Section: New Results

Logarithmic Class Groups

Logarithmic class groups and units, introduced by Jaulent in 1994, are an intriguing $\ell$-adic variation on the classical class and unit groups related to Iwasawa theory and the wild kernels of algebraic $K$-theory. These ${\mathbb{Z}}_{\ell }$-modules of finite type provide direct access to invariants studied in standard conjectures about ${\mathbb{Z}}_{\ell }$-extensions. In [12] we devised a new algorithm to explicitly compute them in subexponential time under standard conjectures (GRH and Gross-Kuz'min) and to validate unconditionaly the computed results (now in exponential time). The algorithm has been implementated in the PARI/GP system.