viii ALEKSEY NIKOLAEVICH PARSHIN
nor an irreducible divisor (a subscheme of codimension 1) but a flag, i.e, a point, an
irreducible curve passing through this point, an irreducible surface passing through
this curve etc. This conception turned out to be extremely fruitful, circulating in
a plenty of subjects of mathematics.
Several years were devoted by Parshin to the development of an analogue of
class field theory for higher-dimensional fields. Here the main novelty was that the
role of the multiplicative group of a field is played by the group Kn (more precisely,
a certain topological variant of
K
n
defined in terms of "symbols" ) . This leads to a
natural generalization of the classical reciprocity law. In the "geometric" case one
gets new relations between residues of differential forms.
During the following years A. N. Parshin discovered and studied other interre-
lations between the problems described above. In particular, the building of class
field theory had required the construction of an analogue of the group of adeles (or
fdeles) for a n-dimensional field. It turned out that this construction can be gen-
eralized to the case of adele groups of algebraic groups. In particular, the "adelic"
description of vector bundles over an algebraic curve (Weil - Serre) has a precise
n-dimensional analogue (for instance, for algebraic surfaces). Thus he discovered
an analogue of Serre's theorem about interrelations between vector bundles and
Bruhat-Tits buildings. In the same circle of ideas lies a "fixed point formula" for
actions of tori on manifolds with arbitrary singularities. In number-theoretical sit-
uation these considerations allow one to study local constants in the functional
equations for L-functions.
Recently many interrelations with other ideas were discovered. For example,
even in the case of algebraic varieties over the field of complex numbers, the gener-
alizations of reciprocity laws turn out to be related with so called "complex linking
numbers", local symbols are related with topological conformal field theory, etc. It
seems that there is a very broad area for study, whose boundary is still not visible.
Some of Parshin's papers do not fit into the two directions described above.
Thus he discovered the possibility of an application of Sh. Kobayashi's hyperbolic
geometry to number theoretical problems (for the first time, to the best of my
knowledge). This was a beginning of now widely ramified direction: proofs of
arithmetical properties of algebraic varieties (or statements of conjectures).
Parshin's interests are much broader that one may judge by his research in
number theory and algebraic geometry. This is demonstrated by his participation
in editing (in Russian) the collected papers of H. Weyl and D. Hilbert, his paper
on complementarity principle in "Voprosy philosophii", or by his "Reflections on
Godel's Theorem" published in "Istoriko-Matematicheskie Issledovaniya".
Aleksey Nikolaevich Parshin is a corresponding member of the Russian Acad-
emy of Sciences and doctor honoris causa of the Universite Paris-Nord.
I.
R. Shafarevich
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