$p^\ell$-Torsion Points In Finite Abelian Groups And Combinatorial Identities

TL;DR

This paper computes moments of $p^ll$-torsion elements in finite abelian groups, linking probabilistic models with combinatorial identities and conjectures in number theory.

Contribution

It introduces a combinatorial approach to analyze moments of torsion elements, connecting heuristic models with algebraic identities and conjectures.

Findings

Compatibility of heuristic models with Poonen-Rains conjecture

Derivation of new combinatorial identities related to symmetric functions

Probabilistic methods provide algebraic context for these identities

Abstract

The main aim of this article is to compute all the moments of the number of $p^\ell$-torsion elements in some type of nite abelian groups. The averages involved in these moments are those de ned for the Cohen-Lenstra heuristics for class groups and their adaptation for Tate-Shafarevich groups. In particular, we prove that the heuristic model for Tate-Shafarevich groups is compatible with the recent conjecture of Poonen and Rains about the moments of the orders of $p$-Selmer groups of elliptic curves. For our purpose, we are led to de ne certain polynomials indexed by integer partitions and to study them in a combinatorial way. Moreover, from our probabilistic model, we derive combinatorial identities, some of which appearing to be new, the others being related to the theory of symmetric functions. In some sense, our method therefore gives for these identities a somehow natural algebraic context.