Computational Algebra Seminar: Etienne -- Hecke operators of finite groups and class groups computation

Speaker: Fabrice Etienne (Bordeaux)  
Title: Hecke operators of finite groups and class groups computation  
Time & Place: 15:00-16:00, Thursday 5 November, SMRI Seminar Rm   
Abstract:  
Let F be a number field, Galois over Q, and let G be the Galois group. Let K and L be 
subfields of F, fixed respectively by the subgroups H and J of G. We can consider 
Z[H\G/J], the free Z-module over the cosets. With a construction that we will explain, 
we can canonically associate to each element of Z[H\G/J] a morphism in Hom(K*, L*). 
The morphisms constructed this way are called Hecke operators.

In this talk, we will see an algorithmic method to compute the class groups of some 
number fields inductively, using the properties of Hecke operators. It is a 
generalisation of a method described by Biasse, Fieker, Hofmann and Page in 2020. 
If time permits, we will also see a method to compute Selmer groups of finite Galois 
modules, that also relies on the properties of Hecke operators