In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field.
John Torrence Tate Jr. was an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He was awarded the Abel Prize in 2010.
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally.
David Bryant Mumford is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University.
In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. As of 2022, only special cases of the conjecture have been proven.
In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by Kolyvagin (1990) in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper Kolyvagin (1988) and the work of Thaine (1988). Euler systems are named after Leonhard Euler because the factors relating different elements of an Euler system resemble the Euler factors of an Euler product.
Igor Rostislavovich Shafarevich was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. Outside mathematics, he wrote books and articles that criticised socialism and other books which were (controversially) described as anti-semitic.
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.
Nicholas Michael Katz is an American mathematician, working in arithmetic geometry, particularly on p-adic methods, monodromy and moduli problems, and number theory. He is currently a professor of Mathematics at Princeton University and an editor of the journal Annals of Mathematics.
Michael Artin is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.
John William Scott "Ian" Cassels, FRS was a British mathematician.
In arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. John Tate (1958) named it for François Châtelet (1946) who introduced it for elliptic curves, and André Weil (1955), who introduced it for more general groups. It plays a basic role in the arithmetic of abelian varieties, in particular for elliptic curves, because of its connection with infinite descent.
In arithmetic geometry, the Selmer group, named in honor of the work of Ernst Sejersted Selmer (1951) by John William Scott Cassels (1962), is a group constructed from an isogeny of abelian varieties.
Dorian Morris Goldfeld is an American mathematician working in analytic number theory and automorphic forms at Columbia University.
In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles (1984). The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on.
Barry Charles Mazur is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology.
In arithmetic geometry, the Tate–Shafarevich groupШ(A/K) of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group WC(A/K) = H1(GK, A) that become trivial in all of the completions of K (i.e. the p-adic fields obtained from K, as well as its real and complex completions). Thus, in terms of Galois cohomology, it can be written as
James S. Milne is a New Zealand mathematician working in arithmetic geometry.
Aleksei Nikolaevich Parshin was a Russian mathematician, specializing in arithmetic geometry. He is most well-known for his role in the proof of the Mordell conjecture.
Christopher McLean Skinner is an American mathematician working in number theory and arithmetic aspects of the Langlands program. He specialises in algebraic number theory.
