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Glasgow Mathematical Journal (1)

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ON MOD p REPRESENTATIONS WHICH ARE DEFINED OVER p: II
L. J. P. KILFORD, GABOR WIESE
Journal:

Glasgow Mathematical Journal / Volume 52 / Issue 2 / May 2010

Published online by Cambridge University Press:

29 March 2010, pp. 391-400

Print publication:

May 2010
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The behaviour of Hecke polynomials modulo p has been the subject of some studies. In this paper 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 ≥ 2 and all characters χ taking values in {±1} splits completely modulo p has density 0, unconditionally for p = 2 and under the Cohen–Lenstra heuristics for p ≥ 3. The method of proof is based on the construction of suitable dihedral modular forms.