On the History of Hilbert’s Twelfth Problem A Comedy of Errors

Hilbert’s 12th problem conjectures that one might be able to generate all abelian extensions of a given algebraic number field in a way that would generalize the so-called theorem of Kronecker and Weber (all abelian extensions of Q can be generated by roots of unity) and the extensions of imaginary quadratic fields (which may be generated from values of modular and elliptic functions related to elliptic curves with complex multiplication). The first part of the lecture is devoted to the false conjecture that Hilbert made for imaginary quadratic fields. This is discussed both from a historical point of view (in that Hilbert’s authority prevented this error from being corrected for 14 years) and in mathematical terms, analyzing the algebro-geometric interpretations of the different statements and their respective traditions. After this, higher-dimensional analogues are discussed. Recent developments in this field (motives, etc., also Heegner points) are mentioned at the end.