Frédéric Jouhet
Cohen-Lenstra Heuristics, finite abelian p-groups and symmetric
functions
The Cohen-Lenstra heuristics for class groups of number fields, defined
in the eighties, and their more recent adaptation by Delaunay to the
Tate-Shafarevich groups of elliptic curves, allow to give precise
predictions on the behavior of these groups. By using combinatorial
methods related to the theory of integer partitions and Hall-Littlewood
symmetric functions, we prove a formula for the moments (related to the
averages coming from the Cohen-Lenstra heuristics model) of the orders
of finite abelian p-groups. This formula, together with the heuristics
philosophy, allow us to give precise predictions on class groups (resp.
Tate-Shafarevich groups and Selmer groups) for some families of number
fields (resp.
elliptic curves). In the case of Tate-Shafarevich groups, these
conjectures are consistent with the recent model developped by Poonen
and Rains on Selmer groups of elliptic curves. Moreover, by combining
the probabilistic model with our results, we obtain combinatorial
identities which seem to be new, although they are related in some
special cases to the theory of Hall-Littlewood polynomials. In some
sense, our approach therefore gives for these identities a somehow
natural algebraic context.
This talk is based on a joint work with Christophe Delaunay (Besancon,
France).