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 in 1968 under Igor Shafarevich. In 1983 he received his Russian doctorate of sciences 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 Shafarevich's finiteness conjecture concerning isomorphism classes of abelian varieties via what is known as Parshin's trick, which gives an embedding of an algebraic curve into the Siegel modular variety. In 1983 Gerd Faltings proved Shafarevich's finiteness conjecture. Shafarevich proved his conjecture for the case with genus g = 1. In 1968 Parshin proved a special case of the following theorem: If is a smooth complex curve and is a finite subset of then there exist only finitely many families of smooth curves of fixed genus g ≥ 2 over. The general case of the preceding theorem was proved by Arakelov. At the same time, Parshin gave a new proof of the Mordell conjecture in function fields. Parshin presented his results in his talk Quelques conjectures de finitude en géométrie diophantienneas an invited speaker at the ICM in 1970 in Nice. Parshin's research deals with generalizations of class field theory in higher dimensions, with integrable systems, and with the history of mathematics. He was an editor for the Russian edition of the collected works of David Hilbert and was a co-editor, with V. I. Arnold, of selected works of Hermann Weyl. Parshin is a corresponding member of the Russian Academy of Sciences. At the ICM in 2010 he was a Plenary Speaker with his talk titled Representations of higher adelic groups and arithmetic.
A. N. Parshin: "Путь. Математика и другие миры" Moscow 2002.
With Shafarevich, Parshin edited several volumes in the series "Algebraic Geometry and Number Theory" in the Encyclopedia of mathematical sciences published by Springer Verlag.
with Shafarevich: The arithmetic of algebraic varieties. Proc. Steklov Institute Math., 1986, No. 3.
with Yuri Zarin: Finiteness problems in algebraic geometry, in Eight papers translated from the Russian. American Mathematical Society Translations Ser. 2, Vol.143, 1989, pp. 35–102, revised version of the original published as an appendix in the Russian edition of Serge LangFundamentals of Diophantine Geometry
Parshin , in Bolibruch, Osipov, Sinai Mathematical Events of the Twentieth Century, Springer 2006, pp. 297–330
Parshin , Mitteilungen DMV, Vol. 18, 2010, pp. 43–48
