Elliptic Units for Complex Cubic Fields

Explore the fascinating world of elliptic units in complex cubic fields through this comprehensive lecture. Delve into the elliptic Gamma function, a meromorphic special function in three variables that satisfies modular functional equations under SL(3, Z). Examine numerical and theoretical evidence suggesting that products of this function's values often yield algebraic numbers with explicit reciprocity laws, connected to derivatives of Hecke L-functions of cubic fields at s = 0. Investigate the relationship between these findings and Stark's conjectures, and discover how this function potentially extends the theory of complex multiplication to complex cubic fields, addressing Hilbert's 12th problem. Learn about the collaborative research presented in arxiv:2311.04110, conducted by the speaker along with Nicolas Bergeron and Pierre Charollois.