Two of the most striking discoveries of 18th and 19th century number theory are the Kronecker-Weber theorem and the theory of complex multiplication. The first asserts that the maximal abelian extension of the field Q of rational numbers is generated by roots of unity – in other words, that all abelian extensions of Q can be constructed by adjoining values at rational arguments of the transcendental function
e(z) := e2pi iz = cos(2 pi z) + i sin(2 pi z).
The second achieves something similar for a quadratic imaginary field K, constructing essentially all of its abelian extensions from values of the modular j-function at arguments of K.

Finding analytic functions that would play the role of trigonometric and modular functions in generating abelian extensions, or class fields, of more general base fields is the somewhat loosely formulated program of explicit class field theory, also known as Kronecker’s Jugendtraum or Hilbert’s twelfth problem.

Partial progress was achieved with the theory of complex multiplication of abelian varieties initiated by Hilbert and his school and brought to maturity in the eponymous 1961 treatise of Shimura and Taniyama. Through this theory, class fields of CM fields are obtained from the values of modular functions at points attached to the moduli of CM abelian varieties in suitable (Hilbert, Siegel, orthogonal,...) Shimura varieties.

Hilbert’s twelfth problem for non-CM base fields remains shrouded in a great deal of mystery, hinting — perhaps — at a rich function theory for arithmetic quotients even when the underlying real symmetric space fails to be endowed with a complex structure and hence cannot uniformise a Shimura variety. This may be what Hilbert intuited when he declared, in his celebrated 1900 address at the Paris ICM,

“I am certain that the theory of analytical functions of several variables in particular would be notably enriched if one should succeed in finding and discussing those functions which play the part for any algebraic number field corresponding to that of the exponential function in the field of rational numbers and of the elliptic modular functions in the imaginary quadratic number field.”

This talk will describe some possible substitutes for trigonometric and modular functions, with applications to explicit class field theory beyond the traditional framework of CM fields.