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Serge Lang (French:; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra .
It plays a basic role in the arithmetic of abelian varieties, in particular for elliptic curves, because of its connection with infinite descent. It can be defined directly from Galois cohomology , as H 1 ( G K , A ) {\displaystyle H^{1}(G_{K},A)} , where G K {\displaystyle G_{K}} is the absolute Galois group of K .
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry.By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. [1]
Steinberg () gave a useful improvement to the theorem.. Suppose that F is an endomorphism of an algebraic group G.The Lang map is the map from G to G taking g to g −1 F(g).. The Lang–Steinberg theorem states [3] that if F is surjective and has a finite number of fixed points, and G is a connected affine algebraic group over an algebraically closed field, then the Lang map is surjective.
The condition was first introduced and studied by Lang. [10] If a field is C i then so is a finite extension. [11] [12] The C 0 fields are precisely the algebraically closed fields. [13] [14] Lang and Nagata proved that if a field is C k, then any extension of transcendence degree n is C k+n.
The Bombieri–Lang conjecture is an analogue for surfaces of Faltings's theorem, which states that algebraic curves of genus greater than one only have finitely many rational points. [ 8 ] If true, the Bombieri–Lang conjecture would resolve the Erdős–Ulam problem , as it would imply that there do not exist dense subsets of the Euclidean ...
Arakelov theory was used by Paul Vojta (1991) to give a new proof of the Mordell conjecture, and by Gerd Faltings in his proof of Serge Lang's generalization of the Mordell conjecture. Pierre Deligne ( 1987 ) developed a more general framework to define the intersection pairing defined on an arithmetic surface over the spectrum of a ring of ...
Lang conjecture Enrico Bombieri (dimension 2), Serge Lang and Paul Vojta (integral points case) and Piotr Blass have conjectured that algebraic varieties of general type do not have Zariski dense subsets of K-rational points, for K a finitely-generated field.
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