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ON MOD p REPRESENTATIONS WHICH ARE DEFINED OVER ![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160404043333003-0208:S001708951000008X_char1.gif?pub-status=live)
p: II
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160404043333003-0208:S001708951000008X_char1.gif?pub-status=live)
Published online by Cambridge University Press: 29 March 2010
Abstract
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.
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- Copyright © Glasgow Mathematical Journal Trust 2010