F. A. Bogomolov, A. M. Vershik, S. V. Vostokov, S. O. Gorchinskiy, A. B. Zheglov, Yu. G. Zarhin, S. V. Konyagin, Vik. S. Kulikov, Yu. V. Nesterenko, D. O. Orlov, D. V. Osipov, I. A. Panin, V. P. Platonov, V. L. Popov, Yu. G. Prokhorov, A. L. Smirnov, “Alexey Nikolaevich Parshin (obituary)”, Uspekhi Mat. Nauk, 78:3(471) (2023), 165–169; Russian Math. Surveys, 78:3 (2023), 549

On 18 June, 2022, the outstanding Soviet and Russian mathematician, one of the brightest members of the Moscow algebraic geometry school, head of the Department of Algebra of the Steklov Mathematical Institute of the Russian Academy of Sciences, academician of the Russian Academy of Sciences Alexey Nikolaevich Parshin passed away.

He was born on 7 November, 1942, in the city of Sverdlovsk. At that time his parents Nikolay Andreevich (a civil engineer) and Lyubov Mikhailovna (a housewife) were in Sverdlovsk in evacuation. The family returned to Moscow already in 1943.

In 1959 Parshin enrolled in the Faculty of Mechanics and Mathematics of Moscow State University, from which he graduated in 1964. During his studies there he attended seminars of two great mathematicians, I. M. Gelfand and I. R. Shafarevich, and Shafarevich became his scientific advisor. After graduating from the university Parshin began his Ph.D. studies at the Steklov Mathematical Institute of the Academy of Sciences of the USSR. In 1967 he defended his Ph.D. thesis, after which he worked in the institute for many years: from 1968 to 1995 as a junior, a senior, and a leading researcher, and from 1995 till his last days as the head of the Department of Algebra, then the Department of Algebra and Number Theory, and then again the Department of Algebra.

Already in his student years Parshin showed his outstanding mathematical capabilities. In his first paper, which was his diploma thesis, Parshin proposed the construction of a ‘non-abelian Jacobian’ of a Riemann surface. In particular, there he defined and studied the properties of iterated integrals. This was done independently of the papers by K. T. Chen. This work was the beginning of a series of Parshin’s remarkable investigations devoted to various generalizations of the classical class field theory.

Results obtained by Parshin during his postgraduate studies subsequently became widely known. At that time he was working on the finiteness conjecture proposed by Shafarevich, according to which there are only finitely many smooth projective algebraic curves of fixed genus greater than 1 over a global field (that is, a finite extension of the field of rational numbers or the field of rational functions in one variable with coefficients in a finite field) with a fixed set of bad reduction (except for obvious cases). Such curves define in a natural way surfaces fibred over a one-dimensional base. Parshin succeeded in proving the finiteness conjecture for algebraic surfaces in the case of a complete base (that is, for families of curves without degenerations), and in the general case of curves over a function field he proved that the height of the set of curves in question is bounded. This was the first important step towards the proof of the finiteness conjecture (for curves over function fields the conjecture was later proved in full generality by S. Yu. Arakelov).

In addition, at the same time Parshin showed that the Shafarevich finiteness conjecture implies the celebrated conjecture stated by L. Mordell in 1922, on the finiteness of the set of rational points on curves of genus greater than 1 over a number field or a function field. This implied, in particular, a new proof of the Mordell conjecture for curves over function fields (the first proof had been given by Yu. I. Manin). The construction used for that was subsequently cited, and is still cited by all experts as ‘Parshin’s trick’. Due to these achievements Parshin was selected as an invited speaker at the International Congress of Mathematicians in Nice in 1970, and in 1971 he was awarded the prize of the Moscow Mathematical Society for Young Mathematicians (together with Arakelov). Moreover, in 1983 the German mathematician G. Faltings succeeded in proving the Mordell conjecture for curves over number fields by using the above results due to Parshin, and for this he was awarded the Fields Medal.

Continuing further in this direction, Parshin defined later canonical heights on arithmetic surfaces and on moduli spaces of abelian varieties, and he used them to prove finiteness theorems for isogenies of abelian surfaces and also the Tate conjecture on homomorphisms for elliptic curves over function fields of finite characteristic. In 1988 he discovered a hypothetical inequality for arithmetic surfaces (an analogue of the Bogomolov–Miyaoka–Yau inequality for algebraic surfaces) and showed that it would imply many famous conjectures: the effective Mordell conjecture, Szpiro’s inequality for elliptic curves, and the abc-conjecture. In addition to these results, in a paper of 1990 dedicated to the 60th birthday of the outstanding mathematician Alexander Grothendieck, Parshin discovered the possibility of applying the Kobayashi hyperbolic geometry to problems in number theory: for example, he proved Lang’s conjecture on integral points on abelian varieties and gave a new proof of Raynaud’s theorem on rational points on abelian varieties, where abelian varieties are defined over the field of rational functions on a complex algebraic curve.

One more direction in arithmetic geometry developed by Parshin for many years, starting from the mid-1970s (including with his students) is the theory of higher-dimensional local fields, higher-dimensional adeles and their applications to higher-dimensional class field theory, that is, to an explicit description of abelian Galois extensions of the fields of rational functions on algebraic varieties defined over finite fields, or of the fields of rational functions on arithmetic schemes. Parshin was one of the founders of this line of research.

One of Parshin’s main motivations for creating the theory of higher-dimensional adeles was the potential applications of this theory to problems about zeta-functions and $L$-functions of higher-dimensional arithmetic schemes, as well as applications to various questions of algebraic geometry. Works related to this range of questions constituted his DSc thesis.

In the 1970s–1980s, using algebraic $K$-theory Parshin developed a class field theory for $n$-dimensional local fields (of equal characteristics) and for algebraic surfaces over finite fields. It should be noted that for the construction of this theory and, in particular, for the geometric definition of an $n$-dimensional local field, Parshin proposed to use not just a point or a divisor as the main local object on higher-dimensional schemes, but a flag of embedded subschemes. This idea turned out subsequently to be extremely fruitful, and it is still used to this day in works of various authors. Using it, Parshin also proposed a construction of adelic groups associated with algebraic surfaces, which is necessary for class field theory, gave a new local definition of a residue, and proved a formula for the sum of residues. In addition, he applied these adelic constructions to the theory of sheaves, providing new computational tools for cohomology, intersection theory, Serre duality, and Chern classes.

In the 1990s Parshin discovered an analogue of J.-P. Serre’s theorem on the connection between vector bundles and Bruhat–Tits buildings in the case of an algebraic surface, after having defined and studied the Bruhat–Tits buildings for the groups ${\rm GL}(m,K)$ over an $n$-dimensional local field $K$.

In the mid-1990s Parshin began to take an interest in the theory of integrable systems. He applied higher-dimensional local fields, introduced and studied by him earlier, to this theory. The circle of his interests included the KdV (Korteweg–de Vries) and KP (Kadomtsev–Petviashvili) equations, their connections with other soliton equations (in the spirit of L. D. Faddeev’s school), and the algebraic theory of KP equations (in the spirit of the school of M. Sato). It was known that a certain class of solutions of these equations can be obtained with the help of algebraic curves (as achieved by the school of S. P. Novikov), as well as with the help of hierarchies and the infinite-dimensional Sato Grassmannian. Parshin organized a seminar for undergraduate and graduate students at Moscow State University and read a course of lectures on these topics. At the end of the 1990s–the beginning of the 2000s he wrote several papers in which he constructed a generalization of the Kadomtsev–Petviashvili hierarchy (in the Lax form) for the case of any number of variables and proved the existence of an infinite number of conservation laws for such a system, and also constructed a generalization of the Krichever correspondence to the case of algebraic surfaces. In these constructions, not only higher-dimensional local fields arose, but also higher-dimensional local skew fields — the skew fields of formal iterated pseudodifferential operators. The higher-dimensional Krichever correspondence and higher-dimensional Sato theory have further been developed by Parshin’s students for many years.

In the middle of the 2000s Parshin returned to the circle of questions connected with the higher-dimensional class field theory and the $L$-functions of two- dimensional arithmetic schemes. At the same time he wrote his first joint papers with his students, and his seminar for undergraduate and graduate students continued on a regular basis in the new building of the Steklov institute. Permanent participants of the seminar in different years were students of Parshin (A. B. Zheglov, D. V. Osipov, E. V. Bedulev, S. O. Gorchinskiy, M. V. Mazo, M. A. Dubovitskaya, S. A. Arnal, R. Ya. Budylin, I. V. Beloshapka), as well as many other young (and not only young) and now well-known mathematicians specializing in algebraic geometry, algebra, and number theory (S. S. Galkin, V. S. Zhgun, S. Yu. Rybakov, C. A. Shramov, A. I. Zykin, A. I. Efimov, and others). In a joint work with Gorchinskiy, Parshin gave a new proof of the holomorphic Lefschetz formula for an action of a torus on the cohomology of bundles with the help of adeles.

In several joint papers with Osipov, Parshin created a harmonic analysis on two-dimensional local fields and on adelic spaces of two-dimensional arithmetic schemes, and, as an application, this analysis was used to obtain a new proof of the Riemann–Roch theorem on an algebraic surface over a finite field. The motivation for these papers was the development of the Tate–Iwasawa method for the analytic continuation of $L$-functions from arithmetic schemes of dimension one to arithmetic schemes of dimension two. In a number of papers (one of which was joint with Arnal) Parshin developed the representation theory of discrete Heisenberg groups (including the classification of infinite-dimensional irreducible representations of finite weights of such groups, the construction of the moduli space of representations as a complex manifold, and the calculation of characters as modular forms on the representation space), which he also connected with two-dimensional local fields. He reported about these results at the International Congress of Mathematicians in Hyderabad in 2010, where he was one of the plenary speakers.

Parshin continued active research in this direction in the 2010s. During the last decade he has studied the connections between the global two-dimensional class field theory and the classical one-dimensional Langlands correspondence, which is intensely interesting to many prominent mathematicians. In particular, Parshin formulated a direct image conjecture generalizing a construction in the classical Langlands program to the case of relative dimension one. He showed that this conjecture implies the classical Hasse-Weil conjecture for $L$-functions of two-dimensional arithmetic schemes. He also continued to develop the theory of representations of discrete Heisenberg groups and their connections with local fields and adeles in a series of articles (including joint papers with Osipov).

The style of Parshin’s mathematical works is distinguished by originality and high scientific level. Parshin had a remarkable mathematical intuition that allowed not only to prove theorems, but also to pose deep questions stimulating the development of mathematics for many years. The conjectures he proved made it possible to solve problems in number theory that had remained unsolved for decades. The methods he created made it possible to investigate arithmetic problems by means of geometric tools that initially seemed unrelated.

Parshin’s creative activity was multifaceted: apart from his mathematical work, he wrote popular mathematical articles for a wide audience, and also performed deep investigations of questions in the history and philosophy of science. In particular, in the 1990s he took an active part in the publication and annotation of works of Hermann Weyl and David Hilbert, in the study and publication of works of P. A. Florenskii, and in the 2000s he also participated in organizing and conducting the seminar “Russian Philosophy (Tradition and Modernity)”. His articles on the history of science and Russian philosophy, collected in the monograph “The way. Mathematics and other worlds”, have found a very broad circle of readers. Parshin took a very active public position: in his last decade he, sparing no effort and time, fought with the reforms of the Russian Academy of Sciences, which began in 2013, and with the bureaucratic imposition of bibliometry. Parshin never stayed away from what was happening around him. He helped other people a lot and shared his ideas.

Parshin was an outstanding, internationally recognized mathematician, a top expert in algebraic geometry and number theory. Over the time of his work in the Steklov Mathematical Institute he made considerable achievements, which were highly praised by his colleagues: on May 26, 2000 he was elected a corresponding member of the Russian Academy of Sciences (RAS) in the Mathematics Division, and on December 22, 2011 he was elected an academician of the RAS in the Division of Mathematical Sciences. For his outstanding services to science Parshin, in addition to the above-mentioned talks at International Congresses of Mathematicians and the prize of the Moscow Mathematical Society for young mathematicians, was awarded the Alexander von Humboldt Prize (Germany) in 1996, an honourary doctorate in the sciences from the University of Paris-XIII in 2002, the Vinogradov Prize of the Russian Academy of Sciences in 2004, and the Chebyshëv Gold Medal of the same academy in 2012. In 2017 he was elected a member of the Academia Europaea.

As the head of the Department of Algebra (and the Department of Algebra and Number Theory, for the period of its existence) of the Steklov Mathematical Institute of RAS, Parshin did a great amount of scientific-organizational work concerned with the development of the department and managing its staff. He was the chairman of the dissertation council on algebra, geometry and number theory at the Steklov Institute, and also a member of the editorial boards of the Russian scientific journals Algebra i Analiz¹, Matematicheskii Sbornik², Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia³, Voprosy Filosofii⁴ and, at various times, the international mathematical journals Journal für die reine und angewandte Mathematik (Crelle’s Journal) and International Journal of Mathematics.

It should be remembered that Parshin became the head of the Department of Algebra (created and previously led for many years by Shafarevich) at the Steklov Mathematical Institute in the mid-1990s, at a very difficult time for Russian science. During that period many prominent algebraic geometers left Russia. Over the past years the Department of Algebra (subsequently divided into the Departments of Algebra and Algebraic Geometry) at the Steklov Mathematical Institute has been replenished by a number of bright young mathematicians and has become a prominent international centre attracting algebraic geometers from Russia and a number of well-known universities all over the world.

Parshin published more than 70 research papers. Eight of his students received a Ph.D. degree, and three among these received D.Sc. degrees. A list of Parshin’s mathematical works can be found on the website https://www.mathnet.ru/eng/person11177.

Everyone who knew Alexey Nikolaevich Parshin will remember him as an outstanding scientist, organizer of science, and a wonderful person.

Citation: F. A. Bogomolov, A. M. Vershik, S. V. Vostokov, S. O. Gorchinskiy, A. B. Zheglov, Yu. G. Zarhin, S. V. Konyagin, Vik. S. Kulikov, Yu. V. Nesterenko, D. O. Orlov, D. V. Osipov, I. A. Panin, V. P. Platonov, V. L. Popov, Yu. G. Prokhorov, A. L. Smirnov, “Alexey Nikolaevich Parshin (obituary)”, Uspekhi Mat. Nauk, 78:3(471) (2023), 165–169; Russian Math. Surveys, 78:3 (2023), 549–554

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