Alexey Parshin

Alexei Parshin in Oberwolfach 2005

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.

Education and career

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 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.[1] The general case (for non-empty S) of the preceding theorem was proved by Arakelov.[1][2] 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).[3] Parshin presented his results 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 talk Representations of higher adelic groups and arithmetic.[4] In 1970 he was an Invited Speaker at the ICM in Nice with talk Quelques conjectures de finitude en géométrie diophantienne.[5]

Awards and honors

Selected publications

Sources

See also

References

  1. 1 2 Caporaso, Lucia (2000). "Remarks about uniform boundedness of rational points over function fields". arXiv:math/0004078Freely accessible.
  2. ↑ Heier, Gordon (2003). "Uniformly effective Shafarevich conjecture on families of hyperbolic curves over a curve with prescribed degeneracy condition". arXiv:math/0311085Freely accessible.
  3. ↑ Parshin, Algebraic curves over function fields. I, Math. USSR Izvestiya Vol. 2, 1968
  4. ↑ Parshin, A. N. "Representations of higher adelic groups and arithmetic." In Proceedings of the International Congress of Mathematicians, vol. 1, pp. 362–392. 2011.
  5. ↑ Paršin, A. N. "Quelques conjectures de finitude en géométrie Diophantienne." In Actes, Congrès intern. math, Tome 1, vol. 1, pp. 467–471. 1970.

External links

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