Igor Shafarevich

Igor Rostislavovich Shafarevich (Russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Russian mathematician who contributed to algebraic number theory and algebraic geometry. He wrote books and articles that criticize socialism, and was an important dissident during the Soviet regime.

Shafarevich died on 19 February 2017 in Moscow, at the age of 93.

Shafarevich made fundamental contributions to several parts of mathematics including algebraic number theory, algebraic geometry and arithmetic algebraic geometry. In algebraic number theory the Shafarevich–Weil theorem extends the commutative reciprocity map to the case of Galois groups which are extensions of abelian groups by finite groups. Shafarevich was the first to give a completely self-contained formula for the pairing which coincides with the wild Hilbert symbol on local fields, thus initiating an important branch of the study of explicit formulas in number theory. Another famous result is Shafarevich's theorem on solvable Galois groups giving the realization of every finite solvable group as a Galois group over the rationals. 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 surfaces. He initiated a Moscow seminar on classification of algebraic surfaces that updated around 1960 the treatment of birational geometry, 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 curves led him independently of John Tate 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 introduced the Grothendieck–Ogg–Shafarevich formula and the Néron–Ogg–Shafarevich criterion. 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 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, A. I. Kostrikin, Igor A. Kostrikin, S. Y. Arakelov, G. V. Belyi, Victor Abrashkin, Andrey N. Tyurin, and V. A. Kolyvagin. He did major work in collaboration with Ilya Piatetski-Shapiro on K3 surfaces. He is a member of the Serbian Academy of Sciences and Arts in the department of Mathematics, Physics and Earth Sciences.

On his 80th birthday, Russian President Vladimir Putin hailed his "fundamental research" in mathematics, and his creation of "a great science school known both in Russia and abroad".

Shafarevich came into conflict with the Soviet authorities in the early 1950s, but was protected by Ivan Petrovsky, the Rector of Moscow University. He belonged to a group of Pochvennichestvo-influenced dissidents who endorsed the Orthodox Christian tradition. Shafarevich published a book, The Socialist Phenomenon (French edition 1975 English edition 1980), which was cited by Solzhenitsyn in his 1978 address to Harvard University.

In the 1970s Shafarevich, with Valery Chalidze, Grigori Podyapolski and Andrei Tverdokhlebov, became one of Sakharov's human rights investigators, and was consequently dismissed from Moscow University. Shafarevich opposed political interference in universities. The algebraic geometer Miles Reid gives the example of Shafarevich asserting that plagiarism and poor work was being ignored in a doctorate obtained by a Communist Party functionary.

Shafarevich's book The Socialist Phenomenon, which was published in the US by Harper & Row in 1980, analyzed numerous examples of socialism, from ancient times, through various medieval heresies, to a variety of modern thinkers and socialist states. From these examples he claimed that all the basic principles of socialist ideology derive from the urge to suppress individuality. The Socialist Phenomenon consists of three major parts:

1. Chiliastic Socialism: Identifies socialist ideas amongst the ancient Greeks, especially Plato, and in numerous medieval heretic groups such as the Cathars, Brethren of the Free Spirit, Taborites, Anabaptists, and various religious groups in the English Civil War, and modern writers such as Thomas More, Campanella, and numerous Enlightenment writers in 18th-century France.
2. State Socialism: Describes the socialism of the Incas, the Jesuit state in Paraguay, Mesopotamia, Egypt, and China.
3. Analysis: Identifies three persistent abolition themes in socialism – the abolition of private property, the abolition of the family, and the abolition of religion (mainly, but not exclusively Christianity)

Shafarevich argued that ancient socialism (such as Mesopotamia and Egypt) was not ideological, as an ideology socialism was a reaction to the emergence of individualism in the Axial Age. He compared Thomas More's (Utopia) and Campanella's (City of the Sun) visions with what is known about the Inca Empire, and concluded that there are striking similarities. He claimed that we become persons through our relationship with God, and argued that socialism is essentially nihilistic, unconsciously motivated by a death instinct. He concluded that we have the choice of either pursuing death or life.

Shafarevich adhered to Russian Orthodox Christianity and incorporated the neo-Platonic views of Eastern Orthodoxy into his understanding of the relation of mathematics and religion.

In his talk to the Göttingen Academy of Sciences upon receiving a prize, Shafarevich presented his view of the relationship between mathematics and religion. He notes the multiple discoveries in mathematics, such as that of non-Euclidean geometry to suggest that pure mathematics reflects an objective reality, not a set of conventional definitions or a formalism. He claimed that mathematics' growth in itself is not directed or organic. In order to have a unity and direction mathematics needs a goal. This goal can either be set by practical applications or by God as the source of the direction of development. Shafarevich opted for the latter, as pure mathematics is not in itself driven by practical applications.

On 21 December 1991 he took part in the first congress of the Russian All-People's Union headed by Sergei Baburin. In October 1992 he became a member of the founding committee of the National Salvation Front. In 1993 he was a candidate for the State Duma with Mikhail Astafyev's Constitutional Democratic Party - Party of Popular Freedom, but failed to get elected.

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 (which ceased in October 1993, and later reopened under the title Zavtra). In 1994 he joined the "All-Russian National Right Wing Centre" chaired by Mikhail Astafyev.

Shafarevich's essay titled Russophobia, expanded into the book Three thousand year old mystery (Трехтысячелетняя загадка) resulted in accusations of antisemitism. He completed the Russophobia essay in 1982 and it was initially circulated as samizdat. In the USSR it was first officially published in 1989.

In the Russophobia essay he argued that great nations experience periods in their history when 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. In Shafarevich's opinion, the role of such a 'small nation' in Russia was played by a small group of intelligentsiya dominated by Jews. They were full of hatred against traditional Russian way of life, playing an active role in the terrorist regimes of Vladimir Lenin and Joseph Stalin.

Its publication led to a request by the United States National Academy of Sciences (NAS) 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. Shafarevich also noted that since NAS enlisted him without his request or knowledge, it is its internal matter to delist him as well. Nevertheless, when the United States invaded Iraq, Shafarevich faxed his resignation.

Accusations of anti-semitism have continued, involving Shafarevich's other publications. Semyon Reznik targets the Russophobia essay for its factual inaccuracies, that Shafarevich misassigned Jewish ethnicity to a number of non-Jewish individuals involved in the execution of Nicholas II, perpetuating the false assertion that there was graffiti in Yiddish at the murder site, and suggested that Shafarevich's phrase "Nicholas II was shot specifically as the Tsar, and this ritual act drew a line under an epoch in Russian history" – may be read as blood libel. Aron Katsenelinboigen, wrote that Shafarevish's work "lives up to the best traditions of anti-Semitic propaganda" .

Later,Shafarevich expanded on his views in his book "Three thousand year old mystery". In this article Shafarevich further claimed that Jews effectively marginalize non-Jews to the point of exclusion in all types of intellectual endeavors. This work was published in Russian in 2002; an introductory section explains the relationship with the Russophobia essay, explaining that the essay developed from an Appendix to an intended work of wider scope which he started writing in samizdat.

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• Shafarevich, Igor (1975), "Socialism in Our Past and Future." In From under the Rubble, with Solzhenitsyn, Alexander; Agursky, Mikhail; Barabanov, Evgeny; Borisov, Vadim; Korsakov, F. Collins: Harvill Press [Regnery Pub. 1989].
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• Shafarevich, Igor (1981), "On Certain Tendencies in the Development of Mathematics", The Mathematical Intelligencer, Vol. 3, Number 4, pp. 182–184.
• Nikulin, V. V.; Shafarevich, Igor (1987), Geometries and Groups, Berlin; Springer-Verlag, ISBN 0387152814
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• Kostrikin, A. I.; Shafarevich, Igor (1991), Noncommutative Rings, Identities, Berlin: Springer-Verlag, ISBN 0387181776
• Parshin, A. N.; Shafarevich, Igor (1995), Number Theory: Fundamental Problems, Ideas, and Theories, Berlin: Springer, ISBN 0387533842
• Arslanov, M. M.; Parshin, A. N.; Shafarevich, Igor (1996), Algebra and Analysis, Berlin: Walter de Gruyter, ISBN 311014803X
• Shafarevich, Igor (2003), Discourses on Algebra, Berlin: Springer, ISBN 3540422536