Igor Shafarevich
Encyclopedia
Igor Rostislavovich Shafarevich is a Soviet and Russian
Russians
The Russian people are an East Slavic ethnic group native to Russia, speaking the Russian language and primarily living in Russia and neighboring countries....

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, founder of a school of algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

 and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

 in the USSR, and a political writer. He was also an important dissident
Dissident
A dissident, broadly defined, is a person who actively challenges an established doctrine, policy, or institution. When dissidents unite for a common cause they often effect a dissident movement....

 figure under the Soviet regime, a public supporter of Andrei Sakharov
Andrei Sakharov
Andrei Dmitrievich Sakharov was a Soviet nuclear physicist, dissident and human rights activist. He earned renown as the designer of the Soviet Union's Third Idea, a codename for Soviet development of thermonuclear weapons. Sakharov was an advocate of civil liberties and civil reforms in the...

's Human Rights Committee from 1970. He supported the criticisms of Aleksandr Solzhenitsyn
Aleksandr Solzhenitsyn
Aleksandr Isayevich Solzhenitsyn was aRussian and Soviet novelist, dramatist, and historian. Through his often-suppressed writings, he helped to raise global awareness of the Gulag, the Soviet Union's forced labor camp system – particularly in The Gulag Archipelago and One Day in the Life of...

 of both Soviet communism
Communism
Communism is a social, political and economic ideology that aims at the establishment of a classless, moneyless, revolutionary and stateless socialist society structured upon common ownership of the means of production...

 and liberal
Liberalism
Liberalism is the belief in the importance of liberty and equal rights. Liberals espouse a wide array of views depending on their understanding of these principles, but generally, liberals support ideas such as constitutionalism, liberal democracy, free and fair elections, human rights,...

 proposals for the future of Russia.

Work in mathematics

Shafarevich made fundamental contributions to several parts of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...


including algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

, algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

 and arithmetic algebraic geometry.
In algebraic number theory the Shafarevich-Weil theorem extends the commutative
reciprocity map to the case of Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

s which are extensions of abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s by finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

s.
Shafarevich was the first to give a completely self-contained formula for the pairing
which coincides with the wild Hilbert symbol
Hilbert symbol
In mathematics, given a local field K, such as the fields of reals or p-adic numbers, whose multiplicative group of non-zero elements is K×, the Hilbert symbol is an algebraic construction, extracted from reciprocity laws, and important in the formulation of local class field theory...

 on local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

s, thus initiating an important branch
of the study of explicit formulas in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

.
Another famous result is the realization of every finite solvable group
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...


as the Galois group over rationals
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

. Another fundamental result is
the Golod-Shafarevich theorem on towers of unramified extensions of number fields.
Shafarevich and his school greatly contributed to the study of algebraic geometry of surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

s.
He initiated a Moscow
Moscow
Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...

 seminar on classification of algebraic surfaces that updated around 1960 the treatment of birational geometry
Birational geometry
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...

, and was largely responsible for the early introduction of the scheme theory approach to algebraic geometry in the Soviet school. His investigation in arithmetic of elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

s led him independently of John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...


to the introduction of the most mysterious group related
to elliptic curves over number fields, the Tate-Shafarevich group (usually called 'Sha', written 'Ш', his Cyrillic initial). He also formulated the Shafarevich conjecture which stated the finiteness of the set of Abelian varieties over a number field having fixed dimension and prescribed set of primes of bad reduction. This conjecture was proved by Gerd Faltings
Gerd Faltings
Gerd Faltings is a German mathematician known for his work in arithmetic algebraic geometry.From 1972 to 1978, he studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathematics and in 1981 he got the venia legendi in mathematics, both from the University...

 as a step in his proof of the Mordell conjecture.

Shafarevich was a student of Boris Delone, and his students included Yuri Manin, A. N. Parshin,
I. Dolgachev, Evgeny Golod
Evgeny Golod
Evgenii Solomonovich Golod is a Russian mathematician who proved the Golod–Shafarevich theorem on class field towers. As an application, he gave a negative solution to the Kurosh–Levitzky problem on the nilpotency of finitely generated nil algebras, and so to a weak form of Burnside's problem....

, A.I. Kostrikin
Alexei Kostrikin
Alexei Ivanovich Kostrikin was a Russian mathematician.-Further reading:*V A Artamonov, Yu A Bahturin, I A Chubarov, et al., Dedication: Alexei Ivanovich Kostrikin, Comm. Algebra 29 , ix-xiv....

, I.A. Kostrikin, S.Y. Arakelov, G.V. Belyi, V. Abrashkin, A. Tyurin and V. A. Kolyvagin
Victor Kolyvagin
Victor Alexandrovich Kolyvagin is a Russian mathematician who wrote a series of papers on Euler systems, leading to breakthroughs on the Birch and Swinnerton-Dyer conjecture, and Iwasawa's conjecture for cyclotomic fields. His work also influenced Andrew Wiles's work on Fermat's Theorem.Kolyvagin...

.
In view of later accusations of anti-Semitism
Anti-Semitism
Antisemitism is suspicion of, hatred toward, or discrimination against Jews for reasons connected to their Jewish heritage. According to a 2005 U.S...

 on his part, it can be noted that his research students included some identified as Jewish, and that later, during his most serious troubles in the 1970s with the Soviet authorities, he did major work in collaboration with Ilya Piatetski-Shapiro on K3 surface
K3 surface
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....

s. He is a member of the Serbian Academy of Sciences and Arts
Serbian Academy of Sciences and Arts
The Serbian Academy of Sciences and Arts is the most prominent academic institution in Serbia today...

 in department of Mathematics, Physics and Earth Sciences.

On his 80th birthday, Russian President Vladimir Putin
Vladimir Putin
Vladimir Vladimirovich Putin served as the second President of the Russian Federation and is the current Prime Minister of Russia, as well as chairman of United Russia and Chairman of the Council of Ministers of the Union of Russia and Belarus. He became acting President on 31 December 1999, when...

 hailed his "fundamental research" in mathematics and his creation of "a large scientific school that is known both in Russia and abroad."

Political activities

According to a statement he released in the mid-1970s, he had had troubles with the Soviet powers in the early 1950s, but was then for a time protected by Ivan Petrovsky
Ivan Petrovsky
Ivan Georgievich Petrovsky, also Petrovskii was a Soviet mathematician in the field of partial differential equations, and studied Petrovsky lacunas. He greatly contributed to solution of Hilbert's 19th and 16th problems...

, then Rector of the Moscow University. He belonged to the wing of the dissident movement that identified itself with the Orthodox Christian
Eastern Orthodox Church
The Orthodox Church, officially called the Orthodox Catholic Church and commonly referred to as the Eastern Orthodox Church, is the second largest Christian denomination in the world, with an estimated 300 million adherents mainly in the countries of Belarus, Bulgaria, Cyprus, Georgia, Greece,...

 tradition in Russia; what is sometimes called the 'romantic nationalist' or Slavophile
Slavophile
Slavophilia was an intellectual movement originating from 19th century that wanted the Russian Empire to be developed upon values and institutions derived from its early history. Slavophiles were especially opposed to the influences of Western Europe in Russia. There were also similar movements in...

 tendency. The position criticised the Communist Party
Communist party
A political party described as a Communist party includes those that advocate the application of the social principles of communism through a communist form of government...

 as part of a broader condemnation of socialism
Socialism
Socialism is an economic system characterized by social ownership of the means of production and cooperative management of the economy; or a political philosophy advocating such a system. "Social ownership" may refer to any one of, or a combination of, the following: cooperative enterprises,...

; Shafarevich published a much-noticed book Socialism (French edition around 1975) of destructive analysis, which was cited centrally by Solzhenitsyn in his 1978 address to Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...

. From 1970 Shafarevich with Valery Chalidze
Valery Chalidze
Valery Chalidze is a Georgian-American author, publisher, and the former Soviet dissident and human rights activist.Chalidze was born in Moscow...

, Grigori Podyapolski and Andrei Tverdokhlebov was one of Sakharov's human rights
Human rights
Human rights are "commonly understood as inalienable fundamental rights to which a person is inherently entitled simply because she or he is a human being." Human rights are thus conceived as universal and egalitarian . These rights may exist as natural rights or as legal rights, in both national...

 investigators. All this was a direct affront to the Soviet system, and Shafarevich was dismissed from Moscow University. Shafarevich continued to challenge the distortions of the academic system: as reported at the time by the algebraic geometer Miles Reid
Miles Reid
Miles Reid is a mathematician who works in algebraic geometry.He studied mathematics at Cambridge University, obtaining his Ph.D. in 1973 under the supervision of Peter Swinnerton-Dyer and Pierre Deligne. He was a research fellow of Christ's College, Cambridge from 1973 to 1978...

, he intervened in a doctoral defence of a communist party functionary, bringing up questions of plagiarism and inadequate work which were simply ignored (Shafarevich as an Academician in the Soviet Academy of Sciences could expect to be heard).

Shafarevich's 1970s book The Socialist Phenomenon was widely circulated in the West. After the Cold War
Cold War
The Cold War was the continuing state from roughly 1946 to 1991 of political conflict, military tension, proxy wars, and economic competition between the Communist World—primarily the Soviet Union and its satellite states and allies—and the powers of the Western world, primarily the United States...

, he started attacking Jews whom he called "small people who deny the historical achievements of Russia", saying his homeland must have "sound democratic statehood, based on the will of the people." His critics call him a radical, anti-Semitic, Orthodox nationalist.

He wrote from this period a number of essays, on topics such as the nationalities issues, Russophobia
Russophobia
Russophobia refers to a diverse spectrum of prejudices, dislikes or fears of Russia, Russians, or Russian culture. Its opposite is Russophilia....

, and Shostakovich
Dmitri Shostakovich
Dmitri Dmitriyevich Shostakovich was a Soviet Russian composer and one of the most celebrated composers of the 20th century....

's treatment. These were very prominent, some being included in Solzhenitsyn's collection From Under the Boulders.

The Socialist Phenomenon

Shafarevich's book The Socialist Phenomenon, published in the US in 1980, argued that the leftist-nihilist utopian impulse is a revival of gnostic religion, rooted in rebellion. In his view, this is an anti-Christian urge that fights obsessively with the normal state of the world, demanding material equality and the eradication of individual and gender
Gender
Gender is a range of characteristics used to distinguish between males and females, particularly in the cases of men and women and the masculine and feminine attributes assigned to them. Depending on the context, the discriminating characteristics vary from sex to social role to gender identity...

 distinctions. Shafarevich wrote that "the death of mankind is not only a conceivable result of the triumph of socialism – it constitutes the goal of socialism."

Shafarevich's views were influenced by Karl Wittfogel's theory of hydraulic society. The mathematician argued that socialism has two archetypes: ancient despotisms (such as Sumeria, Babylonia and Egypt) and millennial sectarian movements of medieval and early modern Europe, along with a Freudian death-instinct. Out of this combination, he said that this ideology works to co-opt the prestige of science and faith in progress. Shafarevich covered in his work what he regarded as socialist tendencies and socialist states that have occurred during the history of mankind. He contrasts Campanella
Tommaso Campanella
Tommaso Campanella OP , baptized Giovanni Domenico Campanella, was an Italian philosopher, theologian, astrologer, and poet.-Biography:...

's (City of the Sun
The City of the Sun
The City of the Sun is a philosophical work by the Italian Dominican philosopher Tommaso Campanella. It is an important early utopian work.The work was written in Italian in 1602, shortly after Campanella's imprisonment for heresy and sedition...

) and Thomas More
Thomas More
Sir Thomas More , also known by Catholics as Saint Thomas More, was an English lawyer, social philosopher, author, statesman and noted Renaissance humanist. He was an important councillor to Henry VIII of England and, for three years toward the end of his life, Lord Chancellor...

's (Utopia
Utopia
Utopia is an ideal community or society possessing a perfect socio-politico-legal system. The word was imported from Greek by Sir Thomas More for his 1516 book Utopia, describing a fictional island in the Atlantic Ocean. The term has been used to describe both intentional communities that attempt...

) visions with the facts known of the Inca Empire
Inca Empire
The Inca Empire, or Inka Empire , was the largest empire in pre-Columbian America. The administrative, political and military center of the empire was located in Cusco in modern-day Peru. The Inca civilization arose from the highlands of Peru sometime in the early 13th century...

 and concludes there are striking similarities. Also, in Shafarevich's (1980:207) opinion, Marxist ideology has not even the climate of scientific inquiry. Marx's most important postulates are contradicted by the very reality:
If a socialist state comes into being only under the conditions created by the development of capitalism, if, as Lenin wrote, "socialism originates in capitalism, develops historically from capitalism, and results from the action of a social force that is engendered by capitalism," then whence did it come and as a result of what social force did it develop in the Inca empire or the states of the ancient Orient? History only reinforces the doubts engendered by the contemporary situation: socialist states have arisen in China, North Korea and Cuba--that is, in the countries where the influence of capitalism can in no way be considered a determining factor.
(Shafarevich 1980:202)

In Russian politics

During Perestroika, Shafarevich opposed both communism and Western democracy. On 21 December 1991 he took part in the first congress of the Russian All-People's Union
Russian All-People's Union
Russian All-People's Union was a Russian nationalist political party, formed in October 1991, in 2001 it merged into Narodnaya Volya. Its leader was Sergey Baburin....

 headed by Sergei Baburin. In October 1992 he became member of the orgkomitet of the National Salvation Front
National Salvation Front (Russia)
The National Salvation Front was a broad coalition of communist, socialist and ultra-nationalist movements against the reforms in Russia. The front was organized on 24 October 1992...

 that united extremist opposition movements.

In 1993 he was a candidate for the State Duma
State Duma
The State Duma , common abbreviation: Госду́ма ) in the Russian Federation is the lower house of the Federal Assembly of Russia , the upper house being the Federation Council of Russia. The Duma headquarters is located in central Moscow, a few steps from Manege Square. Its members are referred to...

 with Mikhail Astafyev's Constitutional Democratic Party - Party of Popular Freedom
Constitutional Democratic Party - Party of Popular Freedom
Constitutional Democratic Party – Party of Popular Freedom was a political party in the USSR and Russia...

, but failed to get elected. In 1994 he joined the All-Russian National Right Centre
Centre-right
The centre-right or center-right is a political term commonly used to describe or denote individuals, political parties, or organizations whose views stretch from the centre to the right on the left-right spectrum, excluding far right stances. Centre-right can also describe a coalition of centrist...

of Astafyev and N. Narotsnitskaya.

Shafarevich was a member of the editorial board of the magazine Nash Sovremennik, and in 1991–1992 of the editorial board of Den of Alexander Prokhanov
Alexander Prokhanov
Alexander Andreyevich Prokhanov is a Soviet and Russian writer. He is a member of the secretariat of the Writers Union of the Russian Federation and the editor-in-chief of ultra-nationalist newspaper "Завтра" ....

 (which ceased in October 1993, and later reopened under the title Zavtra).

Accusations of antisemitism

Shafarevich's essays titled Russophobia
Russophobia
Russophobia refers to a diverse spectrum of prejudices, dislikes or fears of Russia, Russians, or Russian culture. Its opposite is Russophilia....

and "Three thousand year old mystery" (Трехтысячелетняя загадка) resulted in accusations of antisemitism. He completed the Russophobia essay in 1982 and it was initially circulated in samizdat
Samizdat
Samizdat was a key form of dissident activity across the Soviet bloc in which individuals reproduced censored publications by hand and passed the documents from reader to reader...

. In the USSR it was first published in 1989.

In the Russophobia essay, referring to Augustin Cochnin's
Augustin Cochin (historian)
Augustin Cochin was a French historian of the French Revolution. Much of his work was posthumously published in an incomplete state after he was killed in action in World War I....

 opinions, he argued that great nations experience periods in their history, when small reformist elitist groups ('small nations') that have values that differ fundamentally from the values of the majority of the people, gain upper hand in the society. Yitzhak M. Brudny writes that, in Shafarevich's opinion, Jews represented such a 'small nation' in Russia, and full of hatred against traditional Russian way of life, first played an active role in the terrorist regime of Vladimir Lenin
Vladimir Lenin
Vladimir Ilyich Lenin was a Russian Marxist revolutionary and communist politician who led the October Revolution of 1917. As leader of the Bolsheviks, he headed the Soviet state during its initial years , as it fought to establish control of Russia in the Russian Civil War and worked to create a...

 and Joseph Stalin
Joseph Stalin
Joseph Vissarionovich Stalin was the Premier of the Soviet Union from 6 May 1941 to 5 March 1953. He was among the Bolshevik revolutionaries who brought about the October Revolution and had held the position of first General Secretary of the Communist Party of the Soviet Union's Central Committee...

 and then, during the Perestroika
Perestroika
Perestroika was a political movement within the Communist Party of the Soviet Union during 1980s, widely associated with the Soviet leader Mikhail Gorbachev...

 era were trying to implant into Russia the liberal values of the Western world. The ideas expressed in this essay have often been used in radical nationalist circles. He claimed that his sources were writings by Soviet emigrants of Jewish origin. He complained that "Russophobes," who are cunning, hostile, Jewish critics, "dream of transforming Russia into. .. a robot deprived of all elements of human life."

This publication led to the request by the US NAS
United States National Academy of Sciences
The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...

 to Shafarevich to resign his membership, because the NAS charter prohibited stripping an existing membership . In an open letter to the NAS, Shafarevich denied that Russophobia is antisemitic.

Accusations of antisemitism have continued, involving Shafarevich's other publications. The criticisms are not all consistent, however, since critics disagree; where Semyon Reznik
Semyon Reznik
Semyon Reznik — Russian writer, journalist, man of letters and historian, noted in particular for his study of the blood libel and the resurgence of Neonazism in Russia...

 targets the Russophobia essay for its factual inaccuracies, and argues that Shafarevich depicts the murder of Tsar Nicholas II as an example of Jewish ritual murder
Blood libel
Blood libel is a false accusation or claim that religious minorities, usually Jews, murder children to use their blood in certain aspects of their religious rituals and holidays...

, and further notes that Shafarevich has assigned Jewish ethnicity to a number of non-Jewish individuals involved in the execution and perpetuates the false claims of graffiti in Yiddish at the murder site, Aron Katsenelinboigen
Aron Katsenelinboigen
Aron Katsenelinboigen, , a founder of Predispositioning Theory, was born in September 1927 in a Ukrainian town of Izyaslavl.-Background :...

, on the other hand, stated that while there are multiple antisemitic claims in Shafarevich's writings, he stops short of claiming blood libel
Blood libel
Blood libel is a false accusation or claim that religious minorities, usually Jews, murder children to use their blood in certain aspects of their religious rituals and holidays...

.

More recently Shafarevich expanded on his views in his book "Three thousand year old mystery". This work was published in Russian in 2002; an introductory section explains the relationship with the Russophobia essay, in the terms that the essay developed from an Appendix to an intended work of wider scope which he started writing in samizdat. It contains further anti-Semitic allegations, as well as attempts at justification of Stalin's repressions directed at Jews (such as executions of the members of the Jewish Anti-Fascist Committee
Jewish Anti-Fascist Committee
The Jewish Anti-Fascist Committee was formed on Joseph Stalin's order in Kuibyshev in April 1942 with the official support of the Soviet authorities...

, several writers and poets, etc. in 1947–1953).

External links

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