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ON MOD p REPRESENTATIONS WHICH ARE DEFINED OVER p: II

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 de...

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Bibliographic Details
Published in:Glasgow mathematical journal 2010-05, Vol.52 (2), p.391-400
Main Authors: KILFORD, L. J. P., WIESE, GABOR
Format: Article
Language:English
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Summary: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.
ISSN:0017-0895
1469-509X
DOI:10.1017/S001708951000008X