Abstract :
Letkbe a totally real number field of degreenoverQwith ring of integers imagek. The Hilbert modular variety ofkis a desingularization of the (natural) compactification of PSL2(imagek) imagen. This paper focuses on Hilbert modular varieties of Galois quartic fields containing units of norm −1. The main result is that each of these varieties has arithmetic genus greater than one and hence is not rational.
