S. O. Gorchinskiy, Vik. S. Kulikov, V. V. Nikulin, D. O. Orlov, D. V. Osipov, V. L. Popov, N. A. Tyurin, G. B. Shabat, A. I. Shafarevich, V. V. Shokurov, “Igor Rostislavovich Shafarevich (on the centenary of his birthday)”, Uspekhi Mat. Nauk, 78:6(474) (2023), 187–198; Russian Math. Surveys, 78:6 (2023), 1167

The prominent Russian mathematician, philosopher, and political writer Igor Rostislavovich Shafarevich was born on 3 June 1923, in Zhitomir, the home town of his parents. His father, Rostislav Stepanovich, graduated from the Department of Astronomy of the Faculty of Physics and Mathematics of Moscow Imperial University. However, the research career of the talented graduate could not be realized in full measure in the whirlpools of the revolution and civil war: when he returned to Moscow after the birth of his son, he had to teach theoretical mechanics in various colleges and higher education institutions to support the young family during harsh times. His mother, Yuliya Yakovlevna, was a philologist by education and taught Russian language and literature at school, but she believed that her main job was raising their gifted son. For more than 30 years their family occupied a small room in a communal apartment on Bol’shaya Spasskaya St. One day they had to live through a police search of that room, after which the investigators took with them the kid’s books written in German as ‘evidence’. Luckily, the father, who had been summoned to the Interior Ministry offices on Lubyanka St. and had already bid farewell to the family, came back bringing the ‘evidence’ with him, and no charges were brought.

Since his early years, Igor Shafarevich liked reading books and read up with particular interest on ancient history. By his own account, he saw himself as a professional historian and was not greatly fascinated by mathematics lessons. A turning point was at the age of 13, when during a period of illness the boy had to go through school books independently, and all of surprise it turned out that geometry and algebra were as beautiful and attractive as ancient mythology or Greek poetry, and the textbook were scrutinized from cover to cover.

The parents were not enthusiastic of the boy’s request to give him university textbooks, and the consent was conditional on his learning algebra through German books and geometry through French ones. This did not dampen the desire of the talented boy, so textbooks were bought with open-air booksellers, and the schoolboy immersed himself in university mathematics. After having mastered the standard first-year courses Igor applied to L. A. Tumarkin, the dean of the Faculty of Mechanics amd Mathematics of Moscow State University, to allow him to pass examinations on the basic first-year courses. Tumarkin did not just give his permission, but he referred Shafarevich to three prominent professors: Boris Nikolaevich Delone was to administer an exam in analytical geometry, Aleksandr Gennad’evich Kurosh to administer an exam in algebra, and Israel Moiseevich Gelfand to administer an exam in analysis. This was not an accidental choice: all the three were not just very good experts, but also attentive teachers. For example, Gelfand proposed straight away that Shafarevich should read some additional literature, supplying him with a bunch of books, Kurosh advised him to read the Van der Waerden monograph for a broader view of algebra, while Delone surprised the aspiring young researcher with an advice to abandon all other things and learn Galois theory, presenting him with a book by Chebotarev, in which the applicant got lost immediately. In this way he passed the exams in advanced courses on all subjects, and the former schoolboy, who had just graduated from his ninth schoolyear, became a full-scale senior university student at the Faculty of Mechanics and Mathematics. In 1938 he already participated in research seminars, attended special courses (the first of which was delivered by Gelfand — and, as Shafarevich told, for Gelfand this also was his first lecture course at the faculty). In modern terms, Gelfand was also the first scientific advisor of Igor Shafarevich, although Kurosh was also formally one of his diploma advisors. However, the central role in the education of the talented youth was that of Delone: apart from mathematics, this included trips to the conservatory (which became Shafarevich’s life-long passion), hiking, and mountaineering (which also became his hobby for the rest of the life). Delone even made an ‘offence’ by borrowing books for the whole summer from the library of Steklov Mathematical Institute for the university student Shafarevich — although they were only to be borrowed for short periods of time. He took Shafarevich to Steklov Institute, introduced him to colleagues, and even invited him to a meeting of the Council of the Institute (many years later Shafarevich recalled some details of the speech of academician A. N. Krylov at that meeting). And he continued to push his younger friend towards Galois theory. However, this topic was put on hold at that time: in 1940 Shafarevich wrote a diploma thesis on a subject proposed by Gelfand and, together with his classmate V. A. Rokhlin, was admitted to graduate studies.

WW2 undermined all plans: on the second year of his graduate studies Shafarevich was digging trenches near Mozhaisk for two months, and after his return to Moscow Delone and Kurosh petitioned that the outstanding student should be evacuated to Ashkhabad together with the whole university. The professional ties just established were mostly cut because the Mathematical Institute, as a part of the Academy of Sciences, was evacuated to Kazan’. Nonetheless, in 1942 Shafarevich defended sucessfully his Ph.D. thesis and became a doctoral candidate at the Mathematical Institute. From 1943, on returning from evacuation, he also started teaching at Moscow State University.

In Shafarevich’s first paper, published in 1943, he described topological fields on which a topology can be defined by means of a valuation. A criterion was stated in terms of the concepts of topologically nilpotent elements and bounded sets, considered in the paper and used extensively, in particular, in the modern theory of Huber rings. This paper was noticed by international experts: in 1948 the renowned Canadian-American mathematician I. Kaplansky mentioned an ‘elegant result due to Shafarevich’. After defending a Ph.D. thesis on this topic Shafarevich never returned to it.

For a whole decade Shafarevich’s interests were focused on Galois theory and algebraic number theory, the study of which he started by considering extensions of local fields. He proved that, given a prime number $p$ and a local field $K\supset {\mathbb Q}_p$ not containing roots of unity of degree $p$, the Galois group of the maximal $p$-extension is a free pro-$p$-group with $[K:{\mathbb Q}_p]+1$ generators. As main applications, he solved the inverse Galois problem and the embedding problem for such fields and finite $p$-groups. These results were included in Shafarevich’s D.Sc. thesis, which he defended in 1946. A little later, in 1949, he was awarded the prize of the Moscow Mathematical Society for this work. It is interesting to note that at the time Shafarevich did not explicitly use group cohomology, which had just appeared. Subsequently, it became one of his favorite research tools.

One of the central results in algebraic number theory is the general reciprocity law established by Shafarevich in 1950, which solved Hilbert’s ninth problem. More precisely, for $p>2$, given a local field $K\supset {\mathbb Q}_p$ such that $\mu_p\subset K$ and $\alpha,\beta\in K^*$, he found an explicit formula for the Hilbert symbol $(\alpha,\beta)\in \mu_{p}$. This is a far-fetching generalization of the sign in Gauss’s quadratic reciprocity law; on its basis one can prove the general reciprocity law for arbitrary, not necessarily quadratic, roots of unity. A major conceptual feature of Shafarevich’s formula is an analogy with residues of differential forms on a Riemann surface which he made transparent. In general, an analogy between numbers and functions permeated virtually the whole body of Shafarevich’s research, was passed on to his students, and became a central idea of the Moscow school in algebraic geometry.

Shafarevich’s formula was extensively developed by A. I. Lapin, Shafarevich’s first student, and by many of Shafarevich’s followers, including S. V. Vostokov, H. Brückner, K. Iwasawa, A. Wiles, K. Kato, and also Shfarevich’s student V. A. Abrashkin. The formula was significantly generalized, and many interpretations of it were found along the way, inclucing ones in terms of contemporary $p$-adic cohomology theory.

In 1952 Shafarevich became a member of the editorial board of the journal Izvestiya Rossiiskoi Academii Nauk. Seriya Matematichaskaya¹ and he remained there unil 2017, for more than 60 years. From 1957 to 1977 he was a deputy editor-in-chief of the journal.

After his work on the general reciprocity law Shafarevich returned to the inverse Galois problem, this time for global, rather than local, number fields. His efforts were successful, and in 1954 a cycle of papers on this subject was published, culminating in the solution of the inverse Galois problem for solvable groups: given an arbitrary number field $K\supset {\mathbb Q}$ and a finite solvable group $G$, it was shown that a Galois extension $K\subset L$ with the Galois group $G$ exists. Techniques used to prove these results involved some fine arithmetical properties of number fields, and the step-by-step construction of consecutive field extensions required extremely delicate work. In addition, in those papers Shafarevich started to use homological algebra essentially, by availing on a number of D. K. Faddeev’s results on group cohomology.

For the cycle of papers on the inverse Galois problem for algebraic number fields and solvable groups, in 1959 Shafarevich was awarded the Lenin Prize.

The inverse Galois problem for algebraic number fields and unsolvable groups was the subject of subsequent papers by G. V. Belyi, a student of Shafarevich. For his investigations he had to make his famous discovery: he characterized smooth projective curves that can be defined over the field of algebraic numbers as curves covering a projective line with just three branching points.

Shafarevich also treated this range of problems, jointly with his student S. P. Demushkin, in the late 1950–early 1960s. He solved the embedding problem for several important cases of local and global fields, showed that a certain known cohomological obstruction is the only one in the local case, and found an additional obstruction in the global case.

Along with these investigations, in 1957 he published a joint paper with his student A. I. Kostrikin on cohomology of nilpotent Lie algebras. Namely, for a local finite-dimensional augmented algebra $A$ over a field $k$ they found lower and upper estimates for the Poincaré series $P_A(t)=\sum_{i\geqslant 0}\dim\operatorname{Ext}^i_A(k,k)\,t^i$ and posed a question on whether it is rational. Although it turned out that the series $P_A(t)$ might not be rational, in the important case when $k={\mathbb F}_p$ and $A={\mathbb F}_p[G]$, where $G$ is a $p$-group, this series was shown to be rational in 1959 by E. S. Golod, a student of Shafarevich.

In the mid-1950s Shafarevich turned to algebraic geometry. Quite naturally, the first problems he looked at were at the crossroads between number theory and geometry: these problems related to the theory of elliptic curves. He established a bijection between the set of torsors over an elliptic curve $E$ over a field $k$ and the Galois cohomology group $H^1(G_k,E(k^{\mathrm{sep}}))$. This was historically the first cohomological interpretation of such objects, which obtained later far-reacing generalizations and developed into a standard tool of algebraic and arithmetic geometry.

Using this cohomological interpretation Shafarevich solved a long-standing problem in the theory of Diophantine equations: he proved that for each positive integer $d$ there exists a genus-one curve over ${\mathbb Q}$ that has a smooth embedding of degree $d$ into a projective space and admits no embeddings of lower degree. All these results were the first steps in the theory of principal homogeneous spaces, a new field in algebraic geometry which emerged in the 1950s and was developed, independently of Shafarevich, by S. Lang and J. Tate.

Shafarevich discovered the fundamental role of the group of classes of principal homogeneous spaces for an abelian variety $A$ over a number field $K$ that have a point over any completion of $K$. Now this group is denoted by the Cyrillic character Ш (after Shafarevich) in the literature and is called the Shafarevich–Tate group. The calculation of this group and, in particular, showing that it is (conjecturally) finite are among the most difficult and interesting problems in the theory of Diophantine equations. The strongest results here are due to V. A. Kolyvagin, a student of Shafarevich, who proved that the Shafarevich–Tate group is finite,and also proved the Birch–Swinnerton-Dyer conjecture for all (modular) elliptic curves of analytic rank at most one.

In 1958, at the age of 35, Shafarevich became a corresponding member of the Academy of Sciences of the USSR.

In an extended paper from 1961 devoted to I. M. Vinogradov’s 70th birthday he carried out a fundamental investigation of principal homogeneous spaces for an abelian variety $A$ over the field of rational functions $K$ on a curve $C$ over an algebraic number field. Furthermore, the first step was to establish, in the local case, duality between the torsor group for $A$ and the Tate module of the dual abelian variety $\widehat{A}$. Subsequently, this was generalized by many authors, including the student of Shafarevich O. N. Vvedenskii. In the global functional case the Shafarevich–Tate group and the cokernel of the corresponding localization homomorphism were described. Moreover, at the time when there was no theory of étale cohomology, Shafarevich essentially calculated the Euler characteristic of a constructible sheaf in the étale topology on $C$. Independently, such results were obtained by A. Ogg and then generalized by A. Grothendieck. The corresponding formula for the Euler characteristic of an étale sheaf on a curve is called the Grothendieck–Ogg–Shafarevich formula. Apart from numerous applications to constant and unramified abelian varieties over $K$, that paper also presented a criterion for the existence of a good reduction of an abelian variety in terms of representations of the Galois group, which was subsequently called the Néron–Ogg–Shafarevich criterion.

From 1960 until 1995 Shafarevich was the head of the Department of Algebra at Steklov Mathematical Institute. In 1960 he was elected a member of the German National Academy of Natural Sciences Leopoldina.

Already in his works on principal homogeneous spaces appeared a feature characteristic for Shafarevich’s subsequent research: in most papers he looked at geometry as a number theorist and at number theory as a geometer. This is how his plenary lecture at the International Congress of Mathematicians in Stockholm (1962) was organized. The central place in it was taken by two conjectures on algebraic curves defined over a number field or the field of rational functions on a curve over a finite field. Inspired by classical results due to C. Hermite and G. Minkowski, Shafarevich conjectured that there can only be finitely many curves over $K$ with prescribed genus $g\geqslant 2$ and a fixed finite set of bad reduction $S$. Moreover, in the case when $K={\mathbb Q}$, $g\geqslant 1$, and $S=\varnothing$ he conjectured that there exist no curves with such invariants (with some simple adjustments this can also be formulated in the geometric case). A functional analogue of the first conjecture was established by Shafarevich’s students A. N. Parshin and S. Yu. Arakelov. In addition, for an arbitrary global field Parshin reduced the celebrated Mordell conjecture to the first of these conjectures. In combination with some results due to Yu. G. Zarhin, another representative of Shafarevich’s school, these were the key steps towards the proofs of Shafarevich’s finiteness conjecture and Mordell’s conjecture over a number field, which was subsequently found by G. Faltings. Note that the functional analogue of Moredell’s conjecture had previously been proved in two different ways by Parshin and by Yu. I. Manin, a student of Shafarevich.

The second conjecture was generalized and proved by Abrashkin and, independently, by J.-M. Fontaine from France.

Shafarevich returned to the Galois groups of number fields in an important paper from 1963 published in Russian in the journal Publications Mathématiques de l’IHÉS. He considered there the Galois group $G_K(p,S)$ of a maximal pro-$p$-extension of an algebraic number field $K$ which is ramified in a prescribed finite set $S$ of non- Archimedean points of $K$. A formula for the number $d$ of generators of the group $G_K(p,S)$ was obtained in arithmetic terms, and an explicit upper bound for the number $r$ of relations was found. The group $G_K(p,S)$ was also investigated subsequently, in particular, by the German mathematician H. Koch, who was a student of Shafarevich.

Apart from a number of important consequences in algebraic number theory, Shafarevich also discovered a link between the — conjectural at that time — lower bound for $r$ for finite $p$-groups and the tower problem in class field theory. Just a year later, in a joint paper with his student Golod, they established the inequality $r>(d-1)^2/4$ for finite $p$-groups, which provided an affirmative solution to the tower problem, which had stayed open for more than 40 years. Using the methods developed in that paper, Golod obtained a negative solution to the generalized Burnside problem.

In 1964, the book Number theory was published, which was written jointly with Z. I. Borevich and was based on Shafarevich’s lecture courses at Moscow State University. This textbook was immediately translated into English, German, French, and Japanese.

In 1961–1963, while continuing research on algebraic number theory, Shafarevich organized a seminar, whose few participants were mostly his students: these included B. G. Averbukh, Yu. R. Vainberg, A. B. Zhizhchenko, Yu. I, Manin, B. G. Moishezon, A. N. Tyurin, and G. N. Tyurina. Their aim was to scrutinize classical papers by Italian algebraic geometers from the contemporary (for that time) point of view. Independently, similar work was also carried out by O. Zarisky and D. Mumford in the USA and by K. Kodaira in Japan.

As a result of the work of this seminar, a collection of papers was published in 1965, which for a log time remained the only systematic presentation of the theory of surfaces. It made a tremendous impact on the study of algebraic surfaces worldwide, and was translated into English in 1967 and into German in 1968. The seminar and the book were the main incentives for the subsequent development of algebraic geometry in Moscow. The investigations that originated directly from them covered the following directions: rational suraces and rational higher-dimensional varieties; the solution of the Lüroth problem (V. A. Iskovskikh and Manin) and the classification of three-dimensional Fano varieties (Iskovskikh); the theory of vector bundles over algebraic curves and surfaces (Tyurin and F. A. Bogomolov); К3-surfaces (Tyurina, Shafarevich, I. I. Piatetski-Shapiro, V. V. Nikulin, A. N. Rudakov, Vik. S. Kulikov, and others); higher-dimensional birational and complex analytic geometry (B. G. Moishezon); non-rational simply connected surfaces with trivial geometric genus (I. V. Dolgachev); geometry and arithmetics of rational surfaces (Manin); a classification of complex non-algebraic surfaces (Bogomolov).

In 1966, notes of Shafarevich’s lectures at the Tata Institute in Bombey, India were published. There he considered minimal models and the geometry of schemes of dimension two. This provided the first serious opportunity to look at arithmetic surfaces as geometric objects. In particular, Shafarevich posed a question on a geometric approach to arithmetic surfaces that takes into account Archimedean fibres. This prompted his student S. Yu. Arakelov to develop arithmetic intersection theory, later named Arakelov geometry. Arakelov’s approach used some previous constructions, related to Green’s functions, due to Parshin.

After the seminar on algebraic surfaces terminated its work, in 1964–66 Shafarevich organized a seminar at Steklov Mathematical Institute in which various questions related to E. Cartan’s classification of simple transitive pseudogroups of transformations were discussed. It was partly as a result of these activities that two papers with Kostrikin appeared, which for decades ahead defined the programme of classification of simple finite-dimensional Lie algebras over fields of positive characteristic. In those papers they stated a conjecture predicting that for a prime number $p>5$ all simple finite-dimensional Lie ($p$-)algebras over algebraically closed field of characteristic $p$ are either classical or of one of the four Cartan types. Many years of work of various authors, such as R. Block, R. Wilson, H. Strade, and A. Premet, resulted in a proof of this conjecture.

In a paper from 1966, one of the few written in English, with the study of the automorphism group of an affine space over a field of characteristic zero in mind, Shafarevich initiated the development of a new line in algebraic geometry, the theory of infinite-dimensional algebraic varieties. Namely, he regarded $\operatorname{Aut}(\mathbb A^n)$ as an object of infinite-dimensional algebraic geometry, or, using modern terminology, as a group ind-affine variety. Shafarevich established smoothness of the ind-affine variety $\operatorname{Aut}({\mathbb A}^n)$ and the fact that it is generated as a group ind-affine variety by the affine and upper triangular transformations. He also proved that the group ind-affine variety $\operatorname{SAut}({\mathbb A}^n)$ consisting of the automorphisms with unit Jacobian is simple. In a subsequent paper from 2004, dedicated to the memory of his late student A. N. Tyurin, Shafarevich considered the structure of a group ind-affine variety on the group of invertible matrices over the ring of polynomials over a field.

Note that similar results fail for the group of complex points $\operatorname{Aut}({\mathbb A}^n)({\mathbb C})$ on $\operatorname{Aut}({\mathbb A}^n)$ regarded as an abstract group. Namely, I. P. Shestakov and U. U. Umirbaev constructed an example of an element of $\operatorname{Aut}({\mathbb A}^n)({\mathbb C})$ that cannot be decomposed into a product of affine and upper triangular transformations, while V. I. Danilov showed that $\operatorname{SAut}({\mathbb A}^n)({\mathbb C})$ is not a simple group.

Shafarevich also considered infinite-dimensional objects of algebraic geometry in the context of uniformization of algebraic varieties, where pro-algebraic varieties appear, rather than ind-algebraic ones. In a paper from 1966, written with his student Piatetski-Shapiro, they considered an algebraic analogue of uniformization. Namely, given a variety $X$, they asked whether or not there exists a profinite covering by a pro-algebraic variety $\widetilde{X}\to X$ that is unramified over an open subset of $X$ and is quasi-homogeneous, that is, such that the general orbit of the group $\operatorname{Aut}(\widetilde{X})$ is dense. They proved that all quotient spaces of bounded symmetric domains by arithmetic groups have this property. For their proof, they had to investigate closely fields of automorphic functions and algebras of automorphic forms from the algebraic point of view, and to develop some new concepts in the theory of pro-algebraic varieties.

In the context of uniformization of algebraic varieties Shafarevich also addressed subsequently the following question: is it true that for each smooth complex projective variety $X$ its simply connected covering space $\widetilde{X}$ is holomorphically convex? As J. Kollár noted, an affirmative answer to this question is equivalent to the existence of a morphism $X\to \operatorname{Sh}(X)$ into a normal variety $\operatorname{Sh}(X)$ with connected fibres such that a connected subvariety $Z\subset X$ contracts to a point if and only if the image of the homomorphism $\pi_1(Z)\to \pi_1(X)$ is finite. If we consider only the maximal abelian quotient of the fundamental groups, then the Albanese morphism is the relevant version of the Shafarevich map. Although the question of the existence of the Shafarevich map is still open in the general case, significant progress towards an affirmative answer has been made by many authors, including Kollár, L. Katzarkov, P. Eyssidieux, T. Pantev, M. Ramchandran, R. Treger, and others.

During his stay in Paris in 1966 Shafarevich found, jointly with Tate, examples of elliptic curves of arbitrarily high rank over the field ${\mathbb F}_p(t)$. These examples are forms over ${\mathbb F}_p(t)$ of supersingular elliptic curves defined over ${\mathbb F}_p$. In particular, these elliptic curves are isotrivial over ${\mathbb F}_p(t)$. Examples of non-isotrivial elliptic curves of unbounded rank over ${\mathbb F}_p(t)$ were constructed much later by D. Ulmer. The question of whether the ranks of elliptic curves over number fields are unbounded is still open.

In 1966–1967 Shafarevich read a lecture course at the Faculty of Mechanics and Mathematics of Moscow State University, in which he gave a multifaceted presentation of the theory of zeta functions. Lecture notes from the course, taken by Manin, were soon published as a separate monograph.

From 1970 till 1973 Shafarevich was the president of the Moscow Mathematical Society.

At that time he worked intensively on the theory of K3-surfaces. Together with abelian surfaces, they are an analogue of elliptic curves. It can easily be shown that, similarly to complex elliptic curves, complex abelian surfaces and, more generally, complex abelian variaties are uniquely determined by their first integral Hodge structure. However, all attempts to establish a similar result for K3-surfaces encountered significant difficulties, and for a long period of time the problem seemed to be untreatable to experts. Nevertheless, in cooperation with Piatetski-Shapiro, Shafarevich managed to establish an analogue of Torelli’s theorem for K3-surfaces. This was the subject of his lecture at the International Congress of Mathematicians in Nice (1970). Namely, it can be shown that for two arbitrary K3-surfaces $X$ and $X'$ there is a natural bijection

The proof was based on the local Torelli theorem for K3-surfaces, established previously by Tyurina, a student of Shafarevich. In addition, they proved that for K3-surfaces the period map is an open embedding with dense image. The fact that it is also surjective was proved subsequently by Shafarevich’s student Vik. Kulikov. As an application, Torelli’s theorem allows one to describe the automorphism groups of K3-surfaces in terms of the arithmetic properties of lattices. In particular, Nikulin, a student of Shafarevich, classified K3-surfaces with finite automorphism groups. Subsequently, ideas related to Torell’s theorem for K3-surfaces were developed by many authors in various ways. Here we can mention Shafarevich’s students A. N. Todorov and Tyurin, and also papers by D. O. Orlov, M. S. Verbitsky, D. Burns, M. Rapoport, E. Looijenga, C. Peters, and Y.-T. Siu.

In another paper from that period of time, also co-authored with Piatetski- Shapiro, Shafarevich deduced from Torelli’s theorem for complex algebraic K3-surfaces an analogue of Riemann’s hypothesis for K3-surfaces over finite fields. Their proof was significantly different from the one given independently by P. Deligne.

In 1972 Shafarevich published his book Foundations of algebraic geometry, which is still one of the best and most popular textbooks on algebraic geometry in the world.

On the occasion of the Dannie Heineman Prize award ceremony organized in 1973 by the Göttingen Academy of Sciences and Humanities, Shafarevich gave a talk entitled On certain tendencies in the development of mathematics, which was essentially a programme address.

In 1974 he was elected a member of the National Academy of Sciences of the USA and the American Academy of Arts and Sciences.

The analytic methods used for the investigation of complex analytic K3-surfaces seemed to be an unsurmountable barrier to the study of such surfaces over fields of positive characteristic. The first breakthrough in this direction was due to M. Artin, whose paper, according to I. V. Dolgachev, Shafarevich described as one of the most beautiful papers he had ever read. In that paper Artin introduced the notion of periods for supersingular K3-surfaces, that is, those K3-surfaces over an algebraically closed field of positive characteristic all of whose classes in the second étale cohomology are algebraic.

This theory was significantly developed in a large cycle of papers from 1976– 1984 which Shafarevich wrote with his student A. N. Rudakov and, in part, also with T. Zink. There they proved that there are no vector fields on K3-surfaces in positive characteristic, which means that one can consider a formal moduli space. They also discovered unirationality of supersingular K3-surfaces over a field of characteristic $2$. Their main results were a construction of the moduli space for supersingular K3-surfaces, establishing completeness of this variety (that is, the absence of degenerations of supersingular K3-surfaces), analogues of Torelli’s theorem, and a result on the surjectivity of the period map.

Shafarevich published many monographs and textbooks (some of them in co- authorship). The clear and transparent presentation, an abundance of examples, and the gradual progress from simple cases to complicated ones were characteristic features of his books. The book Geometries and groups (joint with Nikulin) published in 1983, is an introduction to contemporary geometry, which is accessible even to high school students. The survey Basic notions of algebra, written in one breath in 1986, immediately became widely known, even to experts in areas far from algebra. The survey Algebraic surfaces (1989; joint with V. A. Iskovskikh) is a perfect textbook for students interested in algebraic geometry.

In 1978 Shafarevich received honourary doctorate from Université Paris-Sud (Orsay). Since 1981 he was a member of the London Royal Society. In 1991 he was elected a member of the Russian Academy of Sciences.

In the latest period of his creative work Shafarevich returned to his three main lines of research: algebra, algebraic geometry, and number theory. In a paper from 1990 he considered the moduli space of finite-dimensional nilpotent commutative algebras $N$ (without unity) satisfying $N^3=0$. He described the singularities and the dimensions of the irreducible components of this variety and established a bijection between the irreducible components and the dimensions $r=\dim(N^2)$. In addition, he studied the stability of such algebras, that is, the closedness of this class of algebras under deformations in the class of all commutative associative algebras. It turned out that for $2<r\leqslant (d-1)(d-2)/6+2$ algebras in the corresponding irreducible components are stable, whereas for $r>(d^2-1)/3$ they are unstable; here $d=\dim(N/N^2)$. In a later paper from 2001, dedicated to the memory of his former student A. I. Kostrikin, he stated a conjecture, supported by a number of examples, that the singularities of the irreducible components of the moduli space of algebras containing semisimple classes have codimension at least $2$.

In two papers from 1996 Shafarevich proved finiteness of the number of K3- surfaces with Picard number 20 and of abelian surfaces with Picard number 4, defined over a number field or an extension of ${\mathbb C}(t)$ of bounded degree. In the next case in terms of difficulty (after the maximal possible Picard number) he proved that there are only finitely many Néron–Severi lattices of K3 surfaces with Picard number 19 and of abelian surfaces with Picard number 3 defined over an extension of the field ${\mathbb C}(t)$ of bounded degree. He stated a general conjecture that there are only finitely many Néron–Severi lattices of K3-surfaces and abelian surfaces defined over a number field or an extension of ${\mathbb C}(t)$ of bounded degree. Since then, this conjecture was studied, and proven in some special cases by M. Orr, A. N. Skorobogatov, and Zarhin.

In a paper written at the age of 90, Shafarevich, who followed some ideas of K. Heegner, presented an elementary proof of the 10th discriminant theorem; that is, that there exist only nine imaginary quadratic fields with class number one. Heegner’s proof, while a complete mathematical argument, was not very clearly presented, and, for more than 15 years, it had remained unclear to experts. Subsequently, proofs of the 10th discriminant theorem were presented by H. Stark and A. Baker, while Heegner’s original ideas were transformed into a generally accepted proof by M. Deuring and B. Birch. Finally, a transparent proof, based on more geometric methods, was given by Shafarevich in a lecture course at Moscow State University delivered in the 1970s. This exact proof was the subject of Shafarevich’s last research paper, which the great master wrote with his characteristic clarity and elegance, combined with the conceptual depth of his approach.

Shafarevich’s interests in life were not limited to mathematics: at the very beginning of his creative career he had to choose between mathematics and history, and later he did not put on hold his humanitarian talents. Of the six volumes of his selected works, the first five contain reflections on a wide range of areas, from classical music to biology, from the history of mathematics to the history of Russia. One of the first studies was on Konrad Lorenz’s ethology, a science of animal behaviour, but in a much broader context: even at that time questions of history and civics seemed to be of great significance for Shafarevich, which in a natural way led him to comparisons between the bases on which the human society is built and the structure of communities of living things. It was in connection with the history of his home country that he initiated the studies summarized in his book Socialism as a phenomenon of world history, which was published in many languages and attracted wide public attention. In that book the author, like a true scientist, accepted the challenge and dispelled the main myth of his time, that marxism is a scientific doctrine. In his later paper Two roads towards one precipice he revealed the deep interlinkage beteen the two seemingly antagonistic formations, capitalism and socialism, which in realuty unite in opposition to the natural flow of life, to human individuality, and, in particular, to nature, which must be conquered, re-engineered, and transformed into a kind of a factory under the control of the privileged few.

Although brought up in an academic environment, Shafarevich did not adopt some of the rules of behaviour common for this community at that time, and for all of his life he stood for what was right as firmly as possible. This was clearly evident in his permanent support of those whose rights were infringed for some reason or another. In 1953 he defended Yuri Manin, his future student, in the question of his admission to Moscow University. Later he defended Boris Moishezon, who, upon having defended his Ph.D. thesis, was under the threat of prosecution for offence of ‘loitering’. Many times he spoke out in defence of talented youth facing biased admission examination at Moscow State University. Apparently, such ‘occasional’ public activity did not meet the demands of Shafarevich conscience, and for a more active involvement he joined the human rights committee chaired by another prominent researcher, academician A. D. Sakharov. However, by choosing the topic of his first report to the committee Shafarevich went beyond the already established patterns of the liberal human rights movement: his report on violations of the rights of believers and, first of all, Orthodox Christians in the Soviet Union, made of him the people’s advocate, the recipient of letters of ordinary people from the whole country, who wrote to him about their woes. So, when another prominent researcher, the future academician O. A. Ladyzhenskaya, introduced Shafarevich to A. I. Solzhenitsyn, this encounter meant for both that they found a friend and a co-worker, who was on the same wave in the main, and could be complementary in details. Solzhenityn’s long-conceived project of publishing a collection of uncensored reflections on the destiny of Russia, entitled symbolically From under the blocks, obtained the second lead author, namely, Shafarevich, who submitted several articles for this collection. While working on that book, Shafarevich became an involuntary witness of Solzhenitsyn’s arrest, and the subsequent exile of the writer delayed the appearance of the collection.

One would say that the humanitarian component of the life of a prominent academic should not affect the mathematical component, but Shafarevich’s joint work with Solzhenitsyn inflicted heavy blows on his mathematical life and had serious consequences not only for Shafarevich himself, but also for future generations of mathematicians: he was suspended from teaching at Moscow State University, and subsequently some of his students were suspended as well.

Twenty years later history repeated itself: after his Russophobia was published, the most engaged part of the international mathematical community attempted to isolate Shafarevich from the ‘world of mathematics’ by requesting, in particular, to cancel invitations of him to mathematical events (which was partly accomplished) and exclude him from the National Academy of Sciences of the USA (which turned out to be impossible; subsequently, Shafarevich withdrew himself from the academy in a protest against the American foreign policy). Without going into details, which could take us far beyond the scope of our text, we note only that neither at that time, nor later did any moral or legal conventions in Russia, or other countries (including Germany) where that book was published in different languages, ever hamper its publication. Already after Shafarevich’s death, one of his former students, who had signed a collective letter against his mentor in connection with Russophobia, judged what he thought to be a ‘bias’ in the selection of facts in that book as a sheer consequence of its author’s compassion for the sufferings of the Russian people in the 20th century.

There is also another subject, concordant with mathematics, that arose in Shafarevich’s papers: in several essays he discussed classical music, mostly the music of Shostakovich, and, according to experts, the level of discussion was highly professional. Even without formal musical education it proved to be possible to extrapolate one’s mathematical intuitions to other harmonies. It was the same with literature: being a connoisseur and admirer of ancient literature, Shafarevich oriented very well in the whole range of world literature and left deep and interesting notes concerning the most significant — although not necessarily best known — Russian literary works of the 20th century.

Of course, he could not be silent about the philosophy and history of mathematics itself: a recent collection of papers edited by A. N. Parshin, a former student of Shafarevich, contains his reminiscences of teachers, colleagues, friends, and students. One of the most interesting and frequently cited texts in that collection, was Shafarevich’s lecture written for the Dannie Heineman Prize award ceremony in Göttingen. He raised there a question of significance for all mathematicians: what is the aim of mathematics as such? The answer he proposed after an interesting discussion sums up Shafarevich’s entire creative work, in mathematics and on humanitarian issues alike, and links the two ways that Shafarevich followed throughout his life.

Apart from Shafarevich’s 35 direct students, experts in algebra, geometry, and number theory, his scientific school counts more than 300 mathematicians, most of whom are widely known researchers of high professional level. All of his students keep memories of their path travelled alongside Igor Shafarevich as of the happiest time of their creative development.

Citation: S. O. Gorchinskiy, Vik. S. Kulikov, V. V. Nikulin, D. O. Orlov, D. V. Osipov, V. L. Popov, N. A. Tyurin, G. B. Shabat, A. I. Shafarevich, V. V. Shokurov, “Igor Rostislavovich Shafarevich (on the centenary of his birthday)”, Uspekhi Mat. Nauk, 78:6(474) (2023), 187–198; Russian Math. Surveys, 78:6 (2023), 1167–1178

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