James S. Milne (born 10 October 1942 in Invercargill, New Zealand) is a New Zealand mathematician working in

arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...

Life

Milne attended the High School in

Invercargill Invercargill ( , mi, Waihōpai is the southernmost and westernmost city in New Zealand, and one of the southernmost cities in the world. It is the commercial centre of the Southland region. The city lies in the heart of the wide expanse of ...

New Zealand New Zealand ( mi, Aotearoa ) is an island country in the southwestern Pacific Ocean. It consists of two main landmasses—the North Island () and the South Island ()—and over 700 List of islands of New Zealand, smaller islands. It is the ...

until 1959, and then studied at the

University of Otago , image_name = University of Otago Registry Building2.jpg , image_size = , caption = University clock tower , motto = la, Sapere aude , mottoeng = Dare to be wise , established = 1869; 152 years ago , type = Public research collegiate u ...

Dunedin Dunedin ( ; mi, Ōtepoti) is the second-largest city in the South Island of New Zealand (after Christchurch), and the principal city of the Otago region. Its name comes from , the Scottish Gaelic name for Edinburgh, the capital of Scotland. Th ...

(B.A. 1964) and

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...

(Masters 1966, Ph.D. 1967 under John Tate). From then to 1969 he was a lecturer at

University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public research university , endowment = £143 million (2020) , budget = � ...

. After that he was at the

University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...

, as Assistant Professor (1969–1972), Associate Professor (1972–1977), Professor (1977–2000), and Professor Emeritus (since 2000). He has also been a visiting professor at King's College London, at the

Institut des hautes études scientifiques An institute is an organisational body created for a certain purpose. They are often research organisations (research institutes) created to do research on specific topics, or can also be a professional body. In some countries, institutes can ...

in Paris (1975, 1978), at the Mathematical Sciences Research Institute in

Berkeley, California Berkeley ( ) is a city on the eastern shore of San Francisco Bay in northern Alameda County, California, United States. It is named after the 18th-century Irish bishop and philosopher George Berkeley. It borders the cities of Oakland and Emer ...

(1986–87), and the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

Princeton, New Jersey Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of whi ...

(1976–77, 1982, 1988). In his dissertation, entitled "The conjectures of Birch and Swinnerton-Dyer for constant abelian varieties over function fields," he proved the conjecture of Birch and Swinnerton–Dyer for constant

abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...

over function fields in one variable over a finite field. He also gave the first examples of nonzero abelian varieties with finite

Tate–Shafarevich group In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of ...

. He went on to study Shimura varieties (certain

hermitian symmetric space In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian ...

s, low-dimensional examples being

modular curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular ...

s) and motives. His students include Piotr Blass, Michael Bester, Matthew DeLong, Pierre Giguere, William Hawkins Jr, Matthias Pfau, Victor Scharaschkin, Stefan Treatman, Anthony Vazzana, and Wafa Wei. Milne is also an avid mountain climber.

Writings

* * ''Abelian Varieties, Jacobian Varieties'', in Arithmetic Geometry Proc. Conference Storrs 1984, Springer 1986 * With Pierre Deligne, Arthur Ogus, Kuang-yen Shih, ''Hodge Cycles, Motives and Shimura Varieties,'' Springer Verlag, Lecture Notes in Mathematics vol. 900, 1982 (therein by Deligne: Tannakian Categories) * ''Arithmetic Duality Theorems,'' Academic Press, Perspectives in Mathematics, 1986 * Editor with Laurent Clozel, Automorphic Forms, ''Shimura Varieties and L-Functions,'' 2 volumes, Elsevier 1988 (Conference University of Michigan, 1988) * ''Elliptic Curves,'' BookSurge Publishing 2006 * ''Shimura Varieties and Motives'' in Jannsen, Kleiman, Serre (editor) motif, Proc. Symp. Pure vol. 55 Math, AMS, 1994

References

* The original article was a Google translation of the corresponding article in German Wikipedia.

External links

Personal website
{{DEFAULTSORT:Milne, James New Zealand mathematicians Living people 1942 births University of Michigan faculty Harvard University alumni University of Otago alumni People from Invercargill Arithmetic geometers