Geodesic
ON MOD p REPRESENTATIONS WHICH ARE DEFINED OVER p: II
Glasgow mathematical journal, Tome 52 (2010) no. 2, pp. 391-400

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.1017/S001708951000008X
Mots-clés : Primary 11F33, secondary 11F25, 11R29 
KILFORD, L. J. P.; WIESE, GABOR. ON MOD p REPRESENTATIONS WHICH ARE DEFINED OVER p: II. Glasgow mathematical journal, Tome 52 (2010) no. 2, pp. 391-400. doi: 10.1017/S001708951000008X
@article{10_1017_S001708951000008X,
     author = {KILFORD, L. J. P. and WIESE, GABOR},
     title = {ON {MOD} p {REPRESENTATIONS} {WHICH} {ARE} {DEFINED} {OVER} p: {II}},
     journal = {Glasgow mathematical journal},
     pages = {391--400},
     year = {2010},
     volume = {52},
     number = {2},
     doi = {10.1017/S001708951000008X},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S001708951000008X/}
}
TY  - JOUR
AU  - KILFORD, L. J. P.
AU  - WIESE, GABOR
TI  - ON MOD p REPRESENTATIONS WHICH ARE DEFINED OVER p: II
JO  - Glasgow mathematical journal
PY  - 2010
SP  - 391
EP  - 400
VL  - 52
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S001708951000008X/
DO  - 10.1017/S001708951000008X
ID  - 10_1017_S001708951000008X
ER  - 
%0 Journal Article
%A KILFORD, L. J. P.
%A WIESE, GABOR
%T ON MOD p REPRESENTATIONS WHICH ARE DEFINED OVER p: II
%J Glasgow mathematical journal
%D 2010
%P 391-400
%V 52
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S001708951000008X/
%R 10.1017/S001708951000008X
%F 10_1017_S001708951000008X

[1] 1.Ahlgren, S., On the irreducibility of Hecke polynomials, Math. Comput. 77 (263) (2008), 1725–1731. Google Scholar | DOI

[2] 2.Baba, S. and Murty, M. R., Irreducibility of Hecke polynomials, Math. Res. Lett. 10 (5–6) (2003), 709–715. Google Scholar | DOI

[3] 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. Google Scholar | DOI

[4] 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] 5.Buzzard, K., On the eigenvalues of the Hecke operator T , J. Number Theory 57 (1) (1996), 130–132. Google Scholar

[6] 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. Google Scholar | DOI

[7] 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. Google Scholar | DOI

[8] 8.Dieulefait, L. and Wiese, G., On Modular Forms and the Inverse Galois problem, arXiv:0905.1288v1 [math.NT] Google Scholar

[9] 9.Kilford, L. J. P., On mod p modular representations which are defined over F , Glas. Mat. Ser. III 43(63, Pt. 1) (2008), 1–6. Google Scholar | DOI

[10] 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. Google Scholar | DOI

[11] 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] 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. Google Scholar | DOI

[13] 13.Schütt, M., CM newforms with rational coefficients. Ramanujan J. 19 (2) (2009), 187–205. Google Scholar | DOI

[14] 14.Soundararajan, K., The number of imaginary quadratic fields with a given class number, Hardy-Ramanujan J. 30 (2007), 13–18. Google Scholar

[15] 15.Weinberger, P. J., Exponents of the class groups of complex quadratic fields, Acta Arith. 22 (1973), 117–124. Google Scholar | DOI

[16] 16.Wiese, G., Dihedral Galois representations and Katz modular forms. Doc. Math. 9 (2004), 123–133. Google Scholar | DOI

Cité par Sources :