// Workers AI · dad joke modeWhat did genus character say? "I'm a type of guy.

In number theory, a genus character of a quadratic number field K is a character of the genus group of K. In other words, it is a real character of the narrow class group of K. Reinterpreting this using the Artin map, the collection of genus characters can also be thought of as the unramified real characters of the absolute Galois group of K (i.e. the characters that factor through the Galois group of the genus field of K).

References

• Chapter II of Siegel, Carl, Advanced Analytic Number Theory
• Bertolini, Massimo; Darmon, Henri (2009), "The rationality of Stark-Heegner points over genus fields of real quadratic fields", Annals of Mathematics, 170: 343–369, doi:10.4007/annals.2009.170.343, ISSN 0003-486X, MR 2521118
• Section 12.5 of Iwaniec, Henryk, Topics in classical automorphic forms
• Section 2.3 of Lemmermeyer, Franz, Reciprocity laws: From Euler to Eisenstein