An introduction to non-abelian class field theory. Automorphic forms of weight 1 and 2-dimensional Galois representations (Q2826579)

The book deals with applications of automorphic forms of weight \(1\) to number theory, and contains nine chapters.NEWLINENEWLINEIn the first chapter several examples are given of non-abelian fields (mostly cubic) in which the splitting primes can be described using the coefficients of certain cusp forms of weight 1, and in the second the same question is considered for the Hilbert class-field of the field \(K=\mathbb Q(\sqrt{-p})\) (where \(p\equiv7\) mod \(8\) is a prime and the class-group of \(K\) is assumed to be cyclic), the case \(p=47\) being treated in detail. The next chapter treats this problem in the case of ray class fields over real quadratic fields, using this time modular forms introduced by \textit{E. Hecke} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1925, 35--44 (1925; JFM 51.0292.04)].NEWLINENEWLINEIn the fourth chapter, a formula is given for the dimension of the space of cusps forms of weight~1 on a Fuchsian group of the first kind (this question is also treated in the Appendix), and in the next chapter Langlands reciprocity conjecture is presented, and results on the Artin conjecture for Artin \(L\)-functions in the case of two-dimensional representations are stated. One finds here also a short discussion of the Serre modularity conjecture, proved by \textit{C. Khare} and \textit{J.-P. Wintenberger} [Invent. Math. 178, No. 3, 485--504 (2009; Zbl 1304.11041); ibid. 178, No. 3, 505--586 (2009; Zbl 1304.11042)] and \textit{M. Kisin} [Invent. Math. 178, No. 3, 587--634 (2009; Zbl 1304.11043)] and a conjecture of \textit{H. M. Stark} [Lect. Notes Math. 601, 278--287 (1977; Zbl 0363.12010)] about cusp forms.NEWLINENEWLINEChapter 6 introduces Maass forms with weights and automorphic hyperfunctions (defined by \textit{M. Sato} [J. Fac. Sci., Univ. Tokyo, Sect. I 8, 139--193 (1959; Zbl 0087.31402)]), and in Chapter 7 relations between five important arithmetical conjectures are presented. In the last two chapters, indefinite theta series of Hecke and Hilbert modular forms of weight one are treated.