# Hilbert’s twelfth problem and deformations of modular forms

## Location

Abstract: Hilbert’s twelfth problem asks for the construction of abelian extensions of number fields via special values of (complex) analytic functions. An early prototype for a solution is the theory of complex multiplication, culminating in the landmark treatise of Shimura and Taniyama which provides a satisfying answer for CM ground fields.

For more general number fields, Stark’s conjecture leads to a conjectural framework for explicit class field theory based on the leading terms of abelian complex L-series at s=1 or s=0. While there has been very little no progress on the original conjecture, Benedict Gross formulated seminal p-adic and ``tame’’ analogues in the mid 1980’s which have turned out to be far more amenable to available techniques, growing out of the proof of the ``main conjectures” by Mazur and Wiles, and culminating in the recent work of Samit Dasgupta and Mahesh Kakde which, by proving a strong refinement of the p-adic Gross-Stark conjecture, leads to what might be touted as a {\em p-adic solution} to Hilbert’s twelfth problem for all totally real fields.

I will compare and contrast this work with a different approach (in collaboration with Alice Pozzi and Jan Vonk) which proves a similar result for real quadratic fields. While limited for now to real quadratic fields, this approach is part of the broader program of extending the full panoply of the theory of complex multiplication to real quadratic, and possibly other non CM base fields.