Math Conjectures

Hilbert's Twelfth Problem

OPENIconicOpen problemProposed 1900 · Standard version

Canonical statement

For every number field KK, construct from analytic data intrinsic to KK special values whose adjunction to KK generates every finite abelian extension of KK, equivalently all of KabK^{\mathrm{ab}}, and give an explicit reciprocity law describing Gal⁡(Kab/K)\operatorname{Gal}(K^{\mathrm{ab}}/K). The construction should generalize roots of unity for K=QK=\mathbb Q and special values of elliptic or modular functions at complex-multiplication points for imaginary quadratic KK.
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For every number field \(K\), construct from analytic data intrinsic to \(K\) special values whose adjunction to \(K\) generates every finite abelian extension of \(K\), equivalently all of \(K^{\mathrm{ab}}\), and give an explicit reciprocity law describing \(\operatorname{Gal}(K^{\mathrm{ab}}/K)\). The construction should generalize roots of unity for \(K=\mathbb Q\) and special values of elliptic or modular functions at complex-multiplication points for imaginary quadratic \(K\).
NT-CFT-037:[Hilbert1902Problems][Langlands1976Jugendtraum][DasguptaKakde2024ExplicitClassField]
Number theoryAlgebra and representation theory

Notes

Hilbert's twelfth problem asks for an explicit analytic construction of the abelian extensions of a number field, not merely an abstract existence theorem. For Q\mathbb Q, roots of unity generate all finite abelian extensions by Kronecker–Weber; for imaginary quadratic fields, complex multiplication supplies generators through special values of elliptic and modular functions. Hilbert presented the general problem in his 1900 list [Hilbert1902Problems], and Langlands later emphasized its connection with reciprocity and automorphic ideas [Langlands1976Jugendtraum].

Global class field theory classifies finite abelian extensions through idèles, but it does not generally exhibit individual generators as intrinsic analytic special values. Modern Brumer–Stark methods have produced unconditional pp-adic explicit class-field constructions for totally real fields [DasguptaKakde2024ExplicitClassField], a major advance that still does not give the requested uniform complex-analytic construction for every number field.

The words “explicit” and “analytic” are not formal mathematical predicates, so this entry records the standard construction problem rather than a single Boolean proposition. What remains is a general mechanism that both generates every finite abelian extension and carries an explicit reciprocity law, simultaneously extending the rational and imaginary-quadratic models.

References (3)

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