Special Values of Class Group L-Functions for CM Fields
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Abstract. Let H be the Hilbert class field of a CM number field K with maximal totally real subfield F of degree n over . We evaluate the second term in the Taylor expansion at s = 0 of the Galoisequivariant L-function S(s) associated to the unramified abelian characters of Gal(H/K). This is an identity in the group ring C[Gal(H/K)] expressing (n)S (0) as essentially a linear combination of logarithms of special values (z), where : n is a Hilbert modular function for a congruence subgroup of SL2 (OF) and z : Gal(H/K) are CM points on a universal Hilbert modular variety. We apply this result to express the relative class number hH/hK as a rational multiple of the determinant of an (hK 1) (hK 1) matrix of logarithms of ratios of special values (z), thus giving rise to candidates for higher analogs of elliptic units. Finally, we obtain a product formula for (z) in terms of exponentials of special values of L-functions.
