Aleksey Nikolaevich Parshin was born on November 7, 1942. In 1959 he en-
tered the Mechanics and Mathematics faculty (Mekh-Mat) of Moscow State Uni-
versity. His outstanding mathematical capabilities became apparent already during
his student years. Then he wrote a paper where he suggested a construction of a
"nonabelian jacobian" of Riemann surface as the quotient of a certain (infinite-
dimensional) manifold modulo the image of the fundamental group of the surface.
This construction was a complex analogue of the theory of "iterated integrals" that
was being developed independently by Chen. Unfortunately, A. N. Parshin never
returned to this intriguing subject. Later it turned out that this theory provides
impressive results for covers with nilpotent Galois groups but even in this case a
lot still has to be done.
This paper was inspired by attempts to find a "nonabelian" generalization of
class field theory. And all posterior papers of A. N. Parshin went back to various
number-theoretical questions, though sometimes they went away to completely dif-
ferent fields. Mostly his research is related to the following two directions. First,
it is a study of two-dimensional schemes that are "fibered" over a one-dimensional
base. Parshin's attention was attracted by so called "finiteness conjecture" accord-
ing to which there are only finitely many of those schemes if the base, degenerations,
and the genus of the fiber are fixed (modulo obvious exceptions). His first results
dealt with the case of schemes over a field, i.e., with algebraic surfaces. Parshin
proved that if the conditions of the "finiteness conjecture" are fulfilled then the cor-
responding surfaces consist of a a finite number of algebraic families (the fact that
these families are zero-dimensional, i.e. boil down to finitely many surfaces, was be-
fore long proved by S. Yu. Arakelov with help of a special deformation technique).
But even Parshin's result allowed already to prove the "finiteness conjecture" for
surfaces over finite fields. In his ICM talk in Nice, A. N. Parshin reported on a
quite unexpected result: the validity of "finiteness conjecture" implies the cele-
brated "Mordell conjecture". In particular, this provides a proof of an analogue of
the Mordell conjecture for schemes over finite fields. Thirteen years later Faltings
proved the "finiteness conjecture" for arithmetical schemes. Thanks to the result
of Parshin, this gave a proof of the Mordell conjecture. A conjectural "Parshin's
inequality" (an analogue of Bogomolov-Miyaoka-Yao inequality for algebraic sur-
faces) lies in the same circle of ideas related to two-dimensional schemes. The
validity of Parshin's inequality would imply a plentitude of arithmetical corollaries.
Another direction that A. N. Parshin continues to develop to this day is related
to a generalization of class field theory and classical arithmetic of one-dimensional
schemes on the whole to the higher-dimensional case. An idea of this generalization
is that the basic notion is neither a point (an irreducible zero-dimensional scheme)
Previous Page Next Page