The arithmetic genus of Hilbert modular varieties over non-Galois cubic fields
It is well known that Hilbert modular varieties of totally real number fields of degree \(>6\) are of general type, so only in low degrees these varieties can be rational. On the other hand the arithmetic genus of such a variety always equals one. By the way, in low degrees too, almost all Hilbert modular varieties are of general type. In the paper under review the author elaborates for totally real cubic fields k the formula for the arithmetic genus due to Hirzebruch- Vignéras [\textit{F. Hirzebruch}, Enseignem. Math., II. Sér. 19, 183-281 (1973; Zbl 0285.14007), \textit{M.-F. Vignéras}, Math. Ann. 224, 189-215 (1976; Zbl 0325.12004)] in which the number of elliptic fixed points and \(\zeta_ k(-1)\) enter. Naturally, the calculations of the elliptic fixed points in terms of relative class numbers is based on the work of \textit{A. Prestel} [Math. Ann. 177, 181-209 (1968; Zbl 0159.113)] following the lines of \textit{D. Weisser} [Math. Ann. 257, 9-22 (1981; Zbl 0467.10021)] where these calculations have been done for Galois number fields. Concerning the relative class numbers and \(\zeta_ k(-1)\) \textit{T. Shintani}'s work [J. Fac. Sci., Univ. Tokyo Sect. IA 23, 393-417 (1976; Zbl 0349.12007)] on the evaluation of zeta-functions is used. Finally, numerical computations show that 24 of the 28 totally real non- Galois cubic fields of smallest discriminant are of arithmetic genus one thus contrasting the Galois case where just four Galois fields share this property.
Hirzebruch-Vignéras formula for the arithmetic genus
scientific article; zbMATH DE number 4177166