Appears in collection : Conference of the Number Theory network RT2N / Conférerence du regroupement thématique Théorie des nombres RT2N

In 1980, Gross proved that if $\chi$ is a Hecke character of $\mathbb{Q}$, and $\chi_K$ is its restriction to a Hecke character of a quadratic imaginary field $K$, then the $p$-adic $L$-series $L_p(\chi_K, s)$ factors as the product of two Kubota--Leopoldt $p$-adic $L$-series. His proof relies on special value formulae for these $L$-series in terms of elliptic and circular units. A natural generalization of Gross's formula to arbitrary CM fields $K$ was formulated by Colmez. However, the corresponding formulae involving units remain conjectural, as instances of the $p$-adic Stark conjecture.