Download PDF by Thomas L. heath: Diophantus of Alexandria. A Study in the History of Greek

Algebraic Geometry

By Thomas L. heath

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Extra info for Diophantus of Alexandria. A Study in the History of Greek Algebra

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2) The steps in this tower are not Galois, but Elkies notes that the polynomial ????(????) is dihedral. More concretely, ????(????−1 − ????) = −????5 − 11 + ????−5 . 1) are Galois (note that the fifth roots of unity belong to the constant field), we consider the extension ℚ(????) of ℚ(????) defined by 1/???? − ???? = ????. Direct verification using MAGMA reveals that the function field ℚ(????, ????) contains a solution ???? to the equation 1/???? − ???? = ???? such that ????5 = ????(????4 − 3????3 + 4????2 − 2???? + 1)/(????4 + 2????3 + 4????2 + 3???? + 1) .

Math. (2) 176 (2012), 589–635. Richard M. Crew, Etale ????-covers in characteristic ????, Compositio Math. 52 (1984), 31–45. A. J. de Jong and F. Oort, Purity of the stratification by Newton polygons, J. Amer. Math. Soc. 13 (2000), 209–241. P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. Carel Faber and Gerard van der Geer, Complete subvarieties of moduli spaces and the Prym map, J. Reine Angew. Math. 573 (2004), 117–137.

The function field ????(????0 (????)) of ????0 (????) is therefore given by ????(????0 (????), ????1 (????)). Moreover, it is known, see [1], that [????(????0 (????), ????1 (????)) : ????(????0 (????))] = ????deg(????) ∏ (1 + ????|???? ???? prime 1 ????deg(????) ) . 1) In principle, the work of finding an explicit description of the function field ????(????0 (????)) is done, once the modular polynomial ???????? (????, ????) has been computed. However, for general ???? the Drinfeld modular polynomial is not known explicitly. Even in the case ???? = ???? it has only been determined recently [2].

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