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On mod p representations which are defined over F_p: II

L. J. P. Kilford and Gabor Wiese

Number 8
Author Gabor Wiese
Year 2009

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 TN,χl,k for all primes l, all weights k>1 and all characters χ  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.

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