Here is how quickly “bitcoin” extends to Universal Teichmuller theory:
jacobian, abelion, Weil Pairing
“Authors would like to sincerely thank Neal Coblitz for pointing out serious mistakes in the abstract version of this paper and for invaluable suggestions.”
http://en.wikipedia.org/wiki/Neal_Koblitz He received his Ph.D. from Princeton University in 1974 under the direction of Nick Katz.
Died 6 August 1998 (aged 92) Princeton, New Jersey, U.S.
After spending a year 1946–1947 at the University of Illinois, Zariski became professor at Harvard University in 1947 where he remained until his retirement in 1969. In 1945, he fruitfully discussed foundational matters for algebraic geometry with André Weil. Weil’s interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski’s interests. The two sets of foundations weren’t reconciled at that point.
At Harvard, Zariski’s students included Shreeram Abhyankar, Heisuke Hironaka, David Mumford, Michael Artin and Steven Kleiman — thus spanning the main areas of advance in singularity theory, moduli theory and cohomology in the next generation. Zariski himself worked on equisingularity theory. Some of his major results, Zariski’s main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that ultimately unified algebraic geometry.
Alexander Grothendieck (German:[ˈɡroːtn̩diːk]; French: [ɡʁɔtɛndik]; 28 March 1928 – 13 November 2014) was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research multiplied the scope of the field and added major elements of commutative algebra, homological algebra, sheaf theory and category theory to its foundations, while his so-called “relative” perspective led to revolutionary advances in many areas of pure mathematics.
While not publishing mathematical research in conventional ways during the 1980s, he produced several influential manuscripts with limited distribution, with both mathematical and biographical content.
Produced during 1980 and 1981, La Longue Marche à travers la théorie de Galois (The Long March Through Galois Theory) is a c. 1600-page handwritten manuscript containing many of the ideas that led to the Esquisse d’un programme. It also includes a study of Teichmüller theory.
J. Pila, “Frobenius Maps of Abelian Varieties
In arithmetic geometry, a Frobenioid is a category with some extra structure that generalizes the theory of line bundles on models of finite extensions of global fields. Frobenioids were introduced by Shinichi Mochizuki (2008).
In mathematics, inter-universal Teichmüller theory is an arithmetic version of Teichmüller theory for number fields with an elliptic curve, introduced by Shinichi Mochizuki (2012a, 2012b, 2012c, 2012d) as an extension of his work on p-adic Teichmüller theory.