On the generation of the coefficient field of a newform by a single Hecke eigenvalue
Koopa Tak-Lun Koo1 ; William Stein1 ; Gabor Wiese2
1 Department of Mathematics University of Washington Box 354350 Seattle, WA 98195 USA
2 Institut für Experimentelle Mathematik Universität Duisburg-Essen Ellernstraße 29 45326 Essen Germany
Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 2, pp. 373-384.

Soit f une forme nouvelle de poids k≥2 sans multiplication complexe. Soit L un sous-corps du corps des coefficients de f. Nous résolvons complètement la question de la densité de l’ensemble des premier p tels que le p-ième coefficient de f engendre L. Cette densité est déterminée par les tordues intérieures de f. Comme cas particulier, on obtient que cette densité est 1 pour L le corps des coefficients de f, pourvu que f n’ait pas de tordue intérieure non-triviale. Nous présentons aussi quelques données nouvelles sur la réductibilité de polynômes de Hecke suggérant des questions pour des recherches à venir.

Let f be a non-CM newform of weight k≥2. Let L be a subfield of the coefficient field of f. We completely settle the question of the density of the set of primes p such that the p-th coefficient of f generates the field L. This density is determined by the inner twists of f. As a particular case, we obtain that in the absence of nontrivial inner twists, the density is 1 for L equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation.

DOI : 10.5802/jtnb.633

Koopa Tak-Lun Koo 1 ; William Stein 1 ; Gabor Wiese 2

1 Department of Mathematics University of Washington Box 354350 Seattle, WA 98195 USA
2 Institut für Experimentelle Mathematik Universität Duisburg-Essen Ellernstraße 29 45326 Essen Germany 
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Koopa Tak-Lun Koo; William Stein; Gabor Wiese. On the generation of the coefficient field of a newform by a single Hecke eigenvalue. Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 2, pp. 373-384. doi : 10.5802/jtnb.633. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.633/

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