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Some key points are the following: r The dimensions (of varieties) where unsolvability of Hilbert’s 10th problem sets in are far beyond those where current, theory-driven research in number theory takes place or concern varieties with no discernible structure. Lang says somewhere that no undecidability is to be expected for abelian varieties. The sense (and plausibility) of this is pretty clear, even if one can contrive undecidability results about abelian varieties, for example, by considering period lattices with nonrecursive generators.

The arithmetization technique was demystified, leading to such notable results as Turing’s on universal machines. Somewhat later, one looked at the finer structure of recursively enumerable sets, developed relative computability, and posed the influential Post problem, whose solution led to the introduction of new techniques of diagonalization. r Rather more slowly, one moved toward the fine structure of definitions in arithmetic. Here G¨odel’s coding of recursions by the Chinese Remainder Theorem was central to the repertoire, while being recognized as too weak on its own to yield undecidability of Hilbert’s 10th problem.

2006). Noncomputable Julia sets. J. Amer. Math. , 19, 551–78. , and Taylor R. (2001). On the modularity of elliptic curves over Q: Wild 3-adic exercises. J. Amer. Math. , 14, 843–939. Bridson, M. R. (2002). The geometry of the word problem. In Invitations to Geometry and Topology, ed. M. R. Bridson and S. M. Salamon, Oxford Graduate Texts Mathematics 7, pp. 29–91. Oxford: Oxford University Press. ———. (2007). Non-positive curvature and complexity for finitely presented groups. In Proceedings of the International Congress of Mathematicians, Madrid 2006, vol.