The Cohen–Lenstra heuristics, moments and $p^j$-ranks of some groups

Christophe Delaunay ¹   ; Frédéric Jouhet ²

¹ Université de Franche-Comté Laboratoire de Mathématiques de Besançon CNRS UMR 6623 Facultés des Sciences et Techniques 16 route de Gray 25030 Besançon, France
² Université de Lyon CNRS Université Lyon 1 Institut Camille Jordan 43, boulevard du 11 novembre 1918 F-69622 Villeurbanne Cedex, France

Acta Arithmetica, Tome 164 (2014) no. 3, pp. 245-263

Résumé

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.

Détail

DOI : 10.4064/aa164-3-3

Keywords: article deals coherence model given cohen lenstra heuristic philosophy class groups their generalizations tate shafarevich groups precisely first extend previous result due fouvry ners which proves conjecture provided cohen lenstra philosophy implies another conjecture consequence work deduce example conjecture probability laws j ranks selmer groups elliptic curves compatible theoretical works other classical conjectures