Find, for every number field k, explicit analytic/arithmetic objects whose values generate the maximal abelian extension \(k^{ab}\), analogous to the role of \(e^{2πix}\) on the unit circle for k = Q (and elliptic curves with complex multiplication for imaginary quadratic fields).
The complex-multiplication case (imaginary quadratic bases) was classical-adjacent by Weber 1908-1912 and launched Hasse's program. Stark–Heegner points, Shimura varieties and p-adic methods grew from attempts. In December 2021 Dasgupta and Kakde proved the Brumer–Stark conjecture, delivering explicit p-adic class field theory for totally real fields — the deepest advance in seventy years. The fully explicit dream remains open for most bases.
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