Aleksei Nikolaevich Parshin

Alexey N. Parshin
Alexei Parshin in Oberwolfach 2005
Native name	Алексей Николаевич Паршин
Born	(1942-11-07) 7 November 1942 (age 74) Sverdlovsk, Soviet Union
Nationality	Russian
Alma mater	Steklov Institute of Mathematics
Scientific career
Fields	Mathematics

Aleksei (or Alexei) Nikolaevich Parshin (Russian: Алексей Николаевич Паршин; born 7 November 1942 in Sverdlovsk) is a Russian mathematician, specializing in number theory and algebraic geometry.

Parshin graduated in 1964 from the Faculty of Mathematics and Mechanics of Moscow State University and then enrolled as a graduate student at the Steklov Institute of Mathematics, where he received his Kand. Nauk (Ph.D.) in 1968. In 1983 he received his Russian doctorate of sciences (Doctor Nauk) from Moscow State University. He is now a professor at the Steklov Institute in Moscow, where he is the head of the Department of Algebra, and he is also a professor at Moscow State University.

Parshin proved in 1968 that the Mordell conjecture is a logical consequence of a finiteness conjecture, formulated by Igor Shafarevich, concerning isomorphism classes of abelian varieties. (Shafarevich presented his finiteness conjecture at the ICM in 1962). In 1983 Gerd Faltings proved Shafarevich's finiteness conjecture (and thereby the Mordell conjecture).

Shafarevich proved his conjecture for the case with genus g = 1. In 1968 Parshin proved a special case (for S = the empty set) of the following theorem: If B is a smooth complex curve and S is a finite subset of B then there exist only finitely many families (up to isomorphism) of smooth curves of fixed genus g ≥ 2 over B \ S. The general case (for non-empty S) of the preceding theorem was proved by Arakelov. At the same time, Parshin gave a new proof (without an application of the Shafarevich finiteness condition) of the Mordell conjecture in function fields (already proved by Yuri Manin in 1963 and by Hans Grauert in 1965). Parshin presented his results at the ICM in 1970 in Nice.