Hilbert's twelfth problem

As of 2026 Hilbert's twelfth problem is solved in several major regimes — over $\mathbb{Q}$ (Kronecker–Weber), over imaginary quadratic fields (classical complex multiplication), over global function fields (Drinfeld, 1974), and, in a $p$-adic analytic formulation, over every totally real number field (Dasgupta–Kakde, 2021–2023) — while remaining open in its original complex-analytic form and for general number fields with both real and complex embeddings. Dasgupta and Kakde proved the Brumer–Stark conjecture and the integral Gross–Stark conjecture for totally real fields $F$ using Ribet's method on group-ring-valued Hilbert modular Eisenstein series and Ritter–Weiss modules, showing that Brumer–Stark $p$-units constructed from $p$-adic multiplicative integrals (together with $[F:\mathbb{Q}]-1$ square roots) generate $F^{\mathrm{ab}}$. Parallel work by Henri Darmon, Samit Dasgupta, Jan Vonk, and collaborators on rigid meromorphic cocycles on the $p$-adic upper half-plane provides a striking $p$-adic analogue of the modular $j$-invariant for real quadratic fields. A purely archimedean (complex-analytic) generator for real quadratic or mixed-signature number fields, or an unconditional proof of the archimedean Stark conjectures, remains out of reach.