It is well known that the square root of any integer can be written as a linear combination of roots of unity. A generalization of this fact is the "Kronecker-Weber Theorem", which states that in fact any element which generates an abelian Galois extension of the field of rational numbers Q can also be written as such a linear combination. The roots of unity may by viewed as the special values of the analytic function e(x) = exp(2*pi*i*x) where x is taken to be a rational number. Broadly speaking, Hilbert's 12th problem is to find an analogous result when Q is replaced by a general algebraic number field F, and in particular to find the analytic functions which play the role of e(x) in this general setting.

Hilbert's 12th problem has been solved in the case where F is an imaginary quadratic field, with the role of e(x) being played by certain modular forms. All other cases are, generally speaking, unresolved. In this talk I will discuss the case where F is a real quadratic field, and more generally, a totally real field. I will describe relevant conjectures of Stark and Gross, as well as current work using a p-adic approach and methods of Shintani. A proof of these conjectures would arguably provide a positive resolution of Hilbert's 12th problem in these cases.