Eyal Goren (McGill University)


Class invariants for CM fields of degree 4.


The problem of effective construction of units in abelian extensions of number fields is at present out of reach. Notwithstanding conjectural constructions, the only exceptions are the constructions for abelian extensions of Q and of a quadratic imaginary field, where the units are the cyclotomic and elliptic units respectively. One reason one seeks such constructions is to find Stark units which appear in Stark's conjectures on special values of L functions.

In this talk, after explaining what is the source of the difficulty, I shall survey what we know at present about the case of CM fields of degree 4, focusing on my work with Ehud de Shalit, Kristin Lauter and Daniel Vallieres. Time allowing, I shall try and put the results in the perspective of the work of Jan Bruinier and Tonghai Yang, indicate some proofs of our results and discuss work in progress.