Work in Mathematics
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 the realization of every finite solvable group as the Galois group over 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, I.A. Kostrikin, S.Y. Arakelov, G. V. Belyi, V. Abrashkin, A. 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 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."
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