For each cusp singularity of the Hilbert modular variety of a totally real algebraic number field there is associated an \( L \)-series which determines the corresponding “parabolic contribution” to the version of Selberg’s trace formula worked out by Shimizu.
For a real quadratic field Meyer [1957, 1967] has given an elementary algorithm by Dedekind sums for the computation of the parabolic contribution (see also Siegel [1965]) from which a simple formula for the parabolic contribution in terms of the continued fraction associated with the cyclic resolution of the corresponding cusp singularity can be derived [Hirzebruch 1971, 1972, 1976]. In fact, the parabolic contribution equals
\[ \frac{1}{12}(3r - b_0 - b_1 - \dots - b_{r-1}) \]
where \( b_0,\dots,b_{r-1} \) are the characteristic numbers of the dual normal bundles of the rational curves which form the exceptional fibre of the cyclic resolution.
The first purpose of this note is to give a simple formula for the total parabolic contribution, i.e., the sum of the parabolic contributions for the different cusps. This formula is deduced from a general identity satisfied by certain \( L \)-series in any real algebraic number field. In the quadratic case one obtains an expression for the total parabolic contribution in terms of the class numbers of imaginary quadratic fields. Further, in the quadratic case the total parabolic contribution vanishes if and only if the discriminant of the field is the sum of two squares and is negative otherwise.
The total parabolic contribution is of additional interest in the quadratic case for two reasons: first, it is one-fourth the signature (adjusted for defects coming from quotient singularities in certain special cases) of the open rational homology 4-manifold obtained as the quotient of the product of two upper-half planes by the non-symmetric Hilbert modular group. This result, announced ot some extent in [Hirzebruch 1971], will be proved here. Thus, from [Hammond 1966] the vanishing of this (adjusted) signature is a necessary and sufficient condition for the existence of modular imbeddings. Second, it will be shown that this signature is twice the difference of the arithmetic genera for the ordinary and mixed Hilbert modular groups.
