On mod p representations which are defined over F_p: II

TL;DR

This paper investigates the distribution of Hecke polynomials modulo p, showing that the set of integers for which these polynomials split completely modulo p has density zero, using dihedral modular forms.

Contribution

It establishes the density zero result for the splitting of Hecke polynomials modulo p, unconditionally for p=2 and conditionally under Cohen-Lenstra heuristics for odd p.

Findings

The set of integers with completely split Hecke polynomials modulo p has density zero.

Unconditional result for p=2, conditional under Cohen-Lenstra heuristics for odd p.

Construction of dihedral modular forms as a key method.

Abstract

The behaviour of Hecke polynomials modulo p has been the subject of some study. In this note we show that, if p is a prime, the set of integers N such that the Hecke polynomials T^{N,\chi}_{l,k} for all primes l, all weights k>1 and all characters \chi taking values in {+1,-1} splits completely modulo p has density 0, unconditionally for p=2 and under the Cohen-Lenstra heuristics for odd p. The method of proof is based on the construction of suitable dihedral modular forms.