New forms and Galois representations of octahedral type. (Formes primitives et représentations galoisiennes de type octaédral.) (French) Zbl 1053.11524

Summary: It is known by a result of Langlands and Weil that one can associate to each representation of the absolute Galois group of \(Q\) with odd determinant and octahedral type a newform of weight one. Using the work of Bayer and Frey, we provide a method for constructing such newforms. The calculation of the coefficients of their Fourier expansions at infinity can then be computed so as to provide tables. The case of forms of even level is studied in detail.


11F80 Galois representations
11R32 Galois theory
11R39 Langlands-Weil conjectures, nonabelian class field theory




[1] Batut C., User’s Guide to Pari-GP 1 (1995)
[2] Bayer P., Math. Z. 207 pp 395– (1991) · Zbl 0712.11035 · doi:10.1007/BF02571397
[3] Buhler J. P., Icosahedral Galois representations (1978) · Zbl 0374.12002
[4] Butler G., Comm. Algebra 11 pp 863– (1983) · Zbl 0518.20003 · doi:10.1080/00927878308822884
[5] Cassou-Noguès et Jehanne P., J. Number Theory 57 pp 366– (1996) · Zbl 0858.11058 · doi:10.1006/jnth.1996.0054
[6] Crespo T., J. Algebra 129 pp 312– (1990) · Zbl 0704.11044 · doi:10.1016/0021-8693(90)90223-B
[7] Deligne P., Ann. Sci. École Norm. Sup. (4) 7 pp 507– (1974)
[8] Dornhoff L., Group representation theory (1971) · Zbl 0227.20002
[9] Hecke E., Lectures on the theory of algebraic numbers (1981) · Zbl 0504.12001
[10] Jehanne A., Acta Arith. 69 pp 259– (1995)
[11] Kiming, I. 1994.”On the experimental verification of the Artin conjecture for 2-dimensional odd Galois representations over Q. Lifting of 2-dimensional projective Galois representations over Q”1–36. Berlin: Springer. [Kiming 1994], dans On Artin’s conjecture for odd 2-dimensional representations, édité par G. Frey, Lecture Notes in Math. 1585 · Zbl 0839.11056
[12] Schur I., J. reine angew. Math. 139 pp 155– (1911)
[13] Serre J.-P., Algebraic number fields, édité par A. Fröhlich pp 193– (1977)
[14] Serre J.-P., Corps locaux (3ème èd.) (1980)
[15] Serre J.-P., Comment. Math. Helv. 59 pp 651– (1984) · Zbl 0565.12014 · doi:10.1007/BF02566371
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.