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ON MOD p REPRESENTATIONS WHICH ARE DEFINED OVER p: II

Published online by Cambridge University Press:  29 March 2010

L. J. P. KILFORD  and
GABOR WIESE
L. J. P. KILFORD
Affiliation:
Department of Mathematics, University Walk, Bristol BS8 1TW, United Kingdom E-mail: l.kilford@gmail.com Web page: http://www.maths.bris.ac.uk/~maljpk/
GABOR WIESE
Affiliation:
Universität Duisburg-Essen, Institut für Experimentelle Mathematik, Ellernstraße 29, 45326 Essen, Germany E-mail: gabor.wiese@uni-due.de Web page: http://maths.pratum.net/
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Abstract

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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.

Keywords

Primary 11F33secondary 11F2511R29
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 2 , May 2010 , pp. 391 - 400
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Ahlgren, S., On the irreducibility of Hecke polynomials, Math. Comput. 77 (263) (2008), 1725–1731.CrossRefGoogle Scholar
2.Baba, S. and Murty, M. R., Irreducibility of Hecke polynomials, Math. Res. Lett. 10 (5–6) (2003), 709–715.CrossRefGoogle Scholar
3.Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system I: The user language, J. Symb. Comp. 24 (3–4) (1997), 235–265. Available at.CrossRefGoogle Scholar
4.Boyd, D. W. and Kisilevsky, H., On the exponent of the ideal class groups of complex quadratic fields, Proc. Amer. Math. Soc. 31 (1972), 433–436.Google Scholar
5.Buzzard, K., On the eigenvalues of the Hecke operator T 2, J. Number Theory 57 (1) (1996), 130–132.Google Scholar
6.Cohen, H. and Lenstra, H. W. Jr, Heuristics on class groups of number fields, in Number theory, Noordwijkerhout 1983 (Jager, H., Editor), Lecture Notes in Mathematics, vol. 1068 (Springer, Berlin, 1984), 33–62.CrossRefGoogle Scholar
7.Conrey, J. B., Farmer, D. W. and Wallace, P. J., Factoring Hecke polynomials modulo a prime, Pacific J. Math. 196 (1) (2000), 123–130.CrossRefGoogle Scholar
8.Dieulefait, L. and Wiese, G., On Modular Forms and the Inverse Galois problem, arXiv:0905.1288v1 [math.NT]Google Scholar
9.Kilford, L. J. P., On mod p modular representations which are defined over F p, Glas. Mat. Ser. III 43(63, Pt. 1) (2008), 1–6.CrossRefGoogle Scholar
10.Koo, K. T-L., Stein, W. and Wiese, G., On the generation of the coefficient field of a newform by a single Hecke eigenvalue, J. Théor. Nombres Bordeaux. 20 (2) (2008), 373–384.CrossRefGoogle Scholar
11.Landau, E., in Handbuch der Lehre von der Verteilung der Primzahlen. 2 Bände. 2d ed. With an appendix by (Bateman, P. T. Editor). (Chelsea Publishing Co., New York, 1953).Google Scholar
12.Rosen, M. and Silverman, J. H., On the independence of Heegner points associated to distinct quadratic imaginary fields, J. Number Theory 127 (1) (2007), 10–36.CrossRefGoogle Scholar
13.Schütt, M., CM newforms with rational coefficients. Ramanujan J. 19 (2) (2009), 187–205.CrossRefGoogle Scholar
14.Soundararajan, K., The number of imaginary quadratic fields with a given class number, Hardy-Ramanujan J. 30 (2007), 13–18.Google Scholar
15.Weinberger, P. J., Exponents of the class groups of complex quadratic fields, Acta Arith. 22 (1973), 117–124.CrossRefGoogle Scholar
16.Wiese, G., Dihedral Galois representations and Katz modular forms. Doc. Math. 9 (2004), 123–133.CrossRefGoogle Scholar