Dihedral congruence primes and class fields of real quadratic fields (Q696924)

The author characterizes the primes \(p\) such that the congruence \(f\equiv f\otimes\mod{\wp}\) holds for primitive cusp forms \(f\) of weight \(k\geq 2\) and quadratic nebentypus \(\chi=\chi_1\chi_2\), and a prime \(\wp\) of \(\overline{\mathbb Q}\) above \(p\), where \(\chi_1\) is even quadratic. Such primes are called dihedral. They are essentially the factors of the absolute norm \(N(\varepsilon_+^{k-1}\pm 1)\) of a totally positive fundamental unit \(\varepsilon_+\) of the real quadratic field \(F_1\) associated with \(\chi_1\). This characterization is a generalization of \textit{H. Hida}'s one [Doc. Math., J. DMV 3, 273-284 (1998; Zbl 0923.11084)] in the `full' nebentypus case \(\chi=\chi_1\). Moreover, using this characterization they extend G. Shimura's method of generating class fields of \(F_1\) by torsion points on modular Abelian varieties.