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Hilbert’s twelfth problem asks for explicit constructions of the abelian extensions of a given number field, similar to what is known for the rational numbers and for imaginary quadratic fields. These abelian extensions are known as class fields because their Galois groups are identified with certain generalized ideal class groups. In the two known cases, the class fields are obtained via the adjunction of roots of unity and of torsion points on elliptic curves with complex multiplication. These are special values of complex analytic functions – the exponential function and elliptic functions with complex multiplication. Hilbert may have envisioned the use of special values of complex analytic functions to construct class fields of more general base fields.

In the 1970s, Harold Stark proposed a strikingly original approach to the generation of class fields, based on his conjectures on the leading term of Artin L-functions at s = 0. In the case of abelian L-functions with a simple zero at s = 0, Stark predicted that the first derivative was the logarithm of a unit in the respective class field, so exponentiating this derivative would give a generator for the abelian extension. In the two known cases, this reduced to the theory of circular and elliptic units, thanks to Dirichlet’s analytic class number formula and Kronecker’s limit formula. Although there is now extensive computational evidence that Stark’s conjecture is correct, there has been little progress on its solution. 

In the 1980s Benedict Gross formulated some p-adic  and tame  analogues of Stark’s conjectures, which gave more information on the p-adic expansions of the conjectural units. Since the p adic L-functions involved in Gross’s conjecture are related to certain Galois modules via the main conjecture in Iwasawa theory, these conjectures have proved more amenable than their complex analogs. Refinements of the Gross-Stark conjecture were later proposed by Darmon and Dasgupta, and the p-adic Gross -Stark conjectures were proved by Darmon, Dasgupta, Pollack and Ventullo around 2011. This line of argument has culminated in the recent work of Samit Dasgupta and Mahesh Kakde which, by proving a large part of the tame conjectures of Gross (along with the refinement of Darmon and Dasgupta in the broader setting of totally real fields) leads to a p−adic solution to Hilbert’s twelfth problem for this large class of fields. 

The goal of this workshop is to take stock of the recent work in this direction and of other progress around the theme of related approaches to explicit class field theory. The key to much of the progress over the years is the careful study of p-adic and tame deformations of modular forms, most notably, of Hilbert modular Eisenstein series. The p-adic interpolation of classical Eisenstein series was introduced by Jean-Pierre Serre to study the congruences of special values of L-functions and the construction of p-adic L-functions for totally real fields, and was further developed by Barry Mazur and Andrew Wiles in their proof of the main conjecture of Iwasawa theory. The workshop will focus on the breakthroughs of Dasguota and Kakde, with a lecture series by the two authors forming a cornerstone of the activity.

Some Recomended Reading
https://sites.math.duke.edu/~dasgupta/papers/Gross.pdf - A historical overview of the Gross-Stark conjecture. 

https://sites.math.duke.edu/~dasgupta/papers/ICM.pdf - A more detailed account of a recent work on the Brumer-Stark conjectures. 

• Public Lecture, Thurs 27 June in The Bayes Centre: Prof Cristian D. Popescu, University of California, San Diego (details to follow)