Aleksei Nikolaevich Parshin[a] (Russian: Алексей Николаевич Паршин; 7 November 1942 – 18 June 2022) was a Russian mathematician, specializing in arithmetic geometry. He is most well-known for his role in the proof of the Mordell conjecture.
Parshin became a junior research fellow at the Steklov Institute of Mathematics in Moscow in 1968, later becoming a senior and leading research fellow.[5][3] He became the head of its Department of Algebra in 1995.[5][3] He also taught at Moscow State University.[3]
In his 1968 thesis, Parshin proved 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.[6][7] Shafarevich proved his finiteness conjecture for the case with genus g = 1. 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.[8] The general case (for non-empty S) of the preceding theorem was proved by Suren Arakelov in 1971.[8][9] 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).[10] In 1983, Gerd Faltings completed the program and proved Shafarevich's finiteness conjecture, thereby proving the Mordell conjecture.[7]
His other research dealt with generalizations of class field theory in higher dimensions, with integrable systems, and with the history of mathematics.[1][3]
Parshin was longtime friends with Russian philosopher Aleksei Losev and started the Russian philosophy seminar at the Dom Loseva Library in Moscow.[14] Parshin was Orthodox Christian and wrote about the relationship between Russian religious philosophy and the modern sciences.[14]
Parshin was an invited speaker at the 1970 International Congress of Mathematicians (ICM) with his talk titled Quelques conjectures de finitude en géométrie diophantienne.[5][15] He was a plenary speaker at the 2010 ICM with his talk titled Representations of higher adelic groups and arithmetic.[5][16]
Selected publications
Parshin, A. N. (2002). Путь. Математика и другие миры [The Way. Mathematics and Other Worlds] (in Russian). Moscow: Добросвет. ISBN978-5-7913-0053-9. (Parshin's writings on Russian science and philosophy)
with Shafarevich, Parshin edited several volumes in "Algebraic Geometry and Number Theory" in the Encyclopedia of Mathematical Sciences series published by Springer Verlag.
Parshin, A. N.; Shafarevich, I. R. (1986). "Arithmetic of algebraic varieties". Proceedings of the Steklov Institute of Mathematics. 168 (3): 75–99.
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 (English version of the appendix Online)
^Heier, Gordon (2003). "Uniformly effective Shafarevich conjecture on families of hyperbolic curves over a curve with prescribed degeneracy condition". arXiv:math/0311085.
