Stark conjectures (refined Gross–Stark–Rubin form): let \(L/K\) be a finite Galois extension of number fields and \(S\) a suitable finite set of places. The leading Taylor coefficients of the Artin L-functions \(L_S(s, \chi)\) at \(s = 0\) are, up to an explicit rational factor, regulators of canonical Stark units in \(L\) — pinning down abelian extensions of \(K\) explicitly.
Stark posed them in the 1970s–80s as refinements of the analytic class number formula: special values of Artin L-functions should encode explicit units, hence explicit class fields — a concrete attack on Hilbert's twelfth problem. The rank-one abelian case over \(\mathbb{Q}\) and imaginary quadratic fields recovers classical theory (cyclotomic and elliptic units); the Brumer–Stark conjecture (away from 2) was proved by Dasgupta–Kakde (2020–2023), but the full conjectures, and any non-abelian case, remain open.
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