modern algebraic geometers, so I wrote the book that should have been written fifty years ago.
• Fields and Galois Theory, Kea Books, 2022. Covers the standard (Emil Artin) approach as well as Grothendieck's.
• Group Theory, Kea Books, 202?. In preparation. Covers not only the standard material on abstract groups, but also algebraic, Lie, and topological groups and their representations.

Some Expository Articles

• Abelian varieties over finite fields (with W. C. Waterhouse), Proc. Symp. Pure Math. 20 (1971), 53-64.
• Tannakian categories (with P. Deligne), in Hodge Cycles, Motives, and Shimura Varieties, LNM 900, Springer, 101-228, 1982.
• Abelian varieties, in Arithmetic Geometry (Proc. Conference on Arithmetic Geometry, Storrs, August 1984) Springer, 1986, 103-150.
• Jacobian varieties, in Arithmetic Geometry (Proc. Conference on Arithmetic Geometry, Storrs, August 1984) Springer, 1986, 167-212.
• Canonical Models of (Mixed) Shimura Varieties and Automorphic Vector Bundles. In: Automorphic Forms, Shimura Varieties, and L-functions, (Proceedings of a Conference held at the University of Michigan, Ann Arbor, July 6-16, 1988), pp283--414.
• The points on a Shimura variety modulo a prime of good reduction. In: The Zeta Function of Picard Modular Surfaces, Publ. Centre de Rech. Math., Montreal (Eds. R. Langlands and D. Ramakrishnan), 1992, pp151--253.
• Motives over finite fields. In: Motives (Eds. Jannsen, Kleiman, Serre), AMS, Proc. Symp Pure Math. 55, 1994, Part 1, pp. 401--459.
• Introduction to Shimura varieties, In Harmonic Analysis, the Trace Formula and Shimura Varieties (James Arthur, Robert Kottwitz, Editors) AMS, 2005, (Lectures at the Summer School held at the Fields Institute, June 2 -- June 27, 2003).
• Motives---Grothendieck's dream (Chinese). Mathematical Advances in Translation, Vol.28, No.3, 193-206, 2009 (Institute of Mathematics, Chinese Academy of Sciences) (translation by Xu Kejian, Qingdao University).
• Shimura varieties and moduli, Handbook of Moduli (Gavril Farkas, Ian Morrison, Editors), International Press of Boston, 2013, Vol II, 462--544.

Some Online Books (number of downloads in 2022)

• Group Theory (51,889).
• Fields and Galois Theory (27,290) [Published 2022].
• Algebraic Geometry (19,043).
• Algebraic Number Theory (26,875).
• Modular Functions and Modular Forms (10,804).
• Abelian Varieties (7,994).
• Lectures on Étale Cohomology (14,084).
• Class Field Theory (12,925).