Where art thou...........Nemo???????

One striking feature of all the non-abelian reciprocity laws is that the formula for the number of solutions is given in terms of symmetries of certain curved spaces—an extraordinary connection between solving algebraic equations and geometric symmetry. In the case of the Shimura-Taniyama reciprocity law, the relevant symmetries are those of the “hyperbolic plane.” The hyperbolic plane can be thought of as a circular disc (without its boundary), but with an unusual notion of distance. For two points near the center of the disc, their “hyperbolic” distance is similar to their usual distance, but distances are increasingly distorted near the edge of the disc. The hyperbolic plane and its symmetries were illustrated in some of Escher’s woodcuts, like the one below. In the hyperbolic world, all the fish in Escher’s print are to be thought of as having the same size.

Circle Limit III M. C. Escher
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I will conclude by discussing one further question about modular arithmetic which has seen recent progress.

Instead of asking for a rule to predict how many solutions an equation will have in arithmetic modulo a varying prime p, one can ask about the statistical behavior of the number of solutions as the prime varies. Going back to the simple case of a quadratic equation in one variable, Lejeune Dirichlet showed in 1837 that for a fixed whole number m, which is not a prefect square, the equation

X 2 ≡ m mod p