The Cohen–Lenstra heuristics, moments and $p^j$-ranks of some groups
This article deals with the coherence of the model given by the Cohen–Lenstra heuristic philosophy for class groups and also for their generalizations to Tate–Shafarevich groups. More precisely, our first goal is to extend a previous result due to É. Fouvry and J. Klüners which proves that a conjecture provided by the Cohen–Lenstra philosophy implies another such conjecture. As a consequence of our work, we can deduce, for example, a conjecture for the probability laws of $p^j$-ranks of Selmer groups of elliptic curves. This is compatible with some theoretical works and other classical conjectures.